On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow

Abstract

Let $X$ be a locally finite irreducible affine building of dimension $\geq 2$, and let $\Gamma \leq \operatorname {Aut}(X)$ be a discrete group acting cocompactly. The goal of this paper is to address the following question: When is $\Gamma$ linear? More generally, when does $\Gamma$ admit a finite-dimensional representation with infinite image over a commutative unital ring? If $X$ is the Bruhat–Tits building of a simple algebraic group over a local field and if $\Gamma$ is an arithmetic lattice, then $\Gamma$ is clearly linear. We prove that if $X$ is of type $\widetilde{A}_2$, then the converse holds. In particular, cocompact lattices in exotic $\widetilde{A}_2$-buildings are nonlinear. As an application, we obtain the first infinite family of lattices in exotic $\widetilde{A}_2$-buildings of arbitrarily large thickness, providing a partial answer to a question of W. Kantor from 1986. We also show that if $X$ is Bruhat–Tits of arbitrary type, then the linearity of $\Gamma$ implies that $\Gamma$ is virtually contained in the linear part of the automorphism group of $X$; in particular, $\Gamma$ is an arithmetic lattice. The proofs are based on the machinery of algebraic representations of ergodic systems recently developed by U. Bader and A. Furman. The implementation of that tool in the present context requires the geometric construction of a suitable ergodic $\Gamma$-space attached to the the building $X$, which we call the singular Cartan flow.

1. Introduction

1.1. Are lattices in affine buildings linear?

Let $X$ be a locally finite affine building with a cocompact automorphism group, and let $\Gamma < \operatorname {Aut}(X)$ be a lattice. The goal of this work is to address the following question: When does $\Gamma$ admit a faithful finite-dimensional linear representations over a field, or more generally over a commutative ring with unity? A group admitting such a representation is called linear.

Let us review some known facts concerning that question.

The most important locally finite affine buildings are the so-called Bruhat–Tits buildings. Such a building $X$ is associated with a semisimple algebraic group $\mathbf{G}$ over a local field $k$. The dimension of $X$ coincides with the $k$-rank of $\mathbf{G}$. When $\mathbf{G}$ is $k$-simple, the building $X$ is irreducible. In that case, if we assume moreover that $\dim (X)\geq 2$, celebrated theorems by Margulis 50 and Venkataramana 78 ensure that lattices in $\mathbf{G}(k)$ are all arithmetic. It is tempting to deduce that lattices in Bruhat–Tits buildings of dimension $\geq 2$ are classified, but this is actually not the case. Indeed, the full automorphism group $\operatorname {Aut}(X)$ of the Bruhat–Tits building $X$ of $\mathbf{G}(k)$ is isomorphic to $\operatorname {Aut}(\mathbf{G}(k))$ (assuming that the $k$-rank of $\mathbf{G}$ is $\geq 2$), which is an extension of the group $\operatorname {Aut}(\mathbf{G})(k)$ of algebraic automorphisms over $k$ by a subgroup $\operatorname {Aut}(k)$ (see 13 and Proposition 9.1 below). As long as $\operatorname {Aut}(k)$ is infinite—which happens if and only if the local field $k$ is of positive characteristic—a lattice in $\operatorname {Aut}(X)$ can potentially have infinite image in $\operatorname {Aut}(k)$ and, hence, fail to satisfy the hypotheses of the arithmeticity theorem. A lattice in $\operatorname {Aut}(X)$ with infinite image in $\operatorname {Aut}(k)$ will be called a Galois lattice. Galois lattices exist in rank $1$ (see Proposition 9.4 below), while their existence in higher rank is an open question at the time of writing.

By the seminal work of Tits (see the last corollary in 73), irreducible locally finite affine buildings of dimension $\geq 3$ are all Bruhat–Tits buildings of simple algebraic groups over local fields. This is not the case in low dimensions.

If $\dim (X)=1$, namely if $X$ is a tree with vertex degrees $\geq 2$, any uniform lattice in $\operatorname {Aut}(X)$ is linear, indeed virtually free. However, a non-uniform lattice can be non-linear.

If $\dim (X)=2$ and $X$ is reducible, then $X$ is the product of two trees. In that case Burger and Mozes 16 constructed celebrated examples of uniform lattices $\Gamma < \operatorname {Aut}(X)$ that are simple groups, and thus cannot be linear, as they fail to be residually finite. It is still unknown whether an irreducible lattice in a product of more than two topologically simple groups, each acting cocompactly on a tree, can be nonlinear (in this context, the condition of irreducibility means that the projection of the lattice to each simple factor is faithful with dense image).

If $\dim (X)=2$ and $X$ is irreducible, but not Bruhat–Tits, then we say that $X$ is exotic. Some of the known constructions of exotic affine buildings and lattices acting on them are referenced in §1.3 below.

The simplest and most studied exotic affine buildings are those of type $\widetilde{A}_2$. They will be of core interest in this paper. Let us recall that (locally finite) $\widetilde{A}_2$-buildings may be characterized among simply connected $2$-dimensional simplicial complexes by the property that the link of every vertex is the incidence graph of a (finite) projective plane; this follows from 29, Th. 7.3. From the classification, it follows that the locally finite Bruhat–Tits buildings of type $\widetilde{A}_2$ are precisely those associated with $\mathrm{PGL}_3(D)$, where $D$ is a finite-dimensional division algebra over a local field $k$ (see 80, Chapter 28). Thus a locally finite $\widetilde{A}_2$-building is exotic if and only if it is not isomorphic to the Bruhat–Tits building of such a group $\mathrm{PGL}_3(D)$.

Our first main result, whose proof occupies the major part of this paper, is the following.

Theorem 1.1.

Let $X$ be a locally finite $\widetilde{A}_2$-building and $\Gamma < \operatorname {Aut}(X)$ be a discrete group acting cocompactly. Assume that $X$ is not isomorphic to the building associated to $\mathrm{PGL}_3(D)$, with $D$ a finite-dimensional division algebra over a local field.

Then, for any commutative unital ring $R$ and any $n \geq 1$, any homomorphism $\Gamma \to \operatorname {GL}_n(R)$ has a finite image.

We shall also establish a similar result for Galois lattices in arbitrary Bruhat–Tits buildings of dimension $\geq 2$.

Theorem 1.2.

Let $\mathbf{G}$ be a $k$-simple algebraic group defined over a local field $k$, of $k$-rank $\geq 2$. Let $\Gamma$ be a lattice in $\operatorname {Aut}(\mathbf{G}(k))$.

If $\Gamma$ has infinite image in $\operatorname {Out}(\mathbf{G}(k))$, then for any commutative unital ring $R$ and any $n\geq 1$, any homomorphism $\Gamma \to \operatorname {GL}_n(R)$ has a finite image.

Combining the latter theorem with the aforementioned Arithmeticity Theorem, we draw the following consequence.

Corollary 1.3.

Let $X$ be the Bruhat–Tits building associated with a $k$-simple algebraic group $\mathbf{G}$ defined over a local field $k$, of $k$-rank $\geq 2$.

A lattice $\Gamma \leq \operatorname {Aut}(X)$ admits a finite-dimensional linear representation with infinite image if and only if $\Gamma$ is virtually contained (necessarily as an arithmetic subgroup) in the image of $\mathbf{G}(k)$ in $\operatorname {Aut}(X)$.

In the case of $\widetilde{A}_2$-buildings, we obtain a similar statement without the hypothesis that the ambient building is Bruhat–Tits.

Corollary 1.4.

Let $X$ be a locally finite $\widetilde{A}_2$-building and $\Gamma \leq \operatorname {Aut}(X)$ be a discrete group acting cocompactly.

Then $\Gamma$ admits a representation with infinite image of dimension $n\geq 1$ over a commutative unital ring if and only if $X$ is Bruhat–Tits and $\Gamma$ is arithmetic.

We mention that by an unpublished work of Yehuda Shalom and Tim Steger we know that every nontrivial normal subgroup of a lattice in an $\widetilde{A}_2$-building is of finite index. Clearly, if the lattice is linear, then finite index normal subgroups are in abundance. The question of finding finite quotients of lattices in possibly exotic affine buildings was already asked by W. Kantor 45, Problem C.6.2. Some computer experiments we held indicate that some nonlinear lattices might have no normal subgroups at all. We suggest the following.

Conjecture 1.5.

Let $X$ be a locally finite affine building of type $\widetilde{A}_2$. Assume that $X$ is not isomorphic to the building associated to $\mathrm{PGL}_3(D)$, where $D$ is a finite-dimensional division algebra over a local field. Let $\Gamma$ be a lattice in $\operatorname {Aut}(X)$. Then $\Gamma$ is virtually simple.

It should be emphasized that, while there are several known constructions of locally finite $\widetilde{A}_2$-buildings with lattices, it is generally a delicate problem to determine whether or not a given such building arising from one of those constructions is Bruhat–Tits. Actually, until very recently, the only examples that could be proved to be exotic are buildings of thickness $\leq 10$, and in most cases the proof of exoticity relied on computer calculations. As an application of Theorem 1.1, presented in Section 10, we will establish an exoticity criterion for a large concrete family of $\widetilde{A}_2$-lattices which was first defined by Ronan 60, §3 and Kantor 45, §C.3 and more recently investigated by Essert 35. In particular, we obtain the following.

Corollary 1.6.

Given $q$ a power of $2$ which is not a power of $8$, there exists an $\widetilde{A}_2$-building $X$ of order $q$ that is not Bruhat–Tits and admits an automorphism group $\Gamma \leq \operatorname {Aut}(X)$ acting freely and transitively on the set of panels of each type.

This result provides the first infinite family of $\widetilde{A}_2$-lattices in exotic buildings of arbitrarily large thickness, and may be viewed as a partial answer to a question of W. Kantor, who wrote in the remarks following Corollary C.3.2 in 45: “It would be very interesting to know which, if any, of these buildings are “classical” ones obtained from $\operatorname {PSL}_3(K)$ for complete local (skew) fields $K$.” After a first version of this paper was circulated, two more sources of infinite families of $\widetilde{A}_2$-lattices in exotic buildings were found; see Theorem C and Corollary E in 58, and Proposition 7 and Remark 8 in 19.

1.2. Proof ingredients

The proofs of both Theorems 1.1 and 1.2 establish the contrapositive assertion, and thus start by assuming that the lattice $\Gamma$ has a linear representation with infinite image over a ring.

The first step of the proof is a reduction showing that the given representation can be assumed to have its image contained in a $k$-simple algebraic group $\mathbf{G}$ over a local field $k$, with an unbounded and Zariski dense image. That step is actually a consequence of an abstract statement of independent interest, valid at a high level of generality, and presented in Appendix A.

At that stage, the proof of Theorem 1.1 reduces to that of the following.

Theorem 1.7.

Let $X$ be a locally finite $\widetilde{A}_2$-building and $\Gamma \leq \operatorname {Aut}(X)$ a discrete group consisting of type preserving automorphisms which act cocompactly. Assume that there exist a local field $k$, a connected adjoint $k$-simple $k$-algebraic group $\mathbf{G}$, and a group homomorphism $\Gamma \to \mathbf{G}(k)$ with an unbounded and Zariski dense image. Then $X$ is a Bruhat–Tits building.

Analogy with Margulis’s Superrigidity Theorem

Our strategy to approach Theorem 1.7 is inspired by the recent proof of the Margulis Superrigidity Theorem by the first author and A. Furman, developed in 5. The tools developed in that work are designed to study a representation $\rho \colon \Gamma \to \mathbf{J}(k)$ of an arbitrary countable group $\Gamma$ to a $k$-simple algebraic group over a local field $k$, with Zariski dense image. A key notion from 5 is that of a gate (see §8.1 below for more details). This gate is an algebraic variety canonically associated with any standard Borel space $Y$ equipped with a measure-class preserving ergodic action of $\Gamma$. The gate, moreover, provides a natural representation from $\operatorname {Aut}_\Gamma (Y)$ to a subquotient of the algebraic group $\mathbf{J}(k)$, which is nontrivial under suitable assumptions on $\Gamma$ and $Y$.

Therefore, in order to prove Theorem 1.7, we need to provide an ergodic $\Gamma$-space $Y$ such that $\operatorname {Aut}_{\Gamma }(Y)$ does not admit any nontrivial representation. The space $Y$ we shall construct will be called the singular Cartan flow. In order to give a better understanding of its construction, let us first explain what this space is in the “classical” case when $\Gamma$ is a lattice in $\operatorname {SL}_3(k)$, where $k$ is a local field.

Let $A$ be the subgroup of $G = \operatorname {SL}_3(k)$ consisting of diagonal matrices, and let $S$ be the singular torus

$$\begin{equation*} S=\left\{ \begin{pmatrix} a & 0& 0\\ 0&a&0\\ 0&0&a^{-2} \end{pmatrix}\mid a\in k^*\right\}. \end{equation*}$$

We consider the $\Gamma$-spaces $G/A$ and $G/S$ (or the actions of $A$ and $S$ on $\Gamma \backslash G$). Ergodicity is then provided by the Howe–Moore Theorem. The gate of the space $Y=G/S$ gives a representation of $N_G(S)/S\simeq \operatorname {PGL}_2(k)$.

Geometric analogues in the exotic case.

Now let us go back to the setting of Theorem 1.7, where $\Gamma$ is a lattice of an $\widetilde{A}_2$-building $X$ that is possibly exotic. In particular, the whole group $\operatorname {Aut}(X)$ could be discrete, and the $\Gamma$-spaces we need to construct can thus hardly be homogeneous spaces of a nondiscrete locally compact enveloping group of $\Gamma$. The spaces that we shall construct are of a geometric nature; the measures will be constructed explicitly by hand, and a crucial step in our proof will be to establish their ergodicity.

In order to construct a geometric analogue of the space $G/A$, we first interpret this space geometrically. The group $A$ acts as a group of translations (with compact kernel) on some model apartment $\Sigma$ of the Bruhat–Tits building $X$ associated to $G$. It can be viewed as the stabilizer of this oriented apartment. Hence, $G/A$ is the $G$-orbit of this oriented apartment in $X$, and since $G$ acts transitively, $G/A$ is the space of all such apartments. We can also go up one step and consider $G$, equipped with the action of $G$ (left multiplication) and $A$ (right multiplication). As a dynamical system, this is analogous to the set of simplicial embeddings of the Coxeter complex $\Sigma$ into $X$, with the action of $G$ by postcomposition and the action of $A$ by precomposition.

Therefore, in our setting, we are led to consider the set $\mathscr{F}$ consisting of all simplicial embeddings of the Coxeter complex $\Sigma$ of type $\widetilde{A}_2$ to $X$. The group $\mathbf{Z}^2$ acts on the set $\mathscr{F}$ via its action on $\Sigma$, while $\Gamma$ acts on $\mathscr{F}$ via its action on $X$. In particular, these two actions commute, and we get a $\mathbf{Z}^2$-action on the compact space $\mathscr{F}/\Gamma$. The latter action is called the Cartan Flow. A measure on $\mathscr{F}$ is constructed in §§6.2 and 6.4. Its construction will be natural enough so that this measure is invariant under both the action of $\Gamma$ and the action of $\mathbf{Z}^2$.

A geometric analogue of the space $G/S$ from the classical picture is more technical to realize. Let us again describe it first in the classical setting. We want to understand $G/S$ both as a $G$-space and a $N_G(S)$-space, with both actions commuting. The group $S$ acts by translations (with compact kernel) on some singular line $\ell \subset X$. The set of all lines parallel to $\ell$ form a regular tree $T$, which is a so-called panel tree of $X$. This tree is the Bruhat–Tits tree of the Levi subgroup $N_G(S)$. Hence, an analogue of the $G\times N_G(S)$-dynamical system $G$ is given by a set of embeddings of $T\times \ell$ into $X$. However, since we want the action of $G$ to be transitive, we must consider only a $G$-orbit of embeddings. This amounts to ensuring that we embed $T$ “as a Bruhat–Tits tree” of $N_G(S)$.

In the geometric setting, we therefore introduce the space $\widetilde{\mathscr{W}}$ of all embeddings of $T\times \ell$ into our building. It is endowed with an action of $\Gamma$ (by postcomposition) and $\mathbf{Z}$ (by precomposition with translates of the line $\ell$). However, we want to restrict to the analogue of the $G$-orbit. In order to do so, we use the projectivity group of the projective plane at infinity of $X$. That group naturally acts by automorphisms on $T$, and we denote by $P$ its closure in $\operatorname {Aut}(T)$ (see below for more about this group). We introduce the singular Cartan flow $\mathscr{W}$ as the orbit of an element by the (commuting) actions of $\Gamma$, $P$, and $\mathbf{Z}$. We again define a natural measure on this space, which takes into account both the geometric measure on $\mathscr{F}$ and the Haar measure on $P$ (see §6.3 for more details).

Ergodicity results

A fundamental step in our proof of Theorem 1.7 is the following (see Theorems 7.1 and 7.3 for more precise statements).

Theorem 1.8.

The Cartan flow and the singular Cartan flow are ergodic.

As mentioned above, those flows are a not necessarily homogeneous space of some locally compact groups; in particular, the ergodicity cannot be deduced from the Howe–Moore Theorem as is done in the classical picture. The relevant tool happens to be the one originally used in the study of the geodesic flow on manifolds with variable negative curvature, namely Hopf’s argument 41 (see also 32 for a nice introduction). In Appendix B we give an abstract version of the Hopf argument which is suited for our purpose. The Cartan flow is not needed for the end of the proof, but it is an intermediate step in the proof of the ergodicity of the singular Cartan flow.

If $Y$ denotes the singular Cartan flow, the gate theory provides us with a linear representation of $\operatorname {Aut}_{\Gamma }(Y)$ (the latter group is $\operatorname {PGL}_2(k)$ in the classical case). It follows from the construction of the singular Cartan flow that the group $\operatorname {Aut}_{\Gamma }(Y)$ naturally contains the group $P$ introduced above, which is the closure of the natural image in $\operatorname {Aut}(T)$ of the projectivity group of the projective plane at infinity of $X$. This projectivity group has a lot more structure than $\Gamma$: it acts on a locally finite tree $T$ (a so-called panel tree of $X$) with a $3$-transitive action on its boundary. The final step in the proof of Theorem 1.7 will be deduced from the following result of independent interest (see Theorem 4.7).

Theorem 1.9.

Let $X$ be a locally finite $\widetilde{A}_2$-building and $P$ be the closure of the projectivity group in the locally compact group $\operatorname {Aut}(T)$. If an open subgroup of $P$ admits a faithful continuous representation $P\to \operatorname {GL}_n(k)$, where $k$ is a local field, then $X$ is a Bruhat–Tits building.

It should be emphasized that, while all the other steps of the proof generalize with only small modifications to Euclidean buildings of types other than $\widetilde{A}_2$, we do not know how to extend Theorem 1.9 to $\widetilde{C}_2$ or $\widetilde{G}_2$ buildings. Indeed, the proof of the latter relies on a result on projective planes due to A. Schleiermarcher (Theorem 3.2), whose analogue for generalized quadrangles and generalized hexagons is still unknown. In this regard, we make the following remark.

Remark 1.10.

Let $X$ be a locally finite 2-dimensional affine building, and let $\Gamma$ be a discrete cocompact group of isometries of $X$. Let $P$ be the closure of the projectivity group of $X$ in the locally compact group $\operatorname {Aut}(T)$. By the methods of our proof of Theorem 1.1 below, one could show that if $P$ is non-linear then also $\Gamma$ is non-linear. While we do not know in general that $P$ is nonlinear for exotic locally finite buildings of type $\widetilde{C}_2$ or $\widetilde{G}_2$, given a concrete example of such a building it might be possible to check the nonlinearity of $P$ directly. For example, in the panel-regular exotic buildings of type $\widetilde{C}_2$ considered in 44 and 35, some of the residues are nonclassical finite generalized quadrangles of order $(q-1, q+1)$ for a prime power $q$. In the case where $q \geq 7$, it would suffice to check that the projectivity groups of those quadrangles contain the full alternating group to ensure that the closure of the projectivity group in the automorphism group of the panel tree is locally symmetric or alternating, hence nonlinear as a consequence of the main results from 56.

Let us now briefly discuss the proof of Theorem 1.2, which is presented in Section 9. It also relies on the Bader–Furman machinery from 5. The starting point of that proof is the observation that the argument for Theorem 1.1 sketched above also provides relevant information when $X$ is the Bruhat–Tits building of $\operatorname {PGL}_3(D)$. Indeed, while in that case the projectivity group is isomorphic to $\operatorname {PGL}_2(D)$, and is thus linear by hypothesis, it turns out that the group $\operatorname {Aut}_\Gamma (Y)$ of automorphisms of the Cartan flow commuting with $\Gamma$ considered above can be strictly larger than the projectivity group. More precisely, if $\Gamma$ is a Galois lattice, then $\Gamma$ has an infinite (hence nonvirtually abelian by property (T)) image in the quotient $\operatorname {Aut}(X)/ \operatorname {PGL}_3(D)$, which is virtually the group of field automorphisms of $X$. It then turns out that $\operatorname {Aut}_\Gamma (Y)/\operatorname {PGL}_2(D)$ is also a nonabelian group of field automorphisms of $\operatorname {PGL}_2(D)$. On the other hand, the gate theory provides us with a continuous faithful representation of the locally compact group $\operatorname {Aut}_\Gamma (Y)$, and by the general structure theory of linear locally compact groups from 25, this implies that $\operatorname {Aut}_\Gamma (Y)/\operatorname {PGL}_2(D)$ must in fact be virtually abelian. This shows that if $\Gamma$ is linear, then it cannot be a Galois lattice. This part of the argument does not require $X$ to be of type $\widetilde{A}_2$, and is actually valid for arbitrary irreducible Bruhat–Tits buildings of dimension $\geq 2$. Indeed the relevant $\Gamma$-spaces are homogeneous spaces of the nondiscrete group $\operatorname {Aut}(X)$, and their ergodicity can be established by a suitable application of the Howe–Moore Theorem.

1.3. A review of exotic $2$-dimensional buildings with lattices

The aforementioned classification of irreducible affine buildings of dimension $\geq 3$ was completed in the early days of the 1980s. The existence of exotic $2$-dimensional affine buildings came soon after. Some of them have no nontrivial automorphisms whatsoever (see, e.g., 61); this is, in particular, the case of generic $\widetilde{A}_2$-buildings (in a suitably defined Baire categorical sense); see 8, Th. 5. Some have a cocompact automorphism group, with a global fixed vertex at infinity; see 77, §7. The existence of such a fixed point prevents that group from being unimodular (see 22, Th. M), and hence cannot contain any lattice. The known exotic $\widetilde{A}_2$-buildings relevant to Theorem 1.1 are those endowed with a proper cocompact action of a discrete group. They are constructed as universal covers of finite simple complexes having finite projective planes as residues, or of suitably defined finite complexes of groups. The constructions that we are aware of are the following (the order of a locally finite $\widetilde{A}_2$-building is the common order of its residue planes, i.e., the finite projective planes appearing as its rank $2$ residues).

•

In 75, §3.1 and 60, Th. 2.5, Tits and Ronan construct four $\widetilde{A}_2$-buildings of order $2$ with cocompact lattices. The lattices act regularly (i.e., sharply transitively) on the set of chambers of their building. It is shown in 74, §3.1 and 48 that two of those are Bruhat–Tits buildings with arithmetic lattices, while the other two are exotic (see also 45, §3.6).

•

In 75, §3.1 and 74, §3.2, Tits constructs 44 buildings of type $\widetilde{A}_2$, of order $8$, with cocompact lattices acting regularly on the set of chambers of their building. It is asserted that 43 of them are exotic.

•

In 60, §3 and 45, §C.3, Ronan and Kantor describe a framework providing, in particular, cocompact lattices of $\widetilde{A}_2$-buildings acting regularly on the panels of each given type. Such lattices are called panel-regular.

•

In 27, Cartwright–Mantero–Steger–Zappa construct $65$ exotic $\widetilde{A}_2$-buildings of order $3$ with cocompact lattices. In those examples, the lattices act regularly on the set of vertices of their builing; in particular, the types of vertices are permuted cyclically.

•

In 7, §3, Barré constructs an exotic $\widetilde{A}_2$-building of order $2$ with a cocompact lattice. The full automorphism group of that building is discrete and has exactly two orbits of vertices.

•

In 35, Essert provides presentations of arbitrary panel-regular lattices in $\widetilde{A}_2$-buildings. When the building is of order $2$, there are exactly two inequivalent presentations of such groups. One of them gives rise to an arithmetic group, the other is exotic (see Remark 10.8 and Lemma 10.9 below). The $\widetilde{A}_2$-buildings of order $3$ and $5$ admitting a panel-regular lattice have recently been classified by S. Witzel (see 81), who determined which are exotic and described their full automorphism group. We show in Section 10 how to use Theorem 1.1 to establish Corollary 1.6 and, more generally, to construct, from a given panel-regular lattice in an exotic $\widetilde{A}_2$ of order $p$, an infinite family of panel-regular lattices in an exotic $\widetilde{A}_2$-building of order $p^n$.

•

In all of the previous examples, the residue planes are always Desarguesian. Recently, N. Radu 57 constructed an example of an exotic $\widetilde{A}_2$-building of order $9$, with a vertex-transitive, virtually torsion-free, discrete automorphism group and with non-Desarguesian residues (isomorphic to the Hughes plane of order $9$).

After the first version of the present paper was circulated, two more sources of exotic $\widetilde{A}_2$-buildings with lattices were identified:

•

In Proposition 7 and Remark 8 in 19, it is shown that for every non-Desarguesian finite projection plane $\mathscr{P}$, there is a locally finite $\widetilde{A}_2$-building $X$ with a discrete group $\Gamma \leq \operatorname {Aut}(X)$ acting cocompactly such that $X$ has a residue plane isomorphic to $\mathscr{P}$. In particular, $X$ must be exotic. Since there exist non-Desarguesian finite projective planes of arbitrarily large order, this provides another infinite family of lattices in exotic $\widetilde{A}_2$-buildings of unbounded order. It is not known whether the lattices constructed in this way are virtually torsion-free.

•

In Theorem C and Corollary E in 58, it is proved that, among the panel-regular lattices in $\widetilde{A}_2$-buildings studied by Essert and mentioned above, the proportion of the exotic ones tends (exponentially fast) to $1$ as the order of the building tends to infinity. In particular, it follows from Theorem 1.1 that the “vast majority” of panel-regular lattices in $\widetilde{A}_2$-buildings considered by Essert are nonlinear.

Remark that in all those examples, the lattices are cocompact: no exotic example with a nonuniform lattice is known.

Exotic buildings of type $\widetilde{C}_2$ and $\widetilde{G}_2$ have been less investigated, but some constructions are known; see 44 and 35. In fact, the most fascinating example of an exotic $2$-dimensional affine building with a cocompact lattice is probably the building of type $\widetilde{G}_2$ constructed by Kantor 44 as the universal cover of a simplicial complex associated with the finite sporadic simple group of Lyons (see also 74, §3.5). The corresponding building is exotic (an assertion attributed to Tits, without proof, in 44). This can be deduced from the fact that the full automorphism group of that building is discrete, a fact established by Grüninger 40, Cor. 3.3.

1.4. Structure of the paper

The next two sections are of an introductory nature and review known material. The next section, §2, serves as a concise introduction to $\widetilde{A}_2$-buildings. In particular, we recall the notions of wall trees and panel trees, which are of great importance in this work. In §3 we recall the notion of the projectivity group and explain how it acts on panel trees. In subsection §3.3 we discuss linear representations and algebraicity of groups acting on trees. In §4 we use the results of the previous section in order to study linear representations of the projectivity group. The main result of this section is Theorem 4.7, which roughly states that the projectivity group of a building admits a linear representation if and only if the building is Bruhat–Tits. This result may be of independent interest. The proof of Theorem 1.7 will be carried eventually by a reduction to this theorem.

The next three sections, §§5, 6, and 7 are devoted to the construction and the study of some new spaces, the most important of which are the the space of marked flats and space of (restricted) marked wall trees. We define these as topological spaces in §5 and introduce their measured structure in §6. Finally, in §7, we prove ergodicity results for these spaces. The ergodicity of the singular Cartan flow, Theorem 7.3, is the technical heart of this paper.

In §8 we combine Theorem 7.3 with the theory of algebraic gates in order to reduce the proof of Theorem 1.7 to Theorem 4.7. We then proceed and prove Theorem 1.1.

The last two sections contain supplementary results. In §9 we consider Galois lattices. The goal of that section is to establish Theorem 1.2. Finally, in §10, we use the results of the paper to construct a concrete infinite family of exotic $\widetilde{A}_2$-lattices, a result recorded in Corollary 1.6.

The paper has two appendices. They contain essential ingredients in the proof of the main theorems of the paper, which are presented as appendices due to their independent characters. The first is Appendix A, which is devoted to the proof of Theorem A.1. This theorem is the main ingredient in the reduction of Theorem 1.1 to Theorem 1.7. In Appendix B we review the classical Hopf argument in an abstract setting. In particular, we prove Theorem B.1, which is essential in the proofs given in §7.

2. $\widetilde{A}_2$-buildings and their boundaries

In this section we recall some basic facts on $\widetilde{A}_2$-buildings and their boundaries. In §2.1 we give a general introduction to $\widetilde{A}_2$-buildings and in §2.2 we discuss their boundaries. In §2.3 we discuss the important notions of wall trees and panel trees. Most of the results we will state here are well known, and we will not provide proofs in these cases. Standard references on the subject are 2, 37 and 62.

2.1. Buildings of type $\widetilde{A}_2$

Let us start by recalling the definition of a building of type $\widetilde{A}_2$ (or $\widetilde{A}_2$-building for short). We denote by $\Sigma$ a model apartment: it is the $2$-dimensional simplicial complex afforded by the tessellation of the Euclidean plane $\mathbf{R}^2$ by equilateral triangles. A wall in $\Sigma$ is a line in $\Sigma$ which is a union of edges. We fix a vertex $0\in \Sigma$.

Definition 2.1.

A building of type $\widetilde{A}_2$ is a triangular complex satisfying the following two conditions, in which we call an apartment a subcomplex of $X$ isomorphic to $\Sigma$:

•

Any two simplices of $X$ are contained in an apartment.

•

If $F$ and $F'$ are two apartments, there exists a simplicial isomorphism $F\to F'$ which fixes pointwise $F\cap F'$.

The $2$-dimensional simplices of an $\widetilde{A}_2$-building are called chambers. The space $X$ is endowed with a natural metric, defined as follows: the distance between two points $x$ and $y$ is the distance between $x$ and $y$ calculated in any apartment containing $x$ and $y$. This is well-defined, and furthermore gives a CAT(0) metric on $X$. Apartments are also called flats, because in the CAT($0$)-metric on $X$, the apartments are maximal subspaces isometric to a Euclidean space.

An $\widetilde{A}_2$-building $X$ is equipped with a coloring $X^{(0)} \to \{0, 1, 2\}$ of its vertex set $X^{(0)}$ such that each chamber has exactly one vertex of each color. The color of a vertex is called its type.

We assume throughout that the building $X$ is thick, i.e., there are more than two chambers adjacent to each edge.

It is worth noting that the local geometry is forced by the geometry of the building. For a start, the building is regular: the cardinality of the set of chambers adjacent to an edge is constant (see for example 52, Theorem 2.4). When that number is finite, we define the order of the $\widetilde{A}_2$-building $X$ as the number $q$ such that the number of chambers adjacent to each edge equals $q+1$. Since we assumed $X$ to be thick, we have $q>1$.

Furthermore, the link at a vertex is always a finite projective plane, and the boundary of the building is a compact (profinite) projective plane. For the convenience of the reader, we recall the definition.⁠¹

¹

Strictly speaking, the definition we give is the definition of the incidence graph of a projective plane; the identification of a projective plane and its incidence graph, implicit throughout this paper, should not cause any confusion.

✖

Definition 2.2.

A projective plane is a bipartite graph, with two types of vertices (called points and lines) such that:

•

Any two lines are adjacent to a unique point.

•

Any two points are adjacent to a unique line.

•

There exist four points, no three of which are adjacent to a common line.

Projective planes are also known as generalized triangles, or thick buildings of type $A_2$.

Another striking feature of $\widetilde{A}_2$- (or more general affine) buildings is the following fact.

Theorem 2.3.

Let $\Omega$ be a subset of $X$ which is either convex or of nonempty interior. If $\Omega$ is isometric to a subset of $\mathbf{R}^2$, then there exists an apartment $F$ such that $\Omega \subset F$.

An important class of subsets of $X$ to which we will apply this theorem is the following.

Definition 2.4.

A sector based at $0$ in $\Sigma$ is a connected component of the complement in $\Sigma$ of the union of the walls passing through $0$.

A sector of $X$ is a subset of $X$ which is isometric to a sector in $\Sigma$.

2.2. Boundaries of buildings

Let $X$ be a building of type $\widetilde{A}_2$. Although we shall primarily view $X$ as a $2$-dimensional simplicial complex, we will occasionally refer to $X$ as a CAT($0$) metric space. Such a metric realization is obtained by giving each of $X$ length $1$, and endowing each $2$-simplex with the metric of a Euclidean equilateral triangle with side length $1$. Since $X$ is a CAT($0$) space, its ideal boundary $\partial X$ is naturally endowed with the structure of a CAT($1$) space. The combinatorial nature of $X$ is such that $\partial X$ inherits a much finer structure, namely that of a compact projective plane, equipped with a family of natural measures.

2.2.1. The projective plane at infinity

A boundary point $v \in \partial X$ is called a vertex at infinity if it is the endpoint of a geodesic ray parallel to a wall in $X$. Two sectors $S$ and $S'$ in $X$ are equivalent if their intersection contains a sector; equivalently, their ideal boundaries $\partial S$ and $\partial S'$ coincide. An equivalence class of sectors (or, equivalently, the ideal boundary of a sector) is called a Weyl chamber.

Two Weyl chambers are called adjacent if they contain a common vertex at infinity. This happens if and only if they are represented by two sectors whose intersection contains a geodesic ray. We let $\Delta$ denote the graph whose vertices are the vertices at infinity such that two vertices define an edge if the corresponding vertices at infinity lie in a common Weyl chamber. The set of edges of $\Delta$ is denoted by $\operatorname {Ch}(\Delta )$. The ideal boundary $\partial X$ may be viewed as a CAT($1$) metric realization of the graph $\Delta$.

We can finally state the fact alluded to above.

Theorem 2.5.

The graph $\Delta$ is a projective plane.

A key point in the proof of this theorem is to prove that two chambers are always in some apartment at infinity. An intermediate step is the following lemma, which is also useful to us.

Lemma 2.6.

Let $x$ be a vertex in $X$, and $C\in \operatorname {Ch}(\Delta )$. There exists a unique sector based at $x$ in the equivalence class of $C$. We denote this sector by $Q(x,C)$.

Two chambers in $\Delta$ are called opposite if they are at maximal distance (i.e., at distance three). Similarly, two vertices are called opposite if they are at maximal distance.

By convexity of apartments, the convex hull of two chambers is contained in any apartment containing both of them. This explains the following.

Lemma 2.7.

Two opposite chambers are contained in a unique apartment.

We note for further use the following standard fact 79, Proposition 29.50.

Lemma 2.8.

For every pair of chambers $x,y\in \operatorname {Ch}(\Delta )$, there exists a chamber $z\in \operatorname {Ch}(\Delta )$ which is opposite to both $x$ and $y$.

2.2.2. Topology on $\Delta$

The projective plane $\Delta$, seen as the boundary of $X$, is endowed with a natural topology: indeed, $\Delta$ is naturally endowed with the structure of a compact projective plane. One way to topologize $\Delta$ consists of observing that the ideal boundary $\partial X$ with its natural $\mathrm{CAT}(1)$ metric is a metric realization of the graph $\Delta$, so that a vertex of $\Delta$ is a point in $\partial X$ and an edge of $\Delta$ is a geodesic segment in $\partial X$. Moreover, the vertex-set of $\Delta$ is a closed subset of the ideal boundary $\partial X$ endowed with the cone topology. Thus the vertex-set of $\Delta$ inherits a compact topology from the cone topology on $\partial X$. Similarly, the edge-set of $\Delta$, denoted by $\operatorname {Ch}(\Delta )$, can be topologized via the map that sends each edge of $\Delta$ to its midpoint in $\partial X$. In that way the vertex-set and edge-set of $\Delta$ are naturally endowed with a topology that is compact and totally disconnected (see 23, Prop. 3.5). A basis of open sets of the topology on $\operatorname {Ch}(\Delta )$ is provided by the sets of the form $\Omega _x(y):=\{C \in \operatorname {Ch}(\Delta ) \mid y\in Q(x,C)\}$, where $x$ and $y$ are vertices of $X$.

We note the following fact for further use.

Lemma 2.9.

Let $C$ be a chamber. The set of chambers which are opposite to $C$ is an open subset of $\operatorname {Ch}(\Delta )$.

Proof.

Fix an apartment $A$ containing $C$, and let $C'$ be the chamber in $A$ opposite to $C$. Let $O$ be the set of chambers opposite to $C$. Then $O$ is the union of sets of the form $\Omega _x(y)$, where $x$ is a vertex in $A$ and $y$ is a vertex in the sector $Q(x,C')$.

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2.3. Wall trees and panel trees

Important tools in our study of triangular buildings are wall trees and panel trees. These closely related notions were introduced by Tits in his study of the structure and classification of affine buildings. A good reference is 80, Chapters 10 and 11.

Let us start by defining wall trees. These trees are defined as equivalence classes of parallel lines. A geodesic line $l$ is said to be singular if it is parallel to a wall in some (hence any) apartment containing it; otherwise it is called regular.

Definition 2.10.

Two geodesic lines $\ell$ and $\ell '$ are called parallel if they are contained in a common apartment, and are parallel in this apartment. The distance $d(\ell ,\ell ')$ between parallel lines is the Euclidean distance between them in any apartment containing them.

Parallelism is an equivalence relation. Furthermore, the distance between parallel lines turns each class into a metric space.

For regular lines, this space is just isometric to $\mathbf{R}$. However, these spaces turn out to be interesting for singular lines 80, Proposition 10.14 and Corollary 10.16.

Proposition 2.11.

Let $\ell$ be a singular line, and $T_\ell$ be the set of lines parallel to $\ell$. Then $T_\ell$ is a thick tree. If $X$ is locally finite of order $q$, then $T_\ell$ is regular of degree $q+1$.

The tree $T_\ell$ defined above is called a wall tree. Note that every geodesic line parallel to $\ell$ has the same two endpoints at infinity as $\ell$. These two endpoints are by definition vertices (or panels) of the building at infinity $\Delta$.

Now we turn to the definition of a panel tree. Let $v$ be a vertex of $\Delta$. Then $v$ is represented by an equivalence class of geodesic rays, two of them being equivalent if they contain subrays that are contained in a common apartment and are parallel in that apartment.

Definition 2.12.

Two geodesic rays are said to be asymptotic if their intersection contains a ray.

If two rays $r,r':[0,+\infty )\to X$ belong to the same equivalence class, their distance is defined as

$$\begin{equation*} \inf _s\lim \limits _{t\to +\infty } d(r(t+s),r'(t)). \end{equation*}$$

It is clear that this distance only depends on the asymptotic classes of $r$ and $r'$. It is possible to check that we indeed have a distance on the set of asymptotic classes of rays in the class of $v$.

Proposition 2.13.

The metric space, denoted $T_v$, of asymptotic classes of rays in the class of $v$, is a thick tree. If $X$ is locally finite of order $q$, then $T_v$ is regular of degree $q+1$.

This proposition can be found in 80, Corollary 11.18. The tree $T_v$ is called the panel tree associated to $v$.

Panel trees are related to wall trees via canonical maps 80, Proposition 11.16.

Proposition 2.14.

Let $\ell$ be a singular geodesic line, and $v$ be an endpoint of $\ell$. Then the map $T_\ell \to T_v$ which associates to a line parallel to $\ell$ its asymptotic class is an isometry.

The set of ends of a panel tree is also well understood. Any ray in $T_v$ can be lifted to a sequence of adjacent parallel geodesic rays in some apartment $F$ containing $v$. To such a sequence one can associate the chamber, in the boundary of $F$, which contains $v$ and goes in the direction of the sequence of rays. This explains the construction of the bijection in the following proposition 80, Proposition 11.22.

Proposition 2.15.

Let $v$ be a vertex in $\Delta$. There is a canonical $\operatorname {Aut}(X)_v$-equivariant bijection between the set of ends of the panel tree $T_v$ and the set $\operatorname {Ch}(v)$ of chambers of $\Delta$ containing $v$.

3. Structure of the projectivity group

The projectivity group is an important invariant of the building at infinity $\Delta$, and plays a crucial role in our proofs. In this chapter we define it and discuss its properties. In the next section, §4, we will study its linear representations. First, in §3.1 we define the projectivity group in the context of a general projective plane, and then, in §3.2, in the restricted setting of an affine building. In the latter setting, we will explain how the projectivity group acts on a panel tree. The last subsection, §3.3, is devoted to the general study of linear representations of groups acting on trees. This is a preparation for §4.

3.1. Projectivity groups of projective planes

In this section, we recall some basic terminology on projectivity groups and record important known results that will be needed in what follows.

Let $\Delta$ be a thick building of type $A_2$ (i.e., $\Delta$ is a projective plane). If $v$ is a vertex of $\Delta$, we denote by $\operatorname {Ch}(v)$ the set of chambers of $\Delta$ adjacent to $v$.

Let $v$ and $v'$ be opposite vertices of $\Delta$. The combinatorial projection to $v$ is the map from $\Delta$ to $\operatorname {Ch}(v)$ which associates to $C\in \Delta$ the unique chamber of $\operatorname {Ch}(v)$ which is at minimal distance from $C$. The fact that we chose $v$ and $v'$ opposite implies that this projection, restricted to $\operatorname {Ch}(v')$, is an involutory bijection.

Given a sequence $(v_0, \dots , v_k)$ of vertices such that $v_i$ is opposite $v_{i-1}$ and $v_{i+1}$ for all $i \in \{1, \dots , k-1\}$, we obtain a bijection of $\operatorname {Ch}(v_0)$ to $\operatorname {Ch}(v_k)$ by composing the successive projection maps from $\operatorname {Ch}(v_{i-1})$ to $\operatorname {Ch}(v_i)$. That bijection is called a perspectivity; it is denoted by $[v_0; v_1 ; \cdots ; v_k]$. In the special case when $v_0 = v_k$, it is called a projectivity at $v_0$. The collection of all projectivities at a vertex $v_0$ is denoted by $\Pi (v_0)$; it is a permutation group of the set $\operatorname {Ch}(v_0)$, called the projectivity group of $\Delta$ at $v_0$ and denoted by $(\Pi (v_0), \operatorname {Ch}(v_0))$. The isomorphism type of the permutation group $(\Pi (v_0), \operatorname {Ch}(v_0))$ does not depend on the choice of the vertex $v_0$. Indeed, given another vertex $v$, there exists a perspectivity from $\operatorname {Ch}(v_0)$ to $\operatorname {Ch}(v)$ (because $\Delta$ is thick; see Lemma 2.8), which conjugates $(\Pi (v_0), \operatorname {Ch}(v_0))$ onto $(\Pi (v), \operatorname {Ch}(v))$. For that reason, it makes sense to define the projectivity group of the building $\Delta$ as any representative $(\Pi , \Omega )$ of the isomorphism class of the projectivity group $(\Pi (v), \operatorname {Ch}(v))$ at a vertex $v$.

Proposition 3.1.

The projectivity group $(\Pi , \Omega )$ of a thick building of type $A_2$ is $3$-transitive.

Proof.

See Proposition 2.4 in 17.

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We remark that projectivity groups can be defined for arbitrary generalized polygons; a result of N. Knarr 46, Lemma 1.2 ensures that they are always $2$-transitive.

The projectivity group $(\Pi , \Omega )$ is an important invariant of a projective plane $\Delta$. For example, it is known that $(\Pi , \Omega )$ is sharply $3$-transitive if and only if $\Delta$ is a projective plane over a commutative field; see Theorem 2.5 in 17. We will need an important variant of that fact. In order to state it, we recall that a $2$-transitive permutation group $(\Pi , \Omega )$ is called a Moufang set if there exists a set $\{U_x\}_{x \in \Omega }$ of subgroups of $\Pi$ such that for each $x \in \Omega$, the group $U_x$ acts sharply transitively on $\Omega \setminus \{x\}$ and $g U_x g^{-1} = U_{g(x)}$ for all $g \in \Pi$. The subgroups $U_x$ for $x \in \Omega$ are called the root groups of the Moufang set.

Theorem 3.2.

Let $\Delta$ be a thick building of type $A_2$. Then $\Delta$ is Moufang if and only if the projectivity group $(\Pi , \Omega )$ is a Moufang set.

Proof.

See the main result from 64.

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We will also need the following elementary criterion, ensuring that a projectivity of a very specific form is nontrivial.

Lemma 3.3.

Let $\Delta$ be a thick building of type $A_2$. Let $C, C_0, C_1, C_2, C_3$ be chambers such that $\Sigma _{i, j} = C_i \cup C \cup C_j$ is a half-apartment of $\Delta$ for all $(i, j) \in \{0, 2\} \times \{1, 3\}$.

For all $i\! \mod 4$, let $v_i$ be the vertex of $C_i$ that is not contained in $C$. If $C_0 \neq C_2$ and $C_1 \neq C_3$, then the projectivity $[v_0; v_1; v_2; v_3; v_0]$ is nontrivial; indeed, its only fixed point in $\operatorname {Ch}(v_0)$ is $C_0$.

Proof.

Observe first that the projectivity $g = [v_0; v_1; v_2; v_3; v_0]$ fixes $C_0$. Let $C'_0 \in \operatorname {Ch}(v_0)$ be any chamber different from $C_0$. We claim that $g = [v_0; v_1; v_2; v_3; v_0]$ does not fix $C'_0$. Indeed, we define inductively, for each $i=0, 1, 2, 3$, the chamber $C'_{i+1}$ as the projection of $C'_i$ to $\operatorname {Ch}(v_{i+1})$, and $D_i$ as the unique chamber which is adjacent to both $C'_i$ and $C'_{i+1}$. It then follows that for all $i = 0, 1, 2$, the chamber $D_{i}$ is adjacent to $D_{i+1}$. Moreover, $D_3$ is adjacent to $C'_4$.

By definition, we have $g(C'_0) = C'_4$. Therefore, if $g(C'_0) = C'_0$, then $D_3$ is adjacent to $D_0$ so that $(D_0, D_1, D_2, D_3, D_0)$ forms a closed gallery of length $4$. Therefore, we must have $D_0= D_1$ or $D_1 = D_2$, which forces, respectively, $C_0 = C_2$ or $C_1 = C_3$.

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3.2. Projectivity groups of Euclidean buildings

Now, let us go back to the situation when $X$ is a thick Euclidean building of type $\widetilde{A}_2$, and $\Delta$ is the set of chambers of its building at infinity. In that particular case, the projectivity group $(\Pi , \Omega )$ of $\Delta$ is more than just a permutation group of a set $\Omega$. Indeed, to each vertex $v \in \Delta$, one associates the panel tree $T_v$, as in Proposition 2.13.

Let $v$ and $v'$ be two opposite vertices of different types in $\Delta$, and let $\ell$ be a geodesic line in $X$ joining these two vertices. By Proposition 2.14, the panel trees $T_v$ and $T_{v'}$ are both canonically isomorphic to the wall tree $T_\ell$. Hence we have a canonical involutory isomorphism between the panel trees $T_v$ and $T_{v'}$.

From that observation, it follows that any perspectivity of $\operatorname {Ch}(v)$ to $\operatorname {Ch}(v')$ defines an isomorphism of panel trees from $T_v$ to $T_{v'}$, and that the projectivity group at $v$ acts by automorphisms on the panel tree $T_v$. We define the projectivity group of $X$ at $v$ as $(\Pi (v), T_v)$ by viewing the usual projectivity group $\Pi (v)$ as a subgroup of $\operatorname {Aut}(T_v)$. Thus the projectivity group of $\Delta$ at $v$ is $(\Pi (v), \partial T_v)$, since the set $\operatorname {Ch}(v)$ is canonically isomorphic to the set of ends of the panel tree $T_v$ by Proposition 2.15. Note that by using isomorphisms as described above, all wall trees and all panel trees are mutually isomorphic (though not canonically).

Definition 3.4.

We fix once and for all a tree $T$ which is isomorphic to all panel and wall trees in $X$ and call it the model tree of $X$. By choosing an actual identification of $T$ with a panel tree $T_v$ and pulling back the projectivity group $\Pi (v)$, we define $\Pi$, the model projectivity group of $X$.

Lemma 3.5.

The projectivity group $\Pi$ acts $3$-transitively on $\partial T$ and acts vertex-transitively on $T$. In particular, $T$ is regular. If, moreover, $X$ is locally finite of order $q$, then $T$ is locally finite of degree $q+1$.

Proof.

The $3$-transitivity of $\Pi$ on $\partial T$ follows from Proposition 3.1. $\Pi$ is vertex-transitive on $T$, since every vertex can be uniquely determined by a triple of distinct ends. The regularity of $T$ follows. The last assertion follows from Propositions 2.11 and 2.14.

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3.3. Algebraic groups and locally finite trees

In this section we recall some general results on groups acting on trees that we need in our analysis of the group of projectivities of an $\widetilde{A}_2$-building.

3.3.1. Linear simple locally compact groups are algebraic

A locally compact group is called linear if it has a continuous, faithful, finite-dimensional representation over a locally compact field.

Theorem 3.6.

Let $G$ be a t.d.l.c. group such that the intersection $G^+$ of all nontrivial closed normal subgroups of $G$ is compactly generated, topologically simple and nondiscrete, and that the quotient $G/G^+$ is compact. If $G$ has a linear open subgroup, then the following assertions hold:

(i)

$G^+$ is algebraic: indeed, $G^+$ is isomorphic to $H(k)/Z$, where $k$ is a local field, $H$ is an absolutely simple, simply connected algebraic group defined and isotropic over $k$, and $Z$ is the center of $H(k)$.

(ii)

$G/G^+$ is virtually abelian.

Proof.

Assertion (i) follows from Corollary 1.4 from 25. Assertion (ii) follows from Theorem 1.1 in 25 and the fact that the compact open subgroups of $G^+$ are not solvable (indeed they are Zariski-dense in a simple algebraic group).

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3.3.2. Boundary-transitive automorphism groups of trees

The following result, due to Burger–Mozes 15, provides a natural set-up where all hypotheses of Theorem 3.6 other than the linearity are satisfied.

Theorem 3.7.

Let $T$ be a thick locally finite tree and $G \leq \operatorname {Aut}(T)$ be a closed subgroup acting $2$-transitively on the set of ends $\partial T$.

Then the intersection $G^+$ of all nontrivial closed normal subgroups of $G$ is compactly generated, topologically simple, and nondiscrete. It acts edge-transitively on $T$ and $2$-transitively on $\partial T$, and the quotient $G/G^+$ is compact. In particular, $G$ is unimodular.

Proof.

See Proposition 3.1.2 from 15 and Theorem 2.2 from 21. For the unimodularity of $G$, observe that $G^+$ must be in the kernel of the modular character of $G$. Thus the modular character of $G$ factors through the quotient $G/G^+$, which is compact. Hence $G$ is indeed unimodular.

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An automorphism of a tree $T$ is called horocyclic if it fixes pointwise a horosphere of $T$. Combining the previous two statements, we obtain a subsidiary result which is a key tool in the proof of our characterization.

Proposition 3.8.

Let $T$ be a thick locally finite tree and $G \leq \operatorname {Aut}(T)$ be a closed subgroup acting $2$-transitively on the set of ends $\partial T$. If $G$ has a linear open subgroup, then the following assertions hold:

(i)

The group $G^+$, defined as in Theorem 3.7, is isomorphic to $H(k)/Z$, where $k$ is a local field, $H$ is an absolutely simple, simply connected algebraic $k$-group of $k$-rank $1$, and $Z$ is the center of $H(k)$.

(ii)

$(G, \partial T)$ is a Moufang set whose root groups are the unipotent radicals of proper $k$-parabolic subgroups of $G^+$.

(iii)

Each element $u \in G$ which is a horocyclic automorphism of $T$ belongs to some root group.

(iv)

The quotient $G/G^+$ is virtually abelian.

Proof.

By Theorems 3.6 and 3.7, we know that $G^+$, defined as in Theorem 3.7, is isomorphic to $H(k)/Z$, where $k$ is a local field and $H$ is an absolutely simple, simply connected algebraic $k$-group of $k$-rank $>0$. Since $G^+$ acts properly and cocompactly on a thick locally finite tree, its $k$-rank must be $1$ (see for example 69, Corollary 4). This proves (i).

It is well known, and easy to see using maximal compact subgroups, that $G^+$ has, up to isomorphism, a unique continuous proper cocompact action by automorphisms on a thick locally finite tree (see, e.g., Lemma 2.6(viii) from 20 for a proof). Thus $T$ is isomorphic to the Bruhat–Tits tree of $G^+$. It follows that $(G^+, \partial T)$ is a Moufang set whose root groups are the unipotent radicals of proper $k$-parabolic subgroups of $G^+$. Since $G^+$ is normal in $G$, it follows that the root groups are permuted by conjugation under $G$. Therefore, the permutation group $(G, \partial T)$ is also a Moufang set. This proves (ii).

In order to establish (iii), we recall that the contraction group of an element $h \in G$ is defined as

$$\begin{equation*} \operatorname {con}(h) = \{ g \in G \; | \; h^n g h^{-n} \to 1\}. \end{equation*}$$

Now let $u \in G$ be a horocyclic element, and let $e \in \partial T$ be the center of the horosphere of $T$ fixed by $u$. Given any hyperbolic element $h \in G$ whose repelling fixed point is $e$, we have $u \in \operatorname {con}(h)$. We can choose such a hyperbolic $h$ in $G^+$ (this is possible because $G^+$ is $2$-transitive on $\partial T$). The image of $\operatorname {con}(h)$ in the compact quotient $G/G^+$ must be trivial, so that $\operatorname {con}(h) \leq G^+$. We now invoke 55, Lemma 2.4, ensuring that $\operatorname {con}(h)$ is a root group of the Moufang set $(G, \partial T)$. This proves (iii). The assertion (iv) holds by Theorems 3.6(ii) and 3.7.

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4. Structure and representations of the group of projectivities

The goal of this section is to prove Theorem 4.7 which roughly states that the projectivity group is linear if and only if the building $X$ is a Bruhat–Tits building. The proof of this theorem will be carried in §4.2. It relies on Proposition 4.1 below, whose proof is our main concern in §4.1.

4.1. Existence of horocyclic elements

This subsection is devoted to the proof of the following technical result.

Proposition 4.1.

Let $X$ be a thick Euclidean building of type $\widetilde{A}_2$. Then the projectivity group contains a nontrivial horocyclic automorphism of the panel tree defined in Definition 3.4.

We need to collect a number of subsidiary facts, assuming throughout that $X$ is a thick Euclidean building of type $\widetilde{A}_2$ whose projective plane at infinity is denoted by $\Delta$. The proof of Proposition 4.1 is deferred to the end of the section.

Lemma 4.2.

Let $e_0, e_1, e_2, e_3$ be vertices of $\Delta$, and $D_0, D_1, D_2, D_3 \subset X$ be geodesic lines such that the two endpoints of the line $D_i$ are $e_i$ and $e_{i+1}$ for all $i$, where the indices are written modulo $4$.

If $D_i \cap D_{i+1}$ is nonempty for all $i$ modulo $4$, then the projectivity $[e_0; e_1; e_2; e_3; e_0]$ fixes the point of the panel tree $T_{e_0}$ represented by $D_0$.

Proof.

Let $x \in T_{e_0}$ be the point represented by $D_0$. Its image under the projectivity $g =[e_0; e_1; e_2; e_3; e_0]$ is the point of $T_{e_0}$ represented by $D_3$. Since the intersection $D_0 \cap D_3$ is nonempty, it contains a geodesic ray tending to $e_0$. Thus $g$ fixes $x$ as claimed.

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Lemma 4.3.

Let $e_0, e_1, e_2$ be vertices of $\Delta$, and $D_0, D_1 \subset X$ be geodesic lines with $D_0 \cap D_1 \neq \varnothing$ such that the two endpoints of the line $D_i$ are $e_i$ and $e_{i+1}$ for $i=0,1$.

Let $x \in T_{e_0}$ be the point represented by $D_0$. Suppose that $x$ is a vertex of $T_{e_0}$, and let $x' \in T_{e_0}$ be a vertex adjacent to $x$. Let $D'_0$ be the geodesic line parallel to $D_0$ and representing $x'$. Also let $D'_1$ be the geodesic line parallel to $D_1$ and representing the same point of $T_{e_1}$ as $D'_{0}$.

Let $c$ be a chamber of $X$ with vertices $u, v, w$. If $u$ and $v$ are both contained in $D_0 \cap D_1$ and if $w$ is contained in $D'_0$, then $w \in D'_1$.

Proof.

We refer to Figure 1. Upon exchanging $u$ and $v$, we can assume that $u$ belongs to the geodesic ray $[v, e_0)$. Therefore $u$ also belongs to the geodesic ray $[v, e_2)$ since $u, v \in D_1$.

Let $u'$ (resp., $v'$) be the first vertex different from $w$ on the geodesic ray $[w, e_2)$ (resp., $[w, e_1)$). We claim that $u'$ is opposite $v'$ in the residue $\operatorname {Res}(w)$ of the vertex $w$.

To establish the claim, we first observe that $u'$ is adjacent to $u$. Indeed, by hypothesis the first vertex different from $v$ on the geodesic segment $[v, e_2)$ is $u$. Since the line $D_1$ is singular, it is a wall in any apartment that contains it. Since the chamber $c \in \operatorname {Ch}(X)$ has a panel on $D_1$, we may find an apartment containing both the line $D_1$ and the chamber $c$. In particular, that apartment contains the geodesic rays $[v, e_2)$ and $[w, e_2)$, which are parallel. It follows that $u$ is adjacent to $u'$, as desired. Similarly, $v$ is adjacent to $v'$.

This observation implies that the vertices $u'$ and $v$ have the same type. Hence $u'$ and $v'$ have different types. Therefore, if $u'$ and $v'$ are not opposite in $\operatorname {Res}(w)$, then they are adjacent. But the unique vertex in $\operatorname {Res}(w)$ which is adjacent to both $u$ and $v'$ is $v$. Thus, if the claim fails, then we have $u'=v$. This implies that the first vertex different from $w$ on $[w, e_2)$ is $v$. Therefore the first vertex different from $v$ on $[v, e_2)$ is opposite $w$ in $\operatorname {Res}(v)$. On the other hand, as seen above, the first vertex different from $v$ on $[v, e_2)$ is $u$. Thus $u$ is opposite $w$ in $\operatorname {Res}(v)$. This is absurd since $u$ is adjacent to $w$. The claim is established.

The claim implies that the union of the geodesic rays $(e_1, w] \cup [w, e_2)$ is a geodesic line intersecting $D'_0$ in the geodesic ray $[w, e_1)$. Therefore that line must coincide with $D'_1$, which confirms that $w \in D'_1$.

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Lemma 4.4.

Let $e_0, e_1, e_2, e_3$ be vertices of $\Delta$, and $D_0, D_1, D_2, D_3 \subset X$ be geodesic lines such that two endpoints of the line $D_i$ are $e_i$ and $e_{i+1}$ for all $i\! \mod 4$.

Suppose that the point $x \in T_{e_0}$ represented by $D_0$ is a vertex and let $x' \in T_{e_0}$ be a vertex adjacent to $x$. Let $D'_0$ be the geodesic line parallel to $D_0$ and representing $x'$. For $i>0$, inductively define $D'_i$ as the geodesic line parallel to $D_i$ and representing the same point of $T_{e_i}$ as $D'_{i-1}$.

For each $n>0$, if $D_0 \cap D_1 \cap D_2 \cap D_3$ is a geodesic segment of length $\geq n$, then $D'_0 \cap D'_1 \cap D'_2 \cap D'_3$ is a geodesic segment of length $\geq n-1$.

Proof.

Since the line $D_0$ represents a vertex of $T_{e_0}$, it lies on the boundary of a half-apartment, and thus entirely consists of vertices and edges of $X$. Let $(v_0, v_1, \dots , v_m)$ be the geodesic segment (where $v_i$ is a vertex) which is the intersection $D_0 \cap D_1 \cap D_2 \cap D_3$ so that $m \geq n$ by hypothesis.

For all $i \in \{0, 1, \dots , m-1\}$, let $v'_i$ be the unique vertex belonging to $D'_0$ which is adjacent to both $v_i$ and $v_{i+1}$. By invoking Lemma 4.3 three times successively, we see that $v'_i$ is contained in $D'_1$, $D'_2$, and $D'_3$. This proves that the segment $[v'_0, v'_{m-1}]$, which is of length $m-1 \geq n-1$, is contained in $D'_0 \cap D'_1 \cap D'_2 \cap D'_3$.

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Lemma 4.5.

Let $e_0, e_1, e_2, e_3 \in \Delta$ be vertices, and $D_0, D_1, D_2, D_3 \subset X$ be geodesic lines such that two endpoints of the line $D_i$ are $e_i$ and $e_{i+1}$ for all $i\! \mod 4$. Suppose that the point $x \in T_{e_0}$ represented by $D_0$ is a vertex.

If $D_0 \cap D_1 \cap D_2 \cap D_3$ is a geodesic segment of length $\geq n$, then the projectivity $[e_0; e_1; e_2; e_3; e_0]$ fixes the ball of radius $n$ around $x$ pointwise.

Proof.

Let $x'$ be a vertex at distance $r \leq n$ from $x$. Let $D'_0$ be the geodesic line parallel to $D_0$ and representing $x'$. For $i>0$, inductively define $D'_i$ as the geodesic line parallel to $D_i$ and representing the same point of $T_{e_i}$ as $D'_{i-1}$.

An immediate induction on $r = d(x, x')$, using Lemma 4.4, shows that $D'_0 \cap D'_1 \cap D'_2 \cap D'_3$ is a geodesic segment of length $\geq n-r$. By Lemma 4.2, this implies that $x'$ is fixed by $[e_0; e_1; e_2; e_3; e_0]$.

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Lemma 4.6.

Let $v \in X$ be a vertex. Let $Q, Q_0, Q_1, Q_2, Q_3$ be sectors based at $v$ such that $A_{i, j} = Q_i \cup Q \cup Q_j$ is a half-apartment of $X$ for all $(i, j) \in \{0, 2\} \times \{1, 3\}$. For all $i\! \mod 4$, let $D_i$ be the boundary line of $A_{i, i+1}$, and let $e_i$ be the common endpoint of $D_{i-1}$ and $D_i$.

Then the projectivity $[e_0; e_1; e_2; e_3; e_0]$ is a horocyclic automorphism of the tree $T_{e_0}$.

Proof.

We refer to Figure 2.

Let $x_0 \in T_{e_0}$ be the vertex represented by $D_0$, and let $(x_0, x_1, \dots )$ be the geodesic ray of $T_{e_0}$ represented by the sector $Q_0$.

We claim that the projectivity $u = [e_0; e_1; e_2; e_3; e_0]$ fixes the ball of radius $n$ around $x_n$ pointwise so that $u$ is indeed horocyclic.

To see this, let $D'_0$ be the geodesic line parallel to $D_0$ and representing $x_n$. For $i>0$, inductively define $D'_i$ as the geodesic line parallel to $D_i$ and representing the same point of $T_{e_i}$ as $D'_{i-1}$.

By construction, the line $D'_i$ is contained in the half-apartment $A_{i, i+1}$. Therefore, it intersects the sector $Q$ in a segment of length $n$. In particular, the intersection $D'_0 \cap D'_1 \cap D'_2 \cap D'_3$ is a geodesic segment of length $\geq n$. The claim then follows from Lemma 4.5.

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Proof of Proposition 4.1.

Let $v \in X$ be a vertex. Let $Q, Q_0, Q_1, Q_2, Q_3$ be sectors based at $v$ such that $A_{i, j} = Q_i \cup Q \cup Q_j$ is a half-apartment of $X$ for all $(i, j) \in \{0, 2\} \times \{1, 3\}$. Since $X$ is thick, we may choose those sectors such that $Q_i \cap Q_{i+2}$ is a geodesic ray, for all $i =0, 1$. In particular, denoting the boundary chamber of $Q_i$ by $C_i \in \operatorname {Ch}(\Delta )$, we have $C_0 \neq C_2$ and $C_1 \neq C_3$.

For all $i\! \mod 4$, let $D_i$ be the boundary line of $A_{i, i+1}$, and let $e_i$ be the common endpoint of $D_{i-1}$ and $D_i$.

By Lemma 4.6, the projectivity $u = [e_0; e_1; e_2; e_3; e_0]$ is a horocyclic automorphism of the tree $T_{e_0}$. By Lemma 3.3, the projectivity $u$ is a nontrivial permutation of $\partial T_{e_0}$ so that $u$ is indeed a nontrivial automorphism of $T_{e_0}$.

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4.2. $\widetilde{A}_2$-buildings with a linear projectivity group are Bruhat–Tits

Theorem 4.7.

Let $X$ be a thick, locally finite, Euclidean building of type $\widetilde{A}_2$. Let $\Pi$ be its projectivity group, viewed as a subgroup of $\operatorname {Aut}(T)$, where $T$ is the model tree of $X$. Let, moreover, $P = \overline{\Pi } \leq \operatorname {Aut}(T)$ denote the closure of $\Pi$ in the group $\operatorname {Aut}(T)$ endowed with the compact-open topology.

If $P$ has an open subgroup admitting a continuous, faithful, finite-dimensional representation over a local field, then $X$ is a Bruhat–Tits building.

By the classification of Bruhat–Tits buildings, one knows that $X$ is then the Euclidean building associated with $\mathrm{PGL}_3(D)$, where $D$ is a finite-dimensional division algebra over a local field (see 80, Chapter 28). Notice, moreover, that the converse to Theorem 4.7 holds: indeed, if $X$ is the Euclidean building associated with $\mathrm{PGL}_3(D)$, then its projectivity group is $\mathrm{PGL}_2(D)$ acting on its Bruhat–Tits tree (this follows from Proposition 2.3 from 46).

Proof of Theorem 4.7.

By Proposition 3.1, the projectivity group $(\Pi , \partial T)$ is $3$-transitive. In particular $P$ is $3$-transitive on $\partial T$. We may therefore invoke Proposition 3.8, which ensures that $(P, \partial T)$ is a Moufang set.

By Proposition 4.1, the group $\Pi$ contains a nontrivial element $u$ which is a horocyclic automorphism of $T$. By Proposition 3.8(iii), that element $u$ belongs to a root group $U_e$ of $P$, for some end $e \in \partial T$. Since $(P, \partial T)$ is Moufang, it follows that $U_e$ is normal in $P_e$. In particular $U_e \cap \Pi _e$ is normal in $\Pi _e$. Since $\Pi$ is $3$-transitive on $\partial T$, it follows that $\Pi _e$ is $2$-transitive on $\partial T \setminus \{e\}$, and hence any nontrivial normal subgroup of $\Pi _e$ is transitive on $\partial T \setminus \{e\}$. We have seen that $U_e \cap \Pi _e$ is a nontrivial normal subgroup of $\Pi _e$ and is thus transitive on $\partial T \setminus \{e\}$. Since the root group $U_e$ is sharply transitive on $\partial T \setminus \{e\}$, we must have $U_e \leq \Pi _e$. It follows that $(\Pi , \partial T)$ is a Moufang set. Therefore the building at infinity $\Delta$ is Moufang by Theorem 3.2. Equivalently $X$ is Bruhat–Tits.

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5. Other topological spaces associated with $\widetilde{A}_2$-buildings

Throughout this section, we let $X$ be a locally finite $\widetilde{A}_2$-building and let $\Gamma \leq \operatorname {Aut}(X)$ be a discrete subgroup consisting of type-preserving automorphisms which acts cocompactly on $X$. The goal of this section is to define auxiliary topological spaces associated with the pair $(X, \Gamma )$ and needed for subsequent developments. In particular, in §5.1 we define the space $\mathscr{F}$ consisting of marked flats and in §5.2 the space $\widetilde{\mathscr{W}}$ consisting of marked wall trees. It turns out that the space $\widetilde{\mathscr{W}}$ is too big for our purposes. We therefore define the notion of restricted marked wall trees and study the space of such objects in §5.3. Unfortunately, this makes our discussion a bit technical. With the hope of helping the reader overcome those technicalities, we provide in §5.4 a broad overview of the spaces we consider and the maps connecting them.

5.1. The space of marked flats

We fix a model flat $\Sigma$ for $X$, a base vertex $0\in \Sigma$, and a sector $Q^+$ based at $0$. The direction of $Q^+$ is called the positive Weyl chamber. The sector of $\Sigma$ based at $0$ and opposite $Q^+$ is denoted by $Q^-$. We let $A$ be the group of simplicial translations of $\Sigma$ (not necessarily type-preserving). The group $A$ acts sharply transitively on the vertices of $\Sigma$. We let $W$ be the group of type-preserving automorphisms of $\Sigma$ fixing $0$. It is called the (spherical) Weyl group of $X$ and is isomorphic to the symmetric group $S_3$.

Definition 5.1.

A marked flat in $X$ is a simplicial (not necessarily type-preserving) embedding $\Sigma \to X$. We denote by $\mathscr{F}$ the set of marked flats in $X$ and endow it with the topology of pointwise convergence.

Remark 5.2.

By Theorem 2.3 the image of a marked flat in $X$ is indeed a flat in $X$.

Note that $\mathscr{F}$ is locally compact as $X$ is locally finite. In fact it is easy to see how $\mathscr{F}$ is decomposed into a disjoint union of compact open subspaces. We define a map

$$\begin{equation*} \mathscr{F}\to X^{(0)} : \phi \mapsto \phi (0), \end{equation*}$$

where as before $X^{(0)}$ denotes the vertex-set of $X$. For each $x\in X^{(0)}$ we let $\mathscr{F}_x$ denote the preimage of $x$ under this map. Clearly the subsets $\mathscr{F}_x$ are compact open and $\mathscr{F}=\bigsqcup \mathscr{F}_x$.

The group $W\ltimes A$ acts on $\mathscr{F}$ by precomposition and $\Gamma$ acts on $\mathscr{F}$ by postcomposition. Thus these two actions commute. We endow $\mathscr{F}/\Gamma$ and $\mathscr{F}/A$ with the quotient topologies. Note that $\mathscr{F}/\Gamma$ is a $W\ltimes A$-space and $\mathscr{F}/A$ is a $\Gamma \times W$-space.

Lemma 5.3.

The space $\mathscr{F}/\Gamma$ is a compact $W\ltimes A$-space.

Proof.

The union of the spaces $\mathscr{F}_x$ taken over a finite set of representatives of the $\Gamma$-orbits in $X^{(0)}$ is compact and maps surjectively onto $\mathscr{F}/\Gamma$. The latter is thus indeed compact.

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Recall that two chambers in $\Delta$ are called opposite if they are at maximal distance in the graph $\Delta$ (i.e., at distance 3). We denote by $\Delta ^{2,\mathrm{op}}\subset \operatorname {Ch}(\Delta ) \times \operatorname {Ch}(\Delta )$ the set of ordered pairs of opposite chambers, and endow it with the topology induced by the product topology. This is a locally compact topology. We consider the map

$$\begin{equation*} \theta \colon \mathscr{F}\to \Delta ^{2,\mathrm{op}} : \phi \mapsto ([\phi (Q^+)],[\phi (Q^-)]) , \end{equation*}$$

where $[\phi (Q^+)]$ denotes the equivalence class of the sector $\phi (Q^+)\subset X$ which is a chamber in $\operatorname {Ch}(\Delta )$ and similarly for $[\phi (Q^-)]$ which is an opposite chamber.

Lemma 5.4.

The map $\theta \colon \mathscr{F}\to \Delta ^{2,\mathrm{op}}$ induces a $\Gamma \times W$-equivariant homeomorphism between $\mathscr{F}/A$ and $\Delta ^{2,\mathrm{op}}$.

Proof.

We first check that $\theta$ is continuous. Assume that $\phi _n\in \mathscr{F}$ are embeddings of $\Sigma$ converging to $\phi$. Then there exists $x$ such that for $n$ large enough we have $\phi _n(0)=x$, and for even larger $n$, $\phi _n$ is constant on the ball of radius $r$ around $x$. In particular, $\phi _n(Q^+)$ and $\phi (Q^+)$ are in $\Omega _x(y)$ for some $y$ at distance $r$ from $x$ (see Section 2.2.2 for the definition of the set $\Omega _x(y)$). Hence $\phi _n(Q^+)$ converges to $\phi (Q^+)$, and similarly for $\phi _n(Q^-)$.

By Lemma 2.7, $\theta$ is surjective and its fibers are exactly the $A$-orbits on $\mathscr{F}$ since $A$ is vertex-transitive on $\Sigma$. By the definition of the quotient topology we thus obtain a continuous bijection $\theta ' \colon \mathscr{F}/A\to \Delta ^{2,\mathrm{op}}$. We need to show that $\theta '$ is an open map. For $x\in X^{(0)}$ we denote by $\mathscr{F}'_x$ the image of $\mathscr{F}_x$ in $\mathscr{F}/A$. Note that $\{\mathscr{F}'_x\}$ is an open cover of $\mathscr{F}/A$. Thus it is enough to show that $\theta '|_{\mathscr{F}'_x}$ is open. By compactness, $\theta '|_{\mathscr{F}'_x}$ is a homeomorphism of $\mathscr{F}'_x$ onto its image, thus it is enough to show that $\theta '|_{\mathscr{F}'_x}$ has an open image. Note that $\theta '(\mathscr{F}'_x)=\theta (\mathscr{F}_x)$ is the union of all $(\Omega _x(y)\times \Omega _x(y'))\cap \Delta ^{2,\mathrm{op}}$, where the vertices $y$ and $y'$ have the following properties: the geodesic segment $[y, y']$ contains $x$, and $[y, y']$ is not contained in any wall of any apartment in $X$. Hence this set is indeed open in $\Delta ^{2,\mathrm{op}}$.

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5.2. The space of marked wall trees

A flat in $X$ may be viewed as the union of all geodesic lines that are parallel to a given regular line in $X$ (see Section 2.3 for the definition of regular and singular lines). We now develop analogous notions for the case of a singular line. Let $\ell$ be a singular geodesic line in $X$ so that the two endpoints of $\ell$ are vertices of $\Delta$. The union of all geodesic lines in $X$ that are parallel to $\ell$ is a closed CAT($0$)-convex subset of $X$. As a CAT($0$) space, it is isometric to the product $T_\ell \times \mathbf{R}$, where $T_\ell$ is the wall tree associated to $\ell$ (see Proposition 2.11). In particular, $T_\ell$ is isomorphic to the model tree $T$ (see Definition 3.4). Conversely, using Theorem 2.3 it is not hard to see that any isometric embedding of the CAT($0$) space $T\times \mathbf{R}$ in $X$ arises in that way.

The subspace $T_\ell \times \mathbf{R} \subset X$ also inherits a simplicial structure from $X$. We shall now describe that simplicial structure abstractly. The model wall tree of $X$ is the simplicial complex denoted by $\Upsilon$ and defined as follows. Fix a base vertex $o \in VT$ in the model tree $T$ of $X$. The vertex set $\Upsilon ^{(0)}$ of $\Upsilon$ is the collection of pairs $(v, s)$, where $v$ is a vertex of $T$ and $s$ is a real number that satisfies the following conditions: if the distance $d(o, v)$ is even, then $s \in \mathbf{Z}$, while if $d(o, v)$ is odd, then $s- \frac{1}{2} \in \mathbf{Z}$. Two distinct vertices, $(v, s)$ and $(w, t)$, form an edge of $\Upsilon$ if and only if $d(v, w) \leq 1$ and $|s-t| \leq 1$. The $2$-simplices of $\Upsilon$ are defined so that every triple of distinct vertices that are pairwise adjacent are contained in a unique $2$-simplex. This defines the model wall tree $\Upsilon$ as a simplicial complex.

Let us mention that a CAT($0$) metric realization of $\Upsilon$ is obtained by giving each edge length $1$ and by endowing each $2$-simplex with the metric of a Euclidean equilateral triangle. In that way $\Upsilon$ carries a complete CAT($0$) metric that is isometric to $T \times \mathbf{R}$, where $T$ is viewed as a metric tree in which edges have length $\frac{\sqrt 3}{2}$. That isometry $\Upsilon \to T \times \mathbf{R}$ is the identity on the vertex set $\Upsilon ^{(0)}$, which has been defined as a subset of $VT \times \mathbf{R}$. Although we shall not need that fact, let us mention moreover that every simplicial embedding $\Upsilon \to X$ is also an isometric embedding with respect to the respective CAT($0$) metric realizations of $\Upsilon$ and $X$.

There are two types of vertices in the spherical building $\Delta$, which we called points and lines in Definition 2.2. From now on, we adopt the convention to denote these two types by $+$ and $-$, respectively. We let $\Delta _+$ (resp., $\Delta _-$) be the set of vertices of type $+$ (resp., of type $-$) in $\Delta$. The natural maps $\operatorname {Ch}(\Delta ) \to \Delta _+$ and $\operatorname {Ch}(\Delta ) \to \Delta _-$, associating to each chamber its two boundary vertices, are $\operatorname {Aut}(X)$-equivariant, continuous and surjective (see 23, Prop. 3.5). We denote by $\Omega _x^+(y)$ and $\Omega _x^-(y)$ the image in $\Delta _+$ and $\Delta _-$ of the basic open sets $\Omega _x(y)$.

Definition 5.5.

A marked wall tree in $X$ is a simplicial embedding $\phi \colon \Upsilon \to X$ of the model wall tree $\Upsilon$ to $X$ such that the limit of $\phi (x,s)$ when $s$ tends to $+\infty$ is a vertex of type $+$. We denote by $\widetilde{\mathscr{W}}$ the set of marked wall trees in $X$ and endow it with the topology of pointwise convergence.

Note that $\widetilde{\mathscr{W}}$ is locally compact as $X$ is locally finite. In fact, it is easy to see how it decomposes into a disjoint union of compact open subspaces. As before, let $o\in VT$ be the base vertex and define a map

$$\begin{align} \widetilde{\mathscr{W}}\to X^{(0)} : \phi \mapsto \phi (o,0). {\tag{1}} \end{align}$$

For each $x\in X^{(0)}$ we let $\widetilde{\mathscr{W}}_x$ denote the preimage of $x$ under this map. Clearly the subsets $\widetilde{\mathscr{W}}_x$ are compact open and $\widetilde{\mathscr{W}}=\bigsqcup \widetilde{\mathscr{W}}_x$.

Let us now describe the full automorphism group $\operatorname {Aut}(\Upsilon )$ of the model wall tree. It is a totally disconnected locally compact group with respect to the topology of pointwise convergence. It embeds continuously as a closed subgroup in $\operatorname {Aut}(T) \times \operatorname {Isom}(\mathbf{R})$ via the metric realization $\Upsilon \to T \times \mathbf{R}$ explained above. In particular, there are canonical projections

$$\begin{equation*} p_T \colon \operatorname {Aut}(\Upsilon ) \to \operatorname {Aut}(T) \qquad \text{and} \qquad p_\mathbf{R}\colon \operatorname {Aut}(\Upsilon ) \to \operatorname {Isom}(\mathbf{R}) \end{equation*}$$

that are both continuous homomorphisms. Moreover, $p_T$ is surjective, whereas the image of $p_\mathbf{R}$ is isomorphic to the infinite dihedral group. However, we emphasize that the continuous surjection $p_T \colon \operatorname {Aut}(\Upsilon ) \to \operatorname {Aut}(T)$ has no section: indeed, no involution that swaps two adjacent vertices in $T$ can be lifted to an automorphism of $\Upsilon$. Let

$$\begin{equation*} \operatorname {Aut}(\Upsilon )^0 = \{ \alpha \in \operatorname {Aut}(\Upsilon ) \mid p_\mathbf{R}(\alpha ) \text{ is a translation}\}. \end{equation*}$$

Since the translation subgroup of $\operatorname {Isom}(\mathbf{R})$ is an open subgroup of index $2$, we see that $\operatorname {Aut}(\Upsilon )^0$ is an open subgroup of index $2$ in $\operatorname {Aut}(\Upsilon )$. We shall also need to consider the subgroup

$$\begin{equation*} S = \operatorname {Ker}(p_T) \cap \operatorname {Aut}(\Upsilon )^0 \vartriangleleft \operatorname {Aut}(\Upsilon ). \end{equation*}$$

Observe that $S \cong (\mathbf{Z},+)$.

For each $\alpha \in \operatorname {Aut}(\Upsilon )$ and $\phi \in \widetilde{\mathscr{W}}$, the map $\phi \circ \alpha$ belongs to $\widetilde{\mathscr{W}}$ if and only if $\alpha \in \operatorname {Aut}(\Upsilon )^0$. Similarly, for each $\gamma \in \operatorname {Aut}(X)$ and $\phi \in \widetilde{\mathscr{W}}$, the map $\gamma \circ \phi$ belongs to $\widetilde{\mathscr{W}}$ if and only if the $\gamma$-action on the spherical building $\Delta$ is type-preserving. Recall that $\Gamma <\operatorname {Aut}(X)$ is type-preserving; thus the $\Gamma$ action on $\Delta$ is also type-preserving. Thus we get a continuous action of $\operatorname {Aut}(\Upsilon )^0$ on $\widetilde{\mathscr{W}}$ by precompositions and an action of $\Gamma$ on $\widetilde{\mathscr{W}}$ by postcompositions. Clearly these actions commute. We endow the quotient spaces $\widetilde{\mathscr{W}}/\Gamma$ and $\widetilde{\mathscr{W}}/\operatorname {Aut}(\Upsilon )^0$ with the quotient topologies. Note that $\widetilde{\mathscr{W}}/\Gamma$ is an $\operatorname {Aut}(\Upsilon )^0$-space and $\widetilde{\mathscr{W}}/\operatorname {Aut}(\Upsilon )^0$ is a $\Gamma$-space.

Lemma 5.6.

The $\Gamma$-action on $\widetilde{\mathscr{W}}$ is proper. Moreover, $\widetilde{\mathscr{W}}/\Gamma$ is a compact $\operatorname {Aut}(\Upsilon )^0$-space.

Proof.

The properness is clear from the definitions. The compactness of $\widetilde{\mathscr{W}}/\Gamma$ follows from the fact that $\Gamma$ has a finite fundamental domain in $X^{(0)}$ and the compactness of the spaces $\widetilde{\mathscr{W}}_x$.

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Let $\Delta _\pm ^\mathrm{op}$ be the set of pairs of opposite vertices $(u,v)$ with $u\in \Delta _+$ and $v\in \Delta _-$. This is an open subset of the compact space $\Delta _+\times \Delta _-$, hence a locally compact space. We define a map

$$\begin{align} \sigma \colon \widetilde{\mathscr{W}}\to \Delta _\pm ^\mathrm{op}: \phi \mapsto (\phi ^+, \phi ^-) = \left(\lim _{s\to +\infty } \phi (o,s),\lim _{s\to -\infty } \phi (o,s)\right), {\tag{2}} \end{align}$$

where $\lim _{s\to +\infty } \phi (o,s)$ represents the vertex at infinity of the ray $(\phi (o, s))_{s \geq 0}$ and similarly for $\lim _{s\to -\infty } \phi (o,s)\in \Delta _-$.

Lemma 5.7.

The $\operatorname {Aut}(\Upsilon )^0$-action on $\widetilde{\mathscr{W}}$ is proper and free, and the map $\sigma \colon \widetilde{\mathscr{W}}\to \Delta _\pm ^\mathrm{op}$ induces a $\Gamma$-equivariant homeomorphism

$$\begin{equation*} \sigma ':\widetilde{\mathscr{W}}/\operatorname {Aut}(\Upsilon )^0\overset{\sim }{\longrightarrow } \Delta _\pm ^\mathrm{op}. \end{equation*}$$

Proof.

The $\operatorname {Aut}(\Upsilon )^0$-action on $\widetilde{\mathscr{W}}$ is free by construction. To see that it is proper it is enough to show that for all $x,y\in X^{(0)}$, the collection

$$\begin{equation*} K=\{\alpha \in \operatorname {Aut}(\Upsilon )^0\mid \alpha (\widetilde{\mathscr{W}}_x) \cap \widetilde{\mathscr{W}}_y \neq \emptyset \} \end{equation*}$$

is a compact subset of $\operatorname {Aut}(\Upsilon )^0$, and this is indeed the case, as if $\alpha \in K$, then $d(\alpha (o,0),(o,0))=d(x,y)$.

To prove that $\sigma '$ is bijective, observe that for all $(u, v) \in \Delta _\pm ^\mathrm{op}$, there exists a marked wall tree $\phi \in \widetilde{\mathscr{W}}$ with $\sigma (\phi ) = (u, v)$. Moreover, for any other marked wall tree $\phi ' \in \widetilde{\mathscr{W}}$, we have $\sigma (\phi ') = (u, v)$ if and only if there exists $\alpha \in \operatorname {Aut}(\Upsilon )^0$ such that $\phi ' = \phi \circ \alpha$. This proves that the map in the lemma is indeed bijective.

Let us now check that the map $\sigma$ is continuous. Let $\phi _n$ be elements of $\widetilde{\mathscr{W}}$, with $\phi _n\to \phi$. Let $u_n=\lim _{s\to +\infty } \phi _n(o,s)$ and $u=\lim _{s\to +\infty } \phi (o,s)$. Then for every $r>0$, for $n$ large enough, $\phi _n$ is constant on a ball of radius $r$ around $(o,0)$. If $y$ is a point in this ball, it follows that $u_n\in \Omega _x^+(y)$ for $n$ large enough. Similarly, $u\in \Omega _x^+(y)$ so that $(u_n)$ converges to $u$.

We are left to show that $\sigma$ is an open map. As in the proof of Lemma 5.4, it suffices by compactness to prove that $\sigma (\widetilde{\mathscr{W}}_x)$ is open, for every $x\in X^{(0)}$. Indeed, $\sigma (\widetilde{\mathscr{W}}_x)$ is a union of all $\Omega _x^+(y)\times \Omega _x^-(y')\cap \Delta _\pm ^\mathrm{op}$, where $y$ and $y'$ are vertices such that $x$ belongs to the geodesic segment $[y, y']$, and hence open.

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Definition 5.8.

We let $\overline{\Delta }_+$ be the set of pairs $(v,f)$, where $v\in \Delta _+$ and $f$ is a simplicial isomorphism from $T$ to the panel tree $T_v$. We let $\pi _+$ be the map $\widetilde{\mathscr{W}}\to \overline{\Delta }_+$ given by $\pi _+(\phi )=(v,f)$, where $v$ is the common limit of $\phi (x,t)$ when $t$ tends to $+\infty$, and $f(x)$ is the class of the geodesic line $\phi (x,\mathbf{R})$ in $T_v$. We similarly define $\overline{\Delta }_-$ and $\pi _-$. Observe that the maps $\pi _+$ and $\pi _-$ are surjective. We endow $\overline{\Delta }_+$ and $\overline{\Delta }_-$ with the quotient topologies obtained under $\pi _+$ and $\pi _-$.

Lemma 5.9.

Fix $u\in \Delta _+$. Let $F_u$ be the set of simplicial isomorphisms from $T$ to $T_u$, endowed with the topology of pointwise convergence. Then the map $F_u\to \overline{\Delta }_+$ defined by $f\mapsto (u,f)$ is a homeomorphism on its image.

Proof.

Let $f_n\in F_u$ be such that $(f_n)$ converges to $f$. Fix a vertex $v\in \Delta _-$ opposite $u$. Let $\phi _n\in \widetilde{\mathscr{W}}$ be such that for every $n$, $\lim _{s\to -\infty }\phi _n(x,s)=v$ and $\pi _+(\phi _n)=(u,f_n)$. Let $x\in T$. Then the map $t \mapsto \phi _n(x,t)$ is a parametrization of a geodesic from $v$ to $u$ which, for $n$ large enough, corresponds to $f(x)$. Upon replacing $\phi _n(x, t)$ by $\phi _n(x, t-t_0)$ for some suitable $t_0 \in \mathbf{R}$, we can assume that the $\phi _n$ are chosen so that $\phi _n(x,0)$ is eventually constant. Then $\phi _n$ converges to some $\phi \in \widetilde{\mathscr{W}}$ with $\pi _+(\phi )=(u,f)$.

Conversely, assume that $\phi _n$ converges to $\phi$ with $\pi _+(\phi _n)=(u,f_n)$ and $\pi _+(\phi )=(u,f)$. Let $x\in T$. Then we have for $n$ large enough $\phi _n(x,0)=\phi (x,0)$. Since $f_n(x)$ (resp., $f(x)$) is the projection of $\phi _n(x,0)$ (resp., $\phi (x,0)$) to $T_u$, we immediately get that $f_n$ converges to $f$.

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It will be convenient for us to consider various fiber products. If $A,B,C$ are topological spaces, with continuous maps $p_A:A\to C$ and $p_B:B\to C$, the fiber product $A\times _C B$ is defined as the set $\{(a,b)\in A\times B\mid p_A(a)=p_B(b)\}$. It is equipped with the topology induced from the product topology.

Lemma 5.10.

The map $\widetilde{\mathscr{W}}/S\to \Delta _\pm ^\mathrm{op}\times _{\Delta _+}\overline{\Delta }_+$, which associates to $\phi$ the pair $((\phi ^+,\phi ^-),\pi _+(\phi ))$, is a homeomorphism, where $\widetilde{\mathscr{W}}/S$ is endowed with the quotient topology.

Similarly, the map $\widetilde{\mathscr{W}}/S\!\to \! \Delta _\pm ^\mathrm{op}\times _{\Delta _-}\overline{\Delta }_-$, which associates to $\phi$ the pair $((\phi ^+,\phi ^-), \pi _-(\phi ))$, is a homeomorphism.

Proof.

We argue for $\Delta _\pm ^\mathrm{op}\times _{\Delta _+}\overline{\Delta }_+$ only, as the other case is similar. The map of the lemma is bijective: every pair of opposite vertices is linked by geodesics forming a tree, and given a pair of opposite vertices, there is an element of $\widetilde{\mathscr{W}}$ realizing a given embedding of the tree, and that element is unique modulo $S$. Since both maps $\widetilde{\mathscr{W}}/S\to \Delta _\pm ^\mathrm{op}$ and $\widetilde{\mathscr{W}}/S\to \overline{\Delta }_+$ are continuous, their product is also continuous.

We now have to check that this map is open. Again, by compactness, it suffices to prove that the image of $\widetilde{\mathscr{W}}_x$ in $\Delta _\pm ^\mathrm{op}\times _{\Delta _+}\overline{\Delta }_+$ is an open set. But this set is exactly the set of pairs whose first coordinate is in $\sigma (\widetilde{\mathscr{W}}_x)$. Hence it is open.

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Lemma 5.11.

The map $\Delta _\pm ^\mathrm{op}\times _{\Delta _+}\overline{\Delta }_+\to \overline{\Delta }_-$, which associates to an element $((u,v),f)$ with $f \colon T\to T_v$ the pair $(v,f')$, where $f' \colon T\to T_u$ is the map obtained by composing $f$ with the identification $T_v\to T_u$, is continuous.

Proof.

The space $\Delta _\pm ^\mathrm{op}\times _{\Delta _+}\overline{\Delta }_+$ is homeomorphic to $\widetilde{\mathscr{W}}/S$ by Lemma 5.10. The map of the lemma is transported by this homeomorphism to the map $\pi _-:\widetilde{\mathscr{W}}/S\to \overline{\Delta }_-$, which is continuous.

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Proposition 5.12.

Let $n \geq 2$ be an integer, and let $V_n(u,v)$ be the set of sequences

$$\begin{equation*} (u_1,u_2,\dots ,u_n) \end{equation*}$$

of vertices which are consecutively opposite and such that $u_1$ is opposite $u$ and $u_n$ is opposite $v$. The set $V_n(u,v)$ is equipped with the topology induced from the product topology. Also let $M(u, v)$ be the set of simplicial isomorphisms of $T_u$ to $T_v$, with the topology of pointwise convergence.

Then the map

$$\begin{equation*} V_n(u,v) \to M(u, v) : (u_1,u_2,\dots ,u_n) \mapsto [u;u_1;u_2;\cdots ;u_n;v] \end{equation*}$$

is continuous.

Proof.

We prove the proposition in the special case where $u$ and $v$ are of type $+$ (so that $n$ is odd). The other cases are similar. Let $V_n$ be the set of sequences of $n$ consecutively opposite vertices, starting from a vertex of type $+$. Note that $V_n$ is naturally identified with the fiber product $\Delta _\pm ^\mathrm{op}\times _{\Delta _-}\Delta _\pm ^\mathrm{op}\times _{\Delta _+}\times \dots \times _{\Delta _-}\Delta _\pm ^\mathrm{op}$ (with $n$ factors).

Let us form the fiber product $V_n\times _{\Delta _+}\overline{\Delta }_+$ associated with the map $V_n \to \Delta _+ : (u_1,\dots ,u_n) \mapsto u_n$. We then consider the map $P_n \colon V_n\times _{\Delta _+}\overline{\Delta }_+\to \overline{\Delta }_+$ which associates to $\big ((u_1,\dots ,u_n), (u_n,f) \big )$ the map $f':T\to T_{u_1}$ obtained by composing the map $f$ with the perspectivity $[u_n;u_{n-1};\cdots ;u_1]$. Applying Lemma 5.11 (and its symmetric statement with $+$ and $-$ exchanged) and using the associativity of the fiber product, we see that the map $P_n$ is continuous.

For every isomorphism $f \colon T\to T_u$, we have

$$\begin{equation*} P_{n+2}\big ((v,u_n,\dots ,u_1, u),(u,f)\big )= (v, [u, u_1, \dots , u_n, v] \circ f). \end{equation*}$$

Let us now choose an isomorphism $f \colon T\to T_u$ that we keep fixed. Given $(u_1,u_2,\dots , u_n) \in V_n(u,v)$, let us denote by $m \in M(u, v)$ the perspectivity $[u;u_1;u_2;\cdots ;u_n;v]$. The map $(u_1,u_2,\dots ,u_n) \mapsto m$ can be factored as the composite of the maps

$$\begin{equation*} (u_1,u_2,\dots ,u_n) \mapsto \big ((v,u_n,\dots ,u_1, u),(u,f)\big ) \underset{P_{n+2}}{\longmapsto } (v, m \circ f) \mapsto m \circ f \mapsto m. \end{equation*}$$

Since each individual map in that sequence is continuous, the proposition follows.

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5.3. The space of restricted marked wall trees

In the next section we will construct measures on the various spaces we consider here. There is a natural measure on $\widetilde{\mathscr{W}}$ that we could construct. However, we will not do it, as this measure fails to satisfy the ergodic properties that we desire. Instead we will consider a certain subspace of $\widetilde{\mathscr{W}}$, to be denoted $\mathscr{W}$, on which the analogous measure will be better behaved. This subsection is devoted to the definition and the study of this subspace and its properties.

For the next definition, recall from Section 5.2 that $\operatorname {Aut}(\Upsilon )$ acts on $\widetilde{\mathscr{W}}$ by precomposition and $\Gamma$ acts on $\widetilde{\mathscr{W}}$ by postcomposition. Moreover, $S$ is a cyclic normal subgroup of $\operatorname {Aut}(\Upsilon )$ acting as integer translations along the line factor of $\Upsilon$.

Definition 5.13.

Two embeddings $\phi ,\psi \in \widetilde{\mathscr{W}}$ are positively equivalent if for every $x\in VT$, there exists $t_x \in \mathbf{R}$ such that $\phi (x, t)=\psi (x, t)$ for all $t > t_x$. The embeddings $\phi$ and $\psi$ are negatively equivalent if for every $x\in VT$, there exists $t_x \in \mathbf{R}$ such that $\phi (x, -t)=\psi (x, -t)$ for all $t > t_x$. We denote the positive and negative equivalences by $\sim _+$ and $\sim _-$, respectively. We denote by $\sim$ the equivalence relation on $\widetilde{\mathscr{W}}$ generated by $\sim _+$, $\sim _-$ and by the orbit equivalence induced by the action of $S\times \Gamma$.

Lemma 5.14.

The equivalence relation $\sim$ is $(\operatorname {Aut}(\Upsilon )^0\times \Gamma )$-invariant. Moreover, every equivalence class is $(S\times \Gamma )$-invariant, and $\operatorname {Aut}(\Upsilon )^0$ acts transitively on $\widetilde{\mathscr{W}}/\sim$.

Proof.

By construction, every individual $\sim$-class is $(S\times \Gamma )$-invariant. Clearly $\sim _+$ and $\sim _-$ are invariant under $\operatorname {Aut}(\Upsilon )^0$. Moreover, since $S\times \Gamma$ is a normal subgroup of $\operatorname {Aut}(\Upsilon )^0 \times \Gamma$, the $(S\times \Gamma )$-orbits are permuted by $\operatorname {Aut}(\Upsilon )^0\times \Gamma$. It follows that $\sim$ is also $(\operatorname {Aut}(\Upsilon )^0\times \Gamma )$-invariant. In particular, $\operatorname {Aut}(\Upsilon )^0$ acts on $\widetilde{\mathscr{W}}/\sim$.

It remains to show that $\operatorname {Aut}(\Upsilon )^0$ is transitive on $\widetilde{\mathscr{W}}/\sim$. Let $\phi _1, \phi _2 \in \widetilde{\mathscr{W}}$. We must find $\alpha \in \operatorname {Aut}(\Upsilon )^0$ such that $\phi _1 \circ \alpha \sim \phi _2$. Set $(\phi _i^+, \phi _i^-) = (u_i, v_i) \in \Delta _\pm ^\mathrm{op}$ for $i=1,2$. We choose $u\in \Delta _-$, which is opposite to both $v_1,v_2$, and $v\in \Delta _+$, which is opposite to both $u,u_2$ (see Lemma 2.8). We next successively choose $\psi _1, \psi _2, \psi _3, \psi _4 \in \widetilde{\mathscr{W}}$ such that:

•

$\psi _1^+ = u$ and $\psi _1 \sim _- \phi _1$;

•

$\psi _2^- = v$ and $\psi _2 \sim _+ \psi _1$;

•

$\psi _3^+ = u_2$ and $\psi _3 \sim _- \psi _2$;

•

$\psi _4^- = v_2$ and $\psi _4 \sim _+ \psi _3$.

Thus we have $\phi _1 \sim _- \psi _1 \sim _+ \psi _2 \sim _- \psi _3 \sim _+ \psi _4$ so that $\phi _1 \sim \psi _4$. Moreover, $(\psi _4^+, \psi _4^-) = (u_2, v_2) = (\phi _2^+, \phi _2^-)$. Since $\operatorname {Aut}(\Upsilon )^0$ is transitive on each fiber of the map $\widetilde{\mathscr{W}}\to \Delta _\pm ^\mathrm{op}$ by Lemma 5.7, we infer that there exists $\alpha \in \operatorname {Aut}(\Upsilon )^0$ with $\psi _4\circ \alpha = \phi _2$. The conclusion follows since $\phi _1 \sim \psi _4$.

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Fix $\gamma \!\in \!\Gamma$ and $u\!\in \!\Delta _-$. The map $\gamma$ induces a map $T_u\!\to \! T_{\gamma u}$. Let $(\gamma u,v_1,\dots ,v_n)$ be a sequence of consecutively opposite vertices. We denote by $[\gamma :u;v_1;\cdots ;v_n]$ the map $T_u\!\to \! T_{v_n}$ obtained by first applying $\gamma$ and then the perspectivity $[\gamma u;v_1;\cdots ;v_n]$.

Lemma 5.15.

Let $\phi ,\phi '\in \widetilde{\mathscr{W}}$, and set $\pi _-(\phi )=(u_-,f)$, $\pi _+(\phi ) = (u_+, f_+)$ and $\pi _-(\phi ')=(u'_-,f')$. Then the following assertions are equivalent:

(i)

$\phi \sim \phi '$.

(ii)

There exist an element $\gamma \in \Gamma$, an integer $n$, and vertices $v_0, v_1,\ldots ,v_n$ in $\Delta$ such that $(\gamma u_-, \gamma u_+ ) = (v_0, v_1)$, the pairs $(v_1,v_2)$, …, $(v_{n-1},v_n)$, $(v_n,u'_-)$ are all opposite, and $f'=[\gamma : u_-;v_1;\cdots ;v_n; u'_-]\circ f$.

Proof.

Assume that $\phi \sim \phi '$. By definition, this means that there exists a sequence $(\phi =\phi _0,\phi _1,\dots ,\phi _{n-1},\phi _n=\phi ')$ such that $\phi _i$ and $\phi _{i+1}$ are equivalent for $\sim _+$ or $\sim _-$ or in the same $(S\times \Gamma )$-orbit. Since $\sim _-$ and $\sim _+$ are preserved by $S\times \Gamma$ (see Lemma 5.14) we may assume that $\phi _1=(s,\gamma )\phi$, and for $i>1$, $\phi _i$ and $\phi _{i+1}$ are either $\sim _+$ or $\sim _-$ equivalent. Since $\sim _+$ and $\sim _-$ are transitive relations we can assume further that they alternate. Thus (i) implies (ii).

Conversely, let $\gamma \in \Gamma$, and let $v_0, v_1,\ldots ,v_n$ be vertices of $\Delta$ such that $(\gamma u_-, \gamma u_+) = (v_0, v_1)$, and $(v_1,v_2)$, …, $(v_{n-1},v_n)$ , $(v_n,u'_-)$ are all opposite pairs, and such that $f'=[\gamma : u_-;v_1;\cdots ;v_n;u'_-]\circ f$. Since $u_-, \gamma u_-$ and $u'_-$ all belong to $\Delta _-$, it follows that $n$ must be odd. Set $\phi _0 = \gamma \circ \phi$. For all $i \in \{1, \dots , n-1\}$, successively choose $\phi _i \in \widetilde{\mathscr{W}}$ satisfying $\big ((\phi _i)_+, (\phi _i)_-\big ) = (v_i, v_{i+1})$ and $\phi _i \sim _+ \phi _{i-1}$ if $i$ is odd, while $\phi _i$ must satisfy $\big ((\phi _i)_-, (\phi _i)_+\big ) = (v_i, v_{i+1})$ and $\phi _i \sim _- \phi _{i-1}$ if $i$ is even. Finally, choose $\phi _n \in \widetilde{\mathscr{W}}$ with $\{(\phi _n)_{+}, (\phi _n)_{-}\} = (v_n, u_-')$ and $\phi _n \sim _+ \phi _{n-1}$. By construction we have $\phi \sim \phi _0 \sim \dots \sim \phi _n$. It remains to observe that the condition $f'=[\gamma : u_-;v_1;\cdots ;v_n;u'_-]\circ f$ ensures that $\phi _n \sim _- \phi '$. Thus (ii) implies (i).

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In the next lemma, we use the projection map $p_T \colon \operatorname {Aut}(\Upsilon ) \to \operatorname {Aut}(T)$ from Section 5.2.

Lemma 5.16.

Fix an $\sim$-equivalence class $\mathscr{W}' \subset \widetilde{\mathscr{W}}$, and let $M'<\operatorname {Aut}(\Upsilon )^0$ be the stabilizer of $\mathscr{W}'$ so that $M'$ contains $S$. Fix $\phi \in \mathscr{W}'$, and set $\pi _-(\phi )=(u,f)\in \overline{\Delta }_-$ and $\pi _+(\phi )=(u',f')\in \overline{\Delta }_+$. Also let

$$\begin{equation*} M'_\phi =f \big (p_T(M') \big )f^{-1}<\operatorname {Aut}(T_u). \end{equation*}$$

Then

$$\begin{align*} M'_\phi =\big \{[\gamma : u; & \; v_1;\cdots ;v_n;u]\mid \ \gamma \in \Gamma , \ v_0, v_1, \dots , v_n \in \Delta _\pm , \ (\gamma u, \gamma u') = (v_0, v_1),\\ & \text{and the pairs } (v_1,v_2),\ldots ,(v_{n-1},v_n),(v_n,u)\text{ are all opposite}\big \}. \end{align*}$$

In particular, the projectivity group $\Pi (u)$ is contained in $M'_\phi$, and $p_T(M')$ is $3$-transitive on $\partial T$.

Proof.

Fix $m'\in M'$. We have $\phi \circ m'\sim \phi$ by the definition of $M'$. Moreover, recalling that $\pi _-(\phi ) = (u, f)$, we have $\pi _-(\phi \circ m') = (u, f \circ \pi _T(m'))$. By Lemma 5.15, there exists an automorphism of $T_u$ of the form $[\gamma :u;v_1;\cdots ;v_n;u]$ for some $\gamma \in \Gamma$ and some vertices $v_0, v_1, \dots , v_n$ of $\Delta$ with $(\gamma u, \gamma u') = (v_0, v_1)$ and $(v_1,v_2)$, …, $(v_{n-1},v_n)$, $(v_n,u)$ all opposite pairs such that $f \circ \pi _T(m') =[\gamma : u;v_1;\cdots ;v_n;u]\circ f$.

Conversely, let $\gamma \in \Gamma$, and let $v_0, v_1, \dots , v_n$ be vertices of $\Delta$ such that $(\gamma u, \gamma u') = (v_0, v_1)$, the pairs $(v_1,v_2)$, …, $(v_{n-1},v_n)$, $(v_n,u)$ are all opposite pairs, and define the automorphism $\alpha = [\gamma : u; \; v_1;\cdots ;v_n;u] \in \operatorname {Aut}(T_u)$. Since $p_T \colon \operatorname {Aut}(\Upsilon )^0 \to \operatorname {Aut}(T)$ is surjective, there exists $m' \in \operatorname {Aut}(\Upsilon )^0$ with $p_T(m') = f^{-1} \circ \alpha \circ f$. We have $\phi \circ m'\sim \phi$ by Lemma 5.15.

That $M'_\phi$ is $3$-transitive on $\partial T_u$ follows from Lemma 3.5. Therefore $p_T(M')$ is $3$-transitive on $\partial T$.

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We wish to consider a single equivalence class in $\widetilde{\mathscr{W}}$. A technical issue is that such an equivalence class will not be closed. The obvious solution is to replace the equivalence relation $\sim$ by its closure (as a subset of $\widetilde{\mathscr{W}}\times \widetilde{\mathscr{W}}$). Note that in general the closure of an equivalence relation need not be an equivalence relation. However, the situation is better when a topological group is acting continuously on the ambient space and transitively on the quotient space.

Definition 5.17.

We let $\simeq$ be the closure of $\sim$ as a subset of $\widetilde{\mathscr{W}}\times \widetilde{\mathscr{W}}$.

Lemma 5.18.

The relation $\simeq$ is an equivalence relation and each $\simeq$-equivalence class is closed in $\widetilde{\mathscr{W}}$. In fact, the $\simeq$-equivalence classes are the closures of the $\sim$-equivalence classes. In particular, every $\simeq$-equivalence class is $S\times \Gamma$-invariant. The quotient space $\widetilde{\mathscr{W}}/\simeq$ is compact, and the natural $\operatorname {Aut}(\Upsilon )^0$-action on $\widetilde{\mathscr{W}}/\simeq$ is continuous.

Proof.

Recall from Lemma 5.14 that $\operatorname {Aut}(\Upsilon )^0$ acts transitively on $\widetilde{\mathscr{W}}/\sim$. Therefore, the orbit map induces an equivariant bijection $\widetilde{\mathscr{W}}/\!\sim \ \to \operatorname {Aut}(\Upsilon )^0/M'$, where $M'$ is the subgroup of $\operatorname {Aut}(\Upsilon )^0$ stabilizing some $\sim$-equivalence class. Let $M$ be the closure of $M'$ in $\operatorname {Aut}(\Upsilon )^0$. We have canonical maps

$$\begin{equation*} \widetilde{\mathscr{W}}\to \widetilde{\mathscr{W}}/\! \sim \ \to \operatorname {Aut}(\Upsilon )^0/M' \to \operatorname {Aut}(\Upsilon )^0/M. \end{equation*}$$

We may now identify $\simeq$ with the fiber equivalence relation of the map $\widetilde{\mathscr{W}}\to \operatorname {Aut}(\Upsilon )^0/M$. Since $S \leq M$ and $p_T(\operatorname {Aut}(\Upsilon )^0) = \operatorname {Aut}(T)$, it follows that $p_T(M) \cong M/S$ is a closed subgroup of $\operatorname {Aut}(T)$ and that $\operatorname {Aut}(\Upsilon )^0/M$ is homeomorphic to $p_T(\operatorname {Aut}(\Upsilon )^0)/p_T(M) = \operatorname {Aut}(T)/p_T(M)$. Since $p_T(M')$ is $3$-transitive on $\partial T$ by Lemma 5.16, the same holds for $M$, and it follows from Theorem 3.7 that $p_T(\operatorname {Aut}(\Upsilon )^0)/p_T(M)$ is compact. All the required assertions now follow.

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The following crucial definition involves a noncanonical choice.

Definition 5.19.

We fix once and for all an $\simeq$-equivalence class in $\widetilde{\mathscr{W}}$ and denote it by $\mathscr{W}$. We call the elements of $\mathscr{W}$ the restricted marked wall trees. We let $M<\operatorname {Aut}(\Upsilon )^0$ be the stabilizer of $\mathscr{W}$. In particular, we have $S \vartriangleleft M$.

Lemma 5.20.

The space $\mathscr{W}$ is closed in $\widetilde{\mathscr{W}}$ and invariant under the action of $M\times \Gamma$. The $\Gamma$-action on $\mathscr{W}$ is proper and $\mathscr{W}/\Gamma$ is a compact $M$-space. The $M$-action on $\mathscr{W}$ is proper and free, and the restriction of the map $\widetilde{\mathscr{W}}\to \Delta _\pm ^\mathrm{op}$ to $\mathscr{W}$ yields $\Gamma$-equivariant homeomorphisms $\mathscr{W}/M \overset{\sim }{\longrightarrow }\Delta _\pm ^\mathrm{op}$ and $\mathscr{W}/S \big / M/S \overset{\sim }{\longrightarrow }\Delta _\pm ^\mathrm{op}$.

Proof.

The first sentence holds by the construction of $\mathscr{W}$. The second sentence follows by Lemma 5.20 and Lemma 5.6. To see the third, use Lemma 5.7 and observe that the embedding $\mathscr{W}\to \widetilde{\mathscr{W}}$ descends to a continuous map $\mathscr{W}/M \to \widetilde{\mathscr{W}}/\operatorname {Aut}(\Upsilon )^0$ which is injective by the definition of $M$. Moreover, this map is surjective in view of Lemma 5.14. Therefore we have a $\Gamma$-equivariant homeomorphism $\mathscr{W}/M \to \widetilde{\mathscr{W}}/\operatorname {Aut}(\Upsilon )^0$. The last required assertions now follow from Lemma 5.7 and the fact that $S$ is a closed normal subgroup of $M$.

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Theorem 5.21.

The group $M$ is a unimodular group and $p_T \colon M \to \operatorname {Aut}(T)$ yields a continuous isomorphism of $M/S$ onto a closed subgroup of $\operatorname {Aut}(T)$ acting $3$-transitively on the set of ends $\partial T$. In particular, $M$ has a closed cocompact normal subgroup $M^+$ containing $S$ such that $M^+/S$ is compactly generated, topologically simple, and nondiscrete. Furthermore, if $X$ is not Bruhat–Tits, then $M^+/S$ has no nontrivial continuous linear representation over any local field.

Proof.

By Lemma 5.18, the group $M$ is the closure of the group $M'$, and by Lemma 5.16, the group $p_T(M)$ contains a copy $\Pi$ of the projectivity group $\Pi (u)$ for some $u\in \Delta _-$. It follows by Lemma 3.5 that $M$ acts $3$-transitively on $\partial T$. As observed before, we have $S \leq M \leq \operatorname {Aut}(\Upsilon )^0$ and $p_T(\operatorname {Aut}(\Upsilon )^0) = \operatorname {Aut}(T)$ so that $p_T(M)$ is a closed subgroup of $\operatorname {Aut}(T)$ acting $3$-transitively on $\partial T$. The group $p_T(M)^+$ is described in Theorem 3.7 where its various properties are recorded, as well as the unimodularity of $p_T(M)$. Since $S$ is discrete, we see that $M$ is unimodular. We set $M^+ = p_T^{-1}(p_T(M)^+)$. We are left to show that the nonlinearity of $M^+/S \cong p_T(M)^+$ in case $X$ is not Bruhat–Tits. To this end we consider the topologically simple group $\overline{\Pi }^+$ which is obtained by applying Theorem 3.7 to the closure $\overline{\Pi }$ of the projectivity group $\Pi$. As $M/M^+$ is profinite and $\overline{\Pi }^+$ topologically simple, we have $\overline{\Pi }^+<p_T(M)^+$ since $\Pi < p_T(M)$. By Theorem 4.7, the group $\overline{\Pi }^+$ has no linear representations over local fields. We deduce that $p_T(M)^+ \cong M^+/S$ shares this property, as the fact that it is topologically simple.

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Definition 5.22.

We call the $M\times S$-space $\mathscr{W}/\Gamma$ the singular Cartan flow.

5.4. The broad picture

In this subsection we pack together the various $\Gamma$-spaces previously constructed into the following commutative diagram of $\Gamma$-equivariant maps:

$$\begin{align} \begin{split} \vcenter{\img[][214pt][134pt][{$\xymatrix{ X^{(0)}/\Gamma& \scrF/\Gamma\ar[l] & \scrS/\Gamma\ar[l] & & \\ X^{(0)} \ar[u] & \scrF\ar[dd] \ar[l] \ar[u] & \scrS\ar[l] \ar[d] \ar[u] & & \\ & & \scrS/S \ar[d]\ar[dl] \ar[r] \ar@/^2pc/[rr] & \overline\Delta_+ \ar[d] & \overline\Delta_- \ar[d] \\ \Ch(\D) & \Delta^{2,\opx} \ar[r] \ar[l]_{p_1,p_2} & \Delta_\pm^\opx\ar[r] \ar@/^2pc/[rr] & \D_+ & \D_- }$}]{Images/imgfe736114f987b3a011260f3d257eac6d.svg}} \end{split} {\tag{3}} \end{align}$$ In order to keep our notation as light as possible we will not name most of the arrows in diagram (3), but in what follows we will make clear what they are.

At the beginning of §5.1 we have fixed the model apartment $\Sigma$ and a positive Weyl chamber $Q^+$ emanating from $0\in \Sigma$. We defined $\mathscr{F}\to X^{(0)}$ by

$$\begin{equation*} \mathscr{F}=\{\text{simplicial embeddings}\nobreakspace \Sigma \to X\} \ni \phi \mapsto \phi (0) \in X^{(0)}. \end{equation*}$$

We further defined the map $\mathscr{F}\to \Delta ^{2,\mathrm{op}}$ by

$$\begin{equation*} \mathscr{F}=\{\text{simplicial embeddings}\nobreakspace \Sigma \to X\} \ni \phi \mapsto ([\phi (Q^+)],[\phi (Q^-)]) \in \Delta ^{2,\mathrm{op}}, \end{equation*}$$

where $[\phi (Q^+)]$ denotes the equivalence class of the section $\phi (Q^+)\subset X$ which is a point in $\Delta$, and similarly for $[\phi (Q^-)]$ which is an opposite point.

At the beginning of §5.2 we fixed the model marked wall tree $\Upsilon$ and a base point $o\in VT$. We now relate $\Sigma$ and $\Upsilon$. We denote by $\ell$ the unique wall through $0$ in $\Sigma$ such that $\ell \cap \overline{Q^+}=\{0\}$ and fix a simplicial embedding $\iota :\Sigma \hookrightarrow \Upsilon$ such that $\iota (\ell )$ is $\{o\}\times \mathbf{R}\subset \Upsilon \subset T\times \mathbf{R}$. Recall that $\mathscr{W}\subset \widetilde{\mathscr{W}}= \{\text{simplicial embeddings}\nobreakspace \Upsilon \to X\}$. We define the map

$$\begin{equation*} \mathscr{W}\to \mathscr{F}: \psi \mapsto \psi |_{\iota (\Sigma )}. \end{equation*}$$

Note that the composition $\mathscr{W}\to \mathscr{F}\to X^{(0)}$ agrees with the restriction to $\mathscr{W}$ of the map (1) given in §5.2. The maps $\mathscr{W}\to \mathscr{F}\to X^{(0)}$ are clearly $\Gamma$-equivariant ($\Gamma$ acts on the target space $X$) and induce corresponding maps $\mathscr{W}/\Gamma \to \mathscr{F}/\Gamma \to X^{(0)}/\Gamma$. The commutativity of the upper part of diagram (3) is obvious.

Note that the composed map $\mathscr{W}\to \mathscr{F}\to \Delta ^{2,\mathrm{op}}$ is $S$-invariant and thus factors through $\mathscr{W}/S$. This explains the map $\mathscr{W}/S\to \Delta ^{2,\mathrm{op}}$ appearing in diagram (3). We define the map $\Delta ^{2,\mathrm{op}}\to \Delta _\pm ^\mathrm{op}$ by associating with a pair of chambers $(C_1,C_2)\in \Delta ^{2,\mathrm{op}}$ the unique pair of elements $\delta _+\in \Delta _+$ and $\delta _-\in \Delta _-$ such that $\delta _+$ and $\delta _-$ are in the boundary of the unique flat that contains both $C_+$ and $C_-$ but disjoint from their closures. By composing $\mathscr{W}/S\to \Delta ^{2,\mathrm{op}}\to \Delta _\pm ^\mathrm{op}$ we also obtain the map $\mathscr{W}/S\to \Delta _\pm ^\mathrm{op}$. Note that the composed map $\mathscr{W}\to \mathscr{W}/S \to \Delta _\pm ^\mathrm{op}$ is the restriction to $\mathscr{W}$ of the map (2) defined in §5.2, namely

$$\begin{equation*} \phi \mapsto (\phi ^+, \phi ^-) = \left(\lim _{s\to +\infty } \phi (o,s),\lim _{s\to -\infty } \phi (o,s)\right) \in \Delta _\pm ^\mathrm{op}. \end{equation*}$$

Similarly, we can restrict to $\mathscr{W}$ the maps $\widetilde{\mathscr{W}}\to \overline{\Delta }_+, \overline{\Delta }_-$ appearing in Definition 5.8 and obtain maps $\mathscr{W}\to \overline{\Delta }_+, \overline{\Delta }_-$. The latter, being $S$-invariant, factor through $\mathscr{W}/S$, and we obtain the maps $\mathscr{W}/S\to \overline{\Delta }_+, \overline{\Delta }_-$ appearing in the diagram. The maps $\overline{\Delta }_+\to \Delta _+$ and $\overline{\Delta }_-\to \Delta _-$ are the obvious ones, and the maps $\operatorname {Ch}(\Delta ) \to \Delta _+$ and $\operatorname {Ch}(\Delta ) \to \Delta _-$ are the ones described at the beginning of §5.2. The commutativity of the diagram is clear from the definitions. Finally, the maps $p_1,p_2\colon \Delta ^{2,\mathrm{op}}\to \operatorname {Ch}(\Delta )$ are the first and second factor projections.

5.5. Equivariance properties of diagram (3)

Note that all the maps in diagram (3) are $\Gamma$-equivariant. The following lemmas describe $\mathscr{F}$ and $\Delta ^{2,\mathrm{op}}$ as spaces of orbits of certain group actions on $\mathscr{W}$.

The group $W$ contains a subgroup $W_0=\{1,\tau \}$, where $\tau$ is the longest element of $W$, which exchanges the positive and the negative Weyl chamber of $\Sigma$. In particular, the map $\mathscr{F}\to \Delta ^{2,\mathrm{op}}$ is $W_0$-equivariant, where $\tau$ acts on $\Delta ^{2,\mathrm{op}}$ by flipping the two coordinates.

Recall our fixed embedding $\iota :\Sigma \hookrightarrow \Upsilon$ from the previous section. Recall also that $M$ acts on $\mathscr{W}$ by precomposition via its action on $\Upsilon$.

Now let $M'$ be the stabilizer of $\iota (\Sigma )$ in $M$. Note that $M'$ is also the stabilizer of $\partial (\iota (\Sigma ))$, since a maximal flat is uniquely defined by its boundary. Let $M_0$ (resp., $M'_0$) be the pointwise fixator of $\iota (\Sigma ))$ (resp., $\partial (\iota (\Sigma ))$). By definition, the groups $M'/M_0$ and $M'/M'_0$ act on $\Sigma$ and $\partial \Sigma$, so that there is an embedding of the group $M'/M_0$ in $\operatorname {Aut}(\Sigma )=W\ltimes A$ and of $M'/M_0'$ in $\operatorname {Aut}(\partial \Sigma )=W$.

Proposition 5.23.

Consider the space $\mathscr{W}/M_0$ (resp., $\mathscr{W}/M_0'$), endowed with the quotient topology and the action of $M'/M_0$ (resp., $M'/M_0'$). Then:

(i)

The groups $M'$, $M_0$, and $M'_0$ are unimodular.

(ii)

The image of $M'/M_0$ (resp., $M'/M'_0$) in $W\ltimes A$ (resp., $W$) is equal to $W_0\ltimes A$ (resp., $W_0$).

(iii)

There is a $\Gamma \times M'/M_0$-equivariant homeomorphism $\mathscr{W}/M_0\to \mathscr{F}$.

(iv)

There is a $\Gamma \times M'/M'_0$-equivariant homeomorphism $\mathscr{W}/M'_0\to \Delta ^{2,\mathrm{op}}$.

Proof.

The image of $\Sigma$ under the projection $\Upsilon \subset T\times \mathbf{R}\to T$ is a geodesic $L$ in $T$. Note that $p_T(M')$ is exactly the stabilizer of $L$, whereas $p_T(M_0)$ (resp., $p_T(M_0')$) is the pointwise fixator of $L$ (resp., of $\partial L$).

Since $p_T(M')$ acts $3$-transitively on $\partial T$, it contains an element which flips the two endpoints of $L$. By our choice of $\iota$, this element acts as $\tau$ on $\Sigma$. Furthermore, the two endpoints corresponding to the $\mathbf{R}$ factor in $\Upsilon \subset T\times \mathbf{R}$ are fixed by $M'$, since $M'\subset \operatorname {Aut}^0(\Upsilon )$. This implies that the image of $M'/M'_0$ is exactly $W_0$. It also follows that the linear part of $M'/M_0$ is equal to $W_0$. Let $A'=M'/M_0\cap A$. To show that $A'=A$, we first note that $M'$ contains $S$, so that the stabilizer of $\ell$ in $A'$ acts transitively on the vertices of $\ell$. Again using the $3$-transitivity of $M$ on $\partial T$, we see that $M'$ contains an element which acts as a translation of length $1$ on $L$. Hence $A'$ acts transitively on the set of all lines in $\Sigma$ which are parallel to $\ell$. Therefore $A'$ is transitive on the vertices of $\Sigma$ so that $A'=A$, since $A$ acts sharply transitively on those vertices. Since $M'/M_0$ contains a reflection whose linear part is $\tau$, it follows that $M'/M_0=W_0\ltimes A$.

To prove that $M'$ is unimodular, it suffices to prove that $M'/S$ is unimodular, since $S$ is discrete. But $M'/S\simeq p_T(M')$ is the stabilizer of $L$ in $T$, which contains a translation of $L$. In particular, $p_T(M')$ contains a cocompact lattice, isomorphic to $\mathbf{Z}$. Therefore it is unimodular. Any open normal subgroup of a unimodular group is also unimodular. Thus $M_0$ and $M'_0$ are unimodular.

The map $\mathscr{W}\to \mathscr{F}$ defined (as above) by $\psi \mapsto \psi _{|\iota (\Sigma )}$ is clearly $M_0$-invariant and $M'$-equivariant, so it factors through a map $\theta \colon \mathscr{W}/M_0\to \mathscr{F}$, which is $M'/M_0$-equivariant, and continuous by definition of the quotient topology. The injectivity of $\theta$ is clear. Let us prove its surjectivity. Using Lemma 2.8, we see that for every $u,v\in \Delta ^{\mathrm{op}}_{\pm }$, $\mathscr{W}$ contains an element $\phi$ such that $(\phi _+,\phi _-)=(u,v)$. It follows that the image of the map $\theta$ contains every marked flat up to translation. Since $M'/M_0$ contains $A$, it is always possible to translate in order to get back to the origin so that the map $\mathscr{W}\to \mathscr{F}$ is indeed surjective. If $\mathscr{W}_x=\{\phi \in \mathscr{W}\mid \phi (o,0)=x\}$, then we see that $\mathscr{W}_x$ is a compact open subset of $\mathscr{W}$. By compactness, $\theta _{|\mathscr{W}_x/M'_0}$ is a homeomorphism on its image, which is $\mathscr{F}_x$, and hence is again open. It follows that $\theta$ is indeed a homeomorphism.

The homeomorphism between $\mathscr{W}/M_0'$ and $\Delta ^{2,\mathrm{op}}$ is obtained by composing the map $\theta$ above with the map $\mathscr{F}\to \Delta ^{2,\mathrm{op}}$. This new map clearly factors through $M_0'$. The proof that it is an equivariant homeomorphism is similar as before.

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Corollary 5.24.

There exists a closed subgroup $N<M^+$ and a continuous surjective homomorphism $N \to \{1,\tau \}$ such that the maps $\mathscr{W}\to \Delta ^{2,\mathrm{op}}$ and $\mathscr{W}/S \to \Delta ^{2,\mathrm{op}}$ are equivariant with respect to this homomorphism.

Proof.

As above, the image of $\Sigma$ under the projection $\Upsilon \subset T\times \mathbf{R}\to T$ is a geodesic line $L$ in $T$. We put $N=M'\cap M^+$. By Theorem 5.21, the group $p_T(M^+)$ is still $3$-transitive on $\partial T$, hence $p_T(N)$ contains an element which flips the two endpoints of $L$. This element acts as $\tau$ on $\mathscr{F}$, hence also on $\Delta ^{2,\mathrm{op}}$.

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6. Measuring the topology

In this section, we retain the setting and the notation of the previous one and explain how to construct natural measures on the various topological spaces considered there. In §6.1, we give the construction of visual measures on $\Delta$ and use it in §6.2 to construct a Radon measure on $\Delta ^{2,\mathrm{op}}$. In §6.3, we use the above to construct a Radon measure on $\Delta _\pm ^\mathrm{op}$. Using the measure on $\Delta _\pm ^\mathrm{op}$ we are able to define a measure on $\mathscr{W}$ and by this on the various spaces appearing in diagram (3). That process will be carried out in §6.4.

6.1. The visual measures on $\operatorname {Ch}(\Delta )$

Our goal is to define natural probability measures $\mu _x$ on $\operatorname {Ch}(\Delta )$, indexed by $x\in X$. We follow the construction of J. Parkinson from 53.

Recall that $A$ is the group of simplicial translations of our model apartment $\Sigma$ and that $Q^+\subset \Sigma$ is the positive Weyl chamber. We denote by $A^+$ the subsemigroup of $A$ formed by elements translating $0$ to a point in $Q^+$ (including possibly points on the walls). This gives a partial order on $A$, defined by $t>t'$ if $t-t'\in A^+$.

Let $\alpha _1$ and $\alpha _2$ be the simple roots associated with the choice of the positive Weyl chamber. In other words, they are the $\mathbf{Z}$-linear forms on $\Sigma$ taking integer values on vertices, whose kernel are walls delimiting $Q^+$ which are positive on $Q^+$ and achieve the value $1$. We also denote by $A^{++}$ the set of $\lambda \in A$ such that $\alpha _1(t)>0$ and $\alpha _2(t)>0$.

Let $t_1$ and $t_2$ be the two generators of $A$ such that every $t\in A$ is written $t=\alpha _1(t)t_1+\alpha _2(t)t_2$. We denote $\ell (t)=\alpha _1(t)+\alpha _2(t)$.

For $t\in A^+$ and $x$ a vertex in $X$, we denote by $V_t(x)$ the set of vertices $y\in X$ such that, in an apartment $F\subset X$ containing $x$ and $y$, we have $t(x)=y$.

There is a natural action of the Weyl group $W$, isomorphic to the symmetric group $S_3$, on $\Sigma$. The group $W$ is generated by the two reflections $s_1$ and $s_2$, which act on $\Sigma$ by reflections with respect to the walls corresponding to $\alpha _1$ and $\alpha _2$, respectively.

Lemma 6.1.

Let $t\in A^{++}$ and $x$ be a vertex in $X$. Then the cardinality $N_t$ of $V_{t}(x)$ does not depend on $x$ and we have

$$\begin{equation*} N_t= K q^{2\ell (t)}, \end{equation*}$$

for some constant $K$ depending on $q$.

Proof.

See 28, Corollary 2.2 or 52, Theorem 5.15.

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We will need a more precise calculation. Fix a chamber $C\in \operatorname {Ch}(\Delta )$ and a vertex $x\in X$ contained in some apartment. Consider the collection $F$ of all apartments in $X$ containing $x$ and $C$. Each element of $F$ could be seen as a marked flat, by choosing its unique type preserving identification with $\Sigma$ taking $0$ to $x$ and $Q^+$ to $Q(x,C)$. For given $t\in A^+$ and $w\in W$ we consider in each apartment in $F$ the image of the element $w\circ t(0)\in \Sigma$ under the associated identification. We denote by $Y_w^t$ the set of all vertices of $X$ obtained this way.

Lemma 6.2.

Denoting $(i,j)=(\alpha _1(t),\alpha _2(t))$, we have

$$\begin{equation*} |Y_{e}^t|=1, \quad |Y_{s_1}^t|= K_1q^{j}, \quad |Y_{s_2}^t|=K_2q^{i}, \quad |Y_{s_1s_2}^t|=K_3q^{2i+j}, \quad |Y_{s_2s_1}^t|=K_4 q^{i+2j}, \end{equation*}$$

for some constants $K_1, \dots , K_4$ depending only on $q$. In particular, for every $w\neq w_0$, the quantity $|Y_w^t|/|V_t(y)|$ tends to $0$ as $i$ and $j$ tend (simultaneously) to $+\infty$.

Proof.

We argue by induction on $\ell (t)=i+j$. The first step of the induction is absorbed by the constants. Let us do the calculation of $|Y_{s_1}^t|$, the other ones being similar. We refer to Figure 3.

Fix $t=it_1+jt_2$, and let $t'=(i+1)t_1+jt_2$. Let $Y=Y_{s_1}^{t}$ and $Y'=Y_{s_1}^{t'}$. We consider the following map $\psi$: to a point $z'$ in $V_{t'}(x)$, we associate the unique point $z=\psi (z')$ in the intersection of $V_t(x)$ and the convex hull of $x$ and $z$. The map $\psi$ is surjective: indeed, if $z\in V_t(x)$, then $x$ and $z$ are in some common apartment, and it is possible to find $z'$ in this apartment such that $\psi (z')=z$. We also claim that the restriction of this map to $Y'$ is also injective. Indeed, if $y_1\in Y$, then any apartment containing $y_1,x$, and $C$ also contains the convex hull of $y_1$ and $C$, and therefore already contains a point $y'_1\in Y'$. On the other hand, any point $y'\in Y'$ such that $\psi (y')=y_1$ is contained in some apartment containing $y_1,x$, and $C$, and therefore must be equal to $y'_1$. This proves the claim. Hence we have $|Y'|=|Y|=K_1q^{2j}$.

Now let $t''=it_1+(j+1)t_2$, and denote $Y''=Y_{s_1}^{t''}$. We still have a surjective map $Y''\to Y$, defined in the same way as the map $Y'\to Y$. However, this map is no longer injective. To count the cardinality of the fibers, choose $y_1\in Y$. This gives a choice of $y'_1\in Y'$, defined as $y'_1=\psi ^{-1}(y_1)$. There are $q$ chambers which have $y_1$ and $y'_1$ as their vertices and are not in the convex hull of $C$ and $y_1$. The third vertex of each of these chambers is a possible choice for $y''_1\in Y''$. Hence $|Y''|=q|Y|=K_1q^{j+1}$.

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Definition 6.3.

Let $x$ be a vertex in $X$. The visual measure based at $x\in X$, denoted $\mu _x$, is the unique measure on $\operatorname {Ch}(\Delta )$ such that for all $y\in V_t(x)$, the set $\Omega _{x}(y)$ has measure $\frac{1}{N_t}$.

The construction of this measure is justified more carefully in 53; it follows from the fact that $\operatorname {Ch}(\Delta )$ can be viewed as the projective limit of sets $V_t(x)$ as $t$ grows in $A^+$. We note also that $g.\mu _x=\mu _{gx}$.

Definition 6.4.

Let $C\in \operatorname {Ch}(\Delta )$. The horofunction based at $C$, denoted $h_C$, is a function $X\times X$ defined as follows. Let $z\in Q(x,C)\cap Q(y,C)$. Assume that $z\in V_s(x)\cap V_{t}(y)$. Then $h_C(x,y) =s-t$.

This definition does not depend on the particular choice of $z$ (see 53, Theorem 3.4). The horofunction $h_C(\cdot ,\cdot )$ satisfies the following cocycle relation 53, Proposition 3.5.

Lemma 6.5.

For all $x,y,z\in X$ and $C\in \operatorname {Ch}(\Delta )$, we have $h_C(x,y)=h_C(x,z)+h_C(z,y)$.

Proposition 6.6.

For $x,y$ vertices in $X$, the measures $\mu _x$ and $\mu _y$ are absolutely continuous relative to each other. Furthermore, for all $C\in \operatorname {Ch}(\Delta )$, we have

$$\begin{equation*} \frac{\mathrm{d}\mu _x}{\mathrm{d}\mu _y}(C)=q^{2\ell ( h_C(x,y))}. \end{equation*}$$

Proof.

See 53, Theorem 3.17.

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Definition 6.7.

We denote by $\mu$ the measure class of $\mu _x$.

Proposition 6.8.

The set $\Delta ^{2,\mathrm{op}}$ is of conull measure (relative to the measure class $\mu \otimes \mu$).

Proof.

For every vertex $x\in X$, let $\Delta ^2_x$ (resp., $\Delta ^{2,\mathrm{op}}_x$) be the subset of $\operatorname {Ch}(\Delta )\times \operatorname {Ch}(\Delta )$ composed of pairs of chambers which are in the boundary of an apartment containing $x$ (resp., and are opposite).

Then $\operatorname {Ch}(\Delta )\times \operatorname {Ch}(\Delta )=\bigcup _{x\in X} \Delta ^2_x$. By countability it is sufficient to prove that for every $x$ we have $\mu _o^2(\Delta ^{2,\mathrm{op}}_x)=\mu _o^2(\Delta ^2_x)$, for a fixed vertex $o\in X$. Since $\mu _x$ and $\mu _o$ are in the same measure class, it is sufficient to prove that $\mu _x^2(\Delta ^{2,\mathrm{op}}_x)=\mu _x^2(\Delta ^2_x)$ for every vertex $x$.

By Fubini, it is enough to prove that for almost every chamber at infinity $C$, the set $\mathcal{B}$ of chambers $C'$ such that $(C,C')\in \Delta ^{2,\mathrm{op}}_x$ is of full measure (in the set of chambers in the same apartment as $C$ and $x$). Let $\mu _{x,C}$ be the restriction of the measure $\mu _x$ to the set $\mathcal{B}'$ of those chambers $C' \in \operatorname {Ch}(\Delta )$ with $(C,C')\in \Delta ^2_x$. We have to prove that $\mu _{x,C}(\mathcal{B})=\mu _{x,C}(\mathcal{B}')$.

Recall the sets $Y_w^t$ discussed in Lemma 6.2 and set $\Omega _x(Y^t_w)=\bigcup _{y\in Y_w^t}\Omega _x(y)$. It is clear that $\mathcal{B}\subseteq \bigcap _{t\in A^+} \Omega _x(Y^t_{w_0})$. We claim that this inclusion is actually an equality. Indeed, if $C'\in \bigcap _{t\in A^+} \Omega _x(Y^t_{w_0})$, then every finite subset of $Q(x,C')\cup Q(x,C)$ is contained in some apartment. Hence by Theorem 2.3 $Q(x,C')\cup Q(x,C)$ is also contained in an apartment, and then it is clear that $C$ and $C'$ are opposite in it.

Note that the intersection $\bigcap _{t\in A^+} \Omega _x(Y^t_{w_0})$ is decreasing in the sense that for every $t,t'\in A^+$ such that $t'-t\in A^+$, we have $\Omega _x(Y^{t'}_{w_0})\subseteq \Omega _x(Y^t_{w_0})$. Hence it is enough to prove that $\mu _{x,C}(\Omega _x(Y^t_{w_0}))$ tends to $\mu _{x,C}(\mathcal{B}')$ when $t\to \infty$. Equivalently, we need to show that for every $w\neq w_0$, the limit of $\mu _{x,C}(\Omega _x(Y^t_{w}))$ is $0$ as $t$ tends to infinity (simultaneously in both directions). This follows from Lemma 6.2, as $\mu _x(\Omega _x(Y^t_w))=|Y_w^t|/|V_t(y)|$.

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6.2. The Radon measure on $\Delta ^{2,\mathrm{op}}$

In this section we construct a natural Radon measure on $\Delta ^{2,\mathrm{op}}$, which is invariant by the actions of $\Gamma$. The existence of such a measure is well known to experts (see for example 51 for a possible definition); its construction is nevertheless usually different than what we propose here.

Lemma 6.9.

For $(C,C')\in \Delta ^{2,\mathrm{op}}$ and $x$ a vertex in $X$, the quantity

$$\begin{equation*} \beta _x(C,C'):=h_C(x,z)+h_{C'}(x,z), \end{equation*}$$

with $z$ a point in the apartment containing $C$ and $C'$, does not depend on $z$.

Furthermore, for every vertex $y\in X$, we have

$$\begin{equation*} \beta _x(C,C')-\beta _y(C,C')=h_C(x,y)-h_{C'}(x,y). \end{equation*}$$

Proof.

Let us prove first that $\beta _x(C,C')$ is independent of $z$. Let $z$ and $z'$ be two points in the apartment $F$ containing $C$ and $C'$. Using the assumption that $C$ and $C'$ are opposite, we have that $h_C(z',z)=-h_{C'}(z',z)$.

Hence by Lemma 6.5 we have

$$\begin{align*} \beta _x(C,C')&=h_C(x,z)+h_{C'}(x,z)\\ &= h_C(x,z')+h_C(z',z)+h_{C'}(x,z')+h_{C'}(z',z)\\ &=h_C(x,z')+h_{C'}(x,z'). \end{align*}$$

Now let us turn to the second equality. Fix $x,y\in X$. Then, using Lemma 6.5 again, we have

$$\begin{align*} \beta _x(C,C')-\beta _y(C,C')&=h_C(x,z)+h_{C'}(x,z)-h_C(y,z)-h_{C'}(y,z)\\ &=h_C(x,z)+h_{C'}(x,z)+h_C(z,y)+h_{C'}(z,y)\\ &=h_C(x,y)+h_{C'}(z,y). \end{align*}$$■

For a fixed vertex $x$ in $X$ we define the measure $m_x$ on $\operatorname {Ch}(\Delta )\times \operatorname {Ch}(\Delta )$ by the formula

$$\begin{equation*} \mathrm{d}m_x(C,C')=q^{-2\ell (\beta _x(C,C'))}\mathrm{d}\mu _{x}(C) \mathrm{d}\mu _x(C'). \end{equation*}$$

By Lemma 6.9 and Proposition 6.6, and also using the fact that $\ell (t+t')=\ell (t)+\ell (t')$ for every $t,t'\in A$, we have

$$\begin{equation*} \frac{\mathrm{d}m_x}{\mathrm{d}m_y}(C,C')=q^{-2\ell ( \beta _x(C,C')) +2\ell ( \beta _y(C,C')) }\frac{\mathrm{d}\mu _x}{\mathrm{d}\mu _y}(C)\frac{\mathrm{d}\mu _x}{\mathrm{d}\mu _y}(C')=1. \end{equation*}$$

Thus the measure $m_x$ is independent on the choice of $x$. We get that this measure is $\Gamma$-invariant, as for all $g\in \Gamma$ we have $g\mu _x=\mu _{gx}$ and $\beta _{gx}(gC,gC')=\beta _x(C,C')$; hence $gm_x=m_{gx}=m_x$.

Using Proposition 6.8 we view $m_x$ as a measure on $\Delta ^{2,\mathrm{op}}$ which, in view of its independence on $x$, we denote $m$. Thus $m$ is a $\Gamma$-invariant measure on $\Delta ^{2,\mathrm{op}}$.

Recall that for a vertex $x\in X$ we define $\mathscr{F}_x$ to be the space simplicial embeddings $f \colon \Sigma \to X$ satisfying $f(0)=x$ and let $\mathscr{F}'_x$ be the image of $\mathscr{F}_x$ under the projection $\mathscr{F}\to \Delta ^{2,\mathrm{op}}$. Note that $\mathscr{F}\to \Delta ^{2,\mathrm{op}}$ is injective in restriction to $\mathscr{F}_x$, thus induces a homeomorphism $\mathscr{F}_x\to \mathscr{F}'_x$ by Lemma 5.4.

Lemma 6.10.

We have $m(\mathscr{F}'_x)<\infty$.

Proof.

By definition, $m(\mathscr{F}'_x)=\int _{(C,C')\in {\mathscr{F}'_x}} q^{2\ell (\beta _x(C,C'))}\mathrm{d}\mu _x(C)\mathrm{d}\mu _x(C')$.

By Lemma 6.9, if $(C,C')\in \mathscr{F}'_x$, we have $\beta _x(C,C')=h_C(x,x)-h_{C'}(x,x)=0$. Hence $m(\mathscr{F}'_x)=(\mu _x\otimes \mu _x)(\mathscr{F}'_x)<\infty$.

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Corollary 6.11.

The measure $m$ is a $\Gamma$-invariant Radon measure on the locally compact space $\Delta ^{2,\mathrm{op}}$ which is in the measure class of $\mu \times \mu$.

Proof.

The only thing that does not immediately follow from the discussion above is the fact that $m$ is finite on compact sets. This follows from Lemma 6.10, as $\{\mathscr{F}'_x\mid x\in X^{(0)}\}$ is an open cover of $\Delta ^{2,\mathrm{op}}$.

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6.3. The Radon measure on $\Delta _\pm ^\mathrm{op}$

The first part of this section is devoted to the construction of a Radon measure $m_\pm$ on $\Delta _\pm ^\mathrm{op}$; see Definition 6.14 and Lemma 6.15 below. The construction is similar to the construction of $m$ given in the previous section. The second part of this section is a preparation for the proof that the measures $m$ and $m_\pm$ are compatible which will be given in the next section; see Theorem 6.21.

For $u\in \Delta _+$ (resp., $v\in \Delta _-$), we define the horofunction $h_u$ (resp., $h_v$) by $h_u(x,y)=\alpha _1(h_C(x,y))$, for any chamber $C$ adjacent to $u$ (resp.,