# Hyperbolic manifolds and pseudo-arithmeticity

In memoriam E. B. Vinberg

## Abstract

We introduce and motivate a notion of pseudo-arithmeticity, which possibly applies to all lattices in with We further show that under an additional assumption (satisfied in all known cases), the covolumes of these lattices correspond to rational linear combinations of special values of . -functions.

## 1. Introduction

The study of hyperbolic of finite volume has many relations with number theory, with a central role in this context being played by the notion of the -manifolds*invariant trace field* (see Reference 17). The work of Vinberg Reference 31 allows to define a comparable invariant (the *adjoint trace field*) for any locally symmetric space. This paper studies finite volume hyperbolic manifolds of dimensions from the perspective of the adjoint trace field, and the algebraic groups that are naturally associated with them (see Theorem 1.2).

Our main purpose is to introduce and motivate a notion of *pseudo-arithmeticity*, which to the best of our knowledge applies to all *currently known* lattices in with In this introduction we state two main results: Theorem .1.5, where we show that the classical hyperbolic manifolds obtained by gluing are pseudo-arithmetic; and Theorem 1.12, in which it is proved that under some mild assumption (defined in Sect. 1.5) the covolumes of pseudo-arithmetic lattices correspond to rational linear combinations of covolumes of arithmetic lattices.

### 1.1. Hyperbolic isometries and algebraic groups

We denote by the hyperbolic and we identify its group of isometries with the Lie group -space, Let . denote the real algebraic group such that and , its identity component. Note the following dichotomy:

- •
for even is connected, so that ;

- •
for odd has two connected components, and we have .

Recall that any complete hyperbolic -manifold can be written as a quotient where , is a torsion-free discrete subgroup (uniquely determined up to conjugacy). The manifold has finite volume exactly when is a lattice in We will usually assume that . is orientable, i.e., Then by Borel’s density theorem . is Zariski-dense in .

Let be a field extension; for an algebraic -group the symbol denotes the obtained by scalar extension (base change induced by -group In this case ). is said to be a * -form* of Assume that . is a number field, with ring of integers Then a . -form of (or of is called )*admissible* (for if the Lie group ) contains exactly one noncompact factor, isomorphic to In this case . is a lattice in .

### 1.2. Trace fields and ambient groups

For any subgroup with we define its ,*(adjoint) trace field* as the subfield of given by

where is the adjoint representation. In case is a lattice, it follows from Weil’s local rigidity that is a number field (see Reference 23, Prop. 1.6.5). The work of Vinberg Reference 31 shows the following.

Since is Zariski-dense, the group is uniquely determined by (up to and it is a commensurability invariant. We call it the -isomorphism),*ambient group* of If . is admissible then is called *quasi-arithmetic*. If moreover is commensurable with it is called ,*arithmetic*. We use the same terminology for the corresponding quotient .

### 1.3. The case of glued manifolds

To date for there are two sources of nonarithmetic lattices in :

- (1)
Hyperbolic reflection groups,Footnote

^{1}some of which can be proved to be nonarithmetic by using Vinberg’s criterion Reference 30.^{1}Note that hyperbolic reflections groups (of finite covolume) cannot exist for and no examples are known for , (see Reference 1, Sect. 1).

- (2)
Hyperbolic manifolds constructed by gluing together “pieces” of arithmetic manifolds along pairwise isometric totally geodesic hypersurfaces.

There are several constructions that fit the description in (2) (see Reference 2Reference 9Reference 27), all of which can be thought as (clever) variations of the original methodFootnote^{2} by Gromov and Piateski-Shapiro Reference 10. Their building blocks are always *arithmetic pieces* i.e., hyperbolic , with totally geodesic boundary taken inside arithmetic manifolds -manifolds (see Sect. 2.4–2.5). We will see in Corollary 2.6 that two such pieces and cannot be glued together unless and have the same field of definition (equivalently, the same trace field).

^{2}

Note that this method has also inspired Vinberg’s recent paper Reference 32, where “hybrid” non-arithmetic reflection groups are constructed. This shows in particular that the two sources (1) and (2) are not disjoint.

The following theorem applies to all nonarithmetic lattices constructed in the sense of (2). By a *multiquadratic* extension we mean a (possibly trivial) field extension of the form ( , ).

The exact description of the trace field – and in particular the degree of – depends on the pieces and the way they are glued together. A precise treatment is the subject of a separate article by the second author Reference 20. See also Reference 21.

### 1.4. Pseudo-arithmetic lattices

Theorem 1.5 motivates the following definitions:

It follows from the definition that (quasi-)arithmetic lattices are pseudo-arithmetic. Theorem 1.5 shows that lattices obtained by gluing arithmetic pieces are pseudo-arithmetic.

In Sect. 4 we will present a method to test if a given reflection group is pseudo-arithmetic or not; this can be thought as an extension of Vinberg’s (quasi-)arithmeticity criterion Reference 30. We have applied our method on the full list of groups provided with the software CoxIter (see Reference 11), which contains about a hundred non-quasi-arithmetic Coxeter groups for all of them turn out to be pseudo-arithmetic. A particularly interesting example is the Coxeter group : presented in Figure 1; it has recently been shown (combining work of Fisher et. al. Reference 8, Sect. 6.2 and of the second author Reference 20) that is *not* commensurable with any lattice obtained by gluing arithmetic pieces. At this point it is natural to ask:

Note that by definition the trace field of a pseudo-arithmetic lattice is totally real. The following question – thus weaker than Question 1.8 – seems easier to answer, yet we do not know its current status.

### 1.5. First type lattices

A lattice is said to be ,*of the first type* if its ambient group is of the form (resp., where ), denotes the projective orthogonal group (resp., its identity component) of a quadratic form defined over the trace field see Sect. ;2.1 for details. This extends the terminology sometimes used for arithmetic lattices. For even, any of -form is of the type and thus for those dimensions all lattices are trivially of the first type. It follows from part (2) of Theorem ,1.5 that lattices obtained by gluing arithmetic pieces are of the first type, since such pieces contain totally geodesic hypersurfaces (see Prop. 2.2). Moreover, every Coxeter group is of the first type (the corresponding quadratic form can be read off the Gram matrix, see Reference 31, §4). To summarize, all *currently known* lattices in with are either:

- •
arithmetic, or

- •
pseudo-arithmetic of the first type.

A lattice simultaneously belongs to both categories if and only if it is arithmetic of the first type (sometimes also called “standard” arithmetic lattices).

In the rest of the paper we will often make implicit use of the following.

### 1.6. Volumes

Let (or be a pseudo-admissible group over ) Then we may associate with . a set of arithmetic lattices in the following way. Let us assume that is explicitly given by i.e., the degree of , is We will assume (as we may) that the quadratic form . is diagonal in the variables with negative coefficient in , For any multi-index . we set and , All the . are totally positive since is totally real by assumption. It follows that all are admissible quadratic forms over For each . we choose an arithmetic subgroup i.e., a subgroup commensurable with , We say that the set of arithmetic lattices . is *subordinated* to .

We will show that – in some precise sense – the homology over of the group is generated by classes associated with the see Theorem ;3.5. A direct consequence for the volume is the following.

For even the result readily follows from the Gauss-Bonnet theorem, and Theorem 1.12 has no interest there (but Theorem 3.5 presumably has). For manifolds obtained by gluing the result was already known; see Reference 7, Sect. 1.4. Note also that for quasi-arithmetic lattices (which corresponds to the case above) the result is proved in Reference 7, Theorem 1.3, and the condition “of the first type” is superfluous.

The covolume of arithmetic lattices is well-understood: it is essentially expressible by means of special values of (see -functionsReference 24Reference 25). In particular the values of the summands can be determined up to rationals. The following example illustrates Theorem 1.12 with the case of the Coxeter group (which was introduced in Sect. 1.4).

Proving a sharp equality in Equation 1.4 seems out of reach with the methods presented in this article. In particular, the method used to prove Theorem 1.12 does not provide any information about the (or even about their signs).

### 1.7. Final remarks

#### 1.7.1

The definition of pseudo-arithmeticity can be transferred verbatim for lattices in ( Nonarithmetic lattices in ). are known to exist for and none of the known examples is quasi-arithmetic. In , there exist nonarithmetic lattices whose trace fields are not quadratic extensions (see Reference 4, Table A.2); in particular they cannot be pseudo-arithmetic.

#### 1.7.2

In Reference 10, Question 0.4 Gromov and Piateski-Shapiro famously asked whether for any lattice of sufficiently large dimension the quotient admits a nice partition into “subarithmetic pieces”. In some weak sense (i.e., at the level of homology), Theorem 3.5 gives a positive answer to this question for the class of pseudo-arithmetic lattices of the first type. It is not clear however if a statement closer to their original formulation can be achieved for those lattices; in particular, if a positive answer to the following question could hold:

## 2. Pseudo-arithmeticity of gluings

The goal of this section is to prove Theorem 1.5.

### 2.1. Rational hyperplanes

Let be an admissible quadratic space over a number field i.e., , is an admissible quadratic form. We denote by or simply , the algebraic group of orthogonal transformations of , We denote by . its adjoint form (it corresponds to the group in the notation of Reference 13). For a field extension the group of , -points can be concretely described as the quotient where , is the group of similitudes of (see Reference 13, Sect. 12.A and Sect. 23.B). In particular, acts on the projective space -rationally and so does its identity component , .

The following is a model for the hyperbolic -space with group of isometries , :

Any hyperplane in corresponds to the image of a subspace with , for some such that Such a hyperplane will be denoted by . We shall say that . is if -rational is, i.e., for some This is equivalent to saying that the projective space . is in -closed A useful characterization is the following. .

### 2.2. Sharp hypersurfaces

Gluings are realized along totally geodesic embedded hypersurfaces of finite volume. To simplify the following discussion, we will call *sharp* such a hypersurface embedded in a hyperbolic manifold of finite volume. Note that if is compact then any totally geodesic hypersurface embedded in is sharp.

### 2.3. Extending similitudes

For a similitude between quadratic spaces, we denote by the map .

### 2.4. Hyperbolic pieces

By a *hyperbolic piece* (of dimension we mean a complete orientable hyperbolic ) of finite volume with a (possibly empty) boundary consisting of finitely many sharp hypersurfaces. We say that a piece is -manifold*singular* if it has nonempty boundary, and *regular* otherwise (in which case it is just a hyperbolic manifold in the sense of Sect. 1.1). A singular piece can always be embedded in a regular one of the same dimension: it suffices to consider the “double” of obtained by gluing together two copies of along each boundary component ( is complete according to Reference 10, 2.10.B).

Let be a singular hyperbolic piece. Its universal cover is isometric to an infinite intersection of half-spaces in see ;Reference 18, Sect. 3.5.1. The fundamental group of identifies with a discrete subgroup stabilizing and such that It is clear that any two choices of universal covers for . in are conjugate by an isometry, and thus, up to conjugacy, the discrete subgroup is uniquely determined by The .*trace field* of is defined as the trace field of (see Sect. 1.2). Furthermore, by Reference 10, 1.7.B we have that is Zariski-dense in and using Theorem ,1.2 there is therefore an intrinsic notion of ambient group for .

### 2.5. Arithmetic pieces

We define an *arithmetic piece* (of dimension as a singular hyperbolic piece of dimension ) that embeds into an arithmetic hyperbolic manifold Thus . where , and is a subgroup of infinite index in (see Reference 10, 2.10.A).