Fields definable in the free group

Abstract

We prove that no infinite field is definable in the theory of the free group.

1. Introduction

After the work of Sela 18 and Kharlampovich-Myasnikov 6 leading to the positive answer to Tarski’s question, there is an increasing model theoretic interest in the first-order theory of nonabelian free groups.

Although Sela proved that any definable set is equivalent to a boolean combination of $\forall \exists$-definable sets we are far from understanding these “basic” sets. According to Sela these sets admit a natural geometric interpretation, but admittedly neither geometers nor logicians have absorbed the sophisticated techniques that occur in his voluminous work. Thus, in principle, it is hard to determine whether a subset of some cartesian power of a nonabelian free group is definable or not.

Moreover, starting from Zilber’s seminal work towards understanding uncountably categorical theories via some naturally defined pregeometries (see 23), questions about what kind of groups are definable or whether an infinite field is definable in a given first order theory have become important within the community of model theorists.

Recently, some positive results along this line of thought have appeared. The following theorem has been proved independently in 7 and 9.

Theorem 1.1.

The only definable proper subgroups of a torsion-free hyperbolic group are cyclic.

When it comes to infinite definable fields in some nonabelian free group, intuitively speaking, one expects to find none. To the best of our knowledge this was first posed as a conjecture in 13. This problem proved very hard to tackle, and the only positive result towards its solution had been the following theorem proved in the thesis 20 of the second-named author.

Theorem 1.2.

Let $\mathbb{F}_n$ be the free group of rank $n$. Let $\phi$ be a formula over $\mathbb{F}_n$. Suppose $\phi (\mathbb{F}_n)\neq \phi (\mathbb{F}_{\omega })$, where $\phi (\mathbb{F})$ denotes the solution set of $\phi$ in $\mathbb{F}$. Then $\phi$ cannot be given definably an abelian group structure.

On the other hand, in many model theoretic questions concerning existing “configurations” in a first order theory one does not need to understand the exact set of solutions of a formula but just its rough “shape”. Indeed in this vein there has been progress. A definable set (or a parametric family) can be endowed with an envelope that contains the definable set, and moreover it carries a natural geometric structure from which we can read properties that when they hold “generically” for the envelope, they also hold for the definable set. The method of envelopes has been proved very useful in proving (weak) elimination of imaginaries (see 14) but also in proving that the first order theory of the free group does not have the finite cover property (see 21). We will utilize envelopes once more in order to confirm the above conjecture.

The main theorem of this paper is:

Theorem 1.

Let $\mathbb{F}$ be a nonabelian free group. Then no infinite field is definable in $\mathbb{F}$.

Our proof is based on the following theorem, which is a consequence of the quantifier elimination procedure. We record it next in the simplest possible form.

Theorem 1.3 (Sela).

Let $\mathbb{F}\coloneq \mathbb{F}(\bar{a})$. Let $\mathcal{T}(G,\mathbb{F})$ be a hyperbolic tower where $G\coloneq \langle \bar{u},\bar{x},\bar{a}\ | \$ $\Sigma (\bar{u},\bar{x},\bar{a})\rangle$ and let $\phi (\bar{x},\bar{a})$ be a first order formula over $\mathbb{F}$. Suppose there exists a test sequence, $(h_n)_{n<\omega }:G\rightarrow \mathbb{F}$ for $\mathcal{T}(G,\mathbb{F})$ such that $\mathbb{F}\models \phi (h_n(\bar{x}),\bar{a})$.

Then for any test sequence, $(h'_n)_{n<\omega }:G\rightarrow \mathbb{F}$, for $\mathcal{T}(G,\mathbb{F})$ there is $n_0$ such that $\mathbb{F}\models \phi (h'_n(\bar{x}),\bar{a})$ for all $n>n_0$.

The notions of a hyperbolic tower and of a test sequence over it will be defined in sections 4 and 6, respectively.

On our way to proving the main theorem, we prove various implicit function theorems. We recall that Merzlyakov’s original theorem stated:

Theorem 1.4 (Merzlyakov).

Let $\mathbb{F}\coloneq \mathbb{F}(\bar{a})$. Let $\Sigma (\bar{x},\bar{y},\bar{a})\subset _{\text{finite}}\langle \bar{x},\bar{y}\rangle *\mathbb{F}$ be a finite set of words. Suppose $\mathbb{F}\models \forall \bar{x}\exists \bar{y}(\Sigma (\bar{x},\bar{y},\bar{a})=1)$. Then there exists a “formal solution” $\bar{w}(\bar{x},\bar{a})\subseteq \langle \bar{x}\rangle *\mathbb{F}$ such that $\Sigma (\bar{x},\bar{w}(\bar{x},\bar{a}),\bar{a})$ is trivial in $\langle \bar{x}\rangle *\mathbb{F}$.

We use the quantifier $\exists ^{<\infty }$ to denote a nonempty and finite set of solutions of a formula $\mathbb{F}\models \exists ^{<\infty } x \phi (x)$ if and only if $0<|\{a\in \mathbb{F}\ | \ \mathbb{F}\models \phi (a)\}|<\infty$.

We obtain the following “generalisation” to an arbitrary formula, but only after strengthening the hypothesis and weakening the conclusion of Merzlyakov’s theorem as follows.

Theorem 2.

Let $\mathbb{F}\coloneq \mathbb{F}(\bar{a})$. Let $\mathbb{F}\models \forall \bar{x}\exists ^{<\infty }\bar{y}\phi (\bar{x},\bar{y},\bar{a})$ and assume there exist a test sequence $(\bar{b}_n)_{n<\omega }$ and a sequence of tuples $(\bar{c}_n)_{n<\omega }$ such that $\mathbb{F}\models \phi (\bar{b}_n,\bar{c}_n,\bar{a})$. Then there exists a tuple of words $\bar{w}(\bar{x},\bar{a})$ in $\langle \bar{x}\rangle *\mathbb{F}$ such that for any test sequence $(\bar{b}'_n)_{n<\omega }$ in $\mathbb{F}$ we have that there exists $n_0$ (that depends on the test sequence) with $\mathbb{F}\models \phi (\bar{b}'_n,\bar{w}(\bar{b}'_n,\bar{a}),\bar{a})$ for all $n>n_0$.

We remark that our main theorem implies, together with the elimination of the “exists infinitely many” quantifier $\exists ^{\infty }$, that no infinite field is definable in any model of this theory.

The proof splits into two parts. Roughly speaking for any definable set $X$ we prove that either $X$ is internal to a finite set of centralizers or it cannot be given definably an abelian group operation; i.e., there is no definable set $Y\subset X\times X\times X$ such that $Y$ is the graph of an abelian group operation on $X$. To conclude that no infinite field is definable we prove that centralizers of nontrivial elements are one-based.

The paper is structured as follows: in the next section we take the opportunity to recall some basic geometric stability theory and introduce the reader to the results that will allow us to conclude our theorem in the case a definable set is “coordinated” by a finite set of centralizers.

The following section contains introductory material that concerns Bass-Serre theory as well as results for a special class of groups called limit groups. The material here is by no means original and is certainly well known. Since in many of our arguments we will use actions on trees or normal forms for groups that admit a graph of groups splitting, we hope that this section will provide an adequate background for the uninitiated.

Section 4 contains many of the core notions which are important in this paper. We start by explaining when a group admits the structure of a tower and then we continue by introducing a construction that leads to the notion of a twin tower. Twin towers will play a fundamental role in our main proof.

In sections 5 and 6 we record and extend some constructions and results of Sela that appear in 16, 17, and 14. Here the reader will find all the technical apparatus that makes our main proof possible. Theorems 6.25, 6.32, and 6.34 lie in the core of our result.

Finally, in the last section we bring everything together and we prove the main result. We split the proof into two cases: the abelian case and the nonabelian case. The abelian case is resolved using geometric stability, while the nonabelian case is resolved using geometric group theory. We have also added an example, which we call the hyperbolic case, where our proof is free of certain technical phenomena, so the reader could clearly see the idea behind it.

2. Some geometric stability

In this section we provide some quick model theoretic background on stable theories. A gentle introduction to stability and forking independence has been given in 10, so to avoid repetition we refer the reader there. Our main focus in this paper will be on geometric stability and in particular on the notion of one-basedness. For more details the reader can consult 11. We work in the monster model $\mathbb{M}$ of a stable theory $T$.

Definition 2.1.

A definable set $X$ (in $\mathbb{M}$) is called weakly normal if for every $a\in X$ only finitely many translates of $X$ under $\operatorname {Aut}(\mathbb{M})$ contain $a$.

Definition 2.2.

The first order theory $T$ is one-based if every definable set (in $\mathbb{M})$ is a boolean combination of weakly normal definable sets.

The simplest example of a one-based theory is the theory of a vector space $(V,+,0, \{r_k\}_{k\in K})$ over a field $K$, where for each $k\in K$, $r_k$ is a function symbol which is interpreted in the structure as scalar multiplication by the element $k\in K$. In the same vein we have:

Fact 2.3.

The theory of any abelian group (in the group language) is one-based.

For the purposes of our paper one-based theories have an important property proved by Pillay in 12.

Fact 2.4 (Pillay).

Let $T$ be one-based. Then no infinite field is interpretable in $T$.

A set is interpretable if it is definable up to a definable equivalence relation. Thus, in particular no infinite field is definable in a one-based theory.

For any definable set $X$ in $\mathbb{M}$ we can define the induced structure on $X$, $X^{ind}$, in the following way: the universe of the structure will be $X$, and for every definable set $Y$ in $\mathbb{M}$ we add a predicate $P_Y$ that corresponds to the intersection $Y\cap X$.

Definition 2.5.

Let $X$ be a definable set in $\mathbb{M}$. Then $X$ is one-based if the first order theory of $X^{ind}$ is one-based.

We say that a family of definable sets $\mathcal{P}$ is $\emptyset$-invariant if the image of any definable set in $\mathcal{P}$ by an automorphism in $\operatorname {Aut}(\mathbb{M})$ is still in $\mathcal{P}$. Moreover, if $B$ is a small subset of $\mathbb{M}$, we say that $\bar{c}\models \mathcal{P}\upharpoonright B$ if $tp(\bar{c}/B)$ contains some definable set that belongs to $\mathcal{P}$. Finally, by $\bar{c} \underset{A}{\forksnot } B$, we mean that $\bar{c}$ is independent from $B$ over $A$ in the sense of forking independence.

Definition 2.6.

A definable set $X$ (over some small subset $A\subset \mathbb{M}$) is $\mathcal{P}$-internal for some $\emptyset$-invariant family of definable sets $\mathcal{P}$ if for any $\bar{c}\in X$ there exists $B\supseteq A$ with $\bar{c} \underset{A}{\forksnot } B$ and $\bar{b}_1,\ldots ,\bar{b}_k$ with $\bar{b}_i\models \mathcal{P}\upharpoonright B$ such that $\bar{c}\in dcl(B,\bar{b}_1,\ldots ,\bar{b}_k)$.

The following theorem has been proved by F. Wagner in 22.

Theorem 2.7 (Wagner).

Let $X$ be a definable set and let $\mathcal{P}$ be a $\emptyset$-invariant family of one-based sets. If $X$ is $\mathcal{P}$-internal, then $X$ is one-based.

3. Actions on trees

The goal of this section is to present a structure theorem for groups acting on trees. In the first subsection we will be interested in group actions on simplicial trees. These actions can be analysed using Bass-Serre theory, and we will explain the duality between the notion of a graph of groups and these actions.

In the second subsection we will record the notion of a real tree and quickly describe some natural group actions on real trees.

3.1. Bass-Serre theory

Bass-Serre theory gives a structure theorem for groups acting on (simplicial) trees, i.e., acyclic connected graphs. It describes a group (that acts on a tree) as a series of amalgamated free products and HNN extensions. The mathematical notion that contains these instructions is called a graph of groups. For a complete treatment we refer the reader to 19.

We start with the definition of a graph.

Definition 3.1.

A graph $G(V,E)$ is a collection of data that consists of two sets $V$ (the set of vertices) and $E$ (the set of edges) together with three maps:

•

an involution $\bar{\phantom {1}}:E\to E$, where $\bar{e}$ is called the inverse of $e$;

•

$\alpha :E\to V$, where $\alpha (e)$ is called the initial vertex of $e$; and

•

$\tau :E\to V$, where $\tau (e)$ is called the terminal vertex of $e$,

so that $\bar{e}\neq e$ and $\alpha (e)=\tau (\bar{e})$ for every $e\in E$.

Definition 3.2 (Graph of groups).

A graph of groups $\mathcal{G}\coloneq (G(V,E)$, $\{G_u\}_{u\in V}$, $\{G_e\}_{e\in E}$, $\{f_e\}_{e\in E})$ consists of the following data:

•

a graph $G(V,E)$;

•

a family of groups $\{G_u\}_{u\in V}$; i.e., a group is attached to each vertex of the graph;

•

a family of groups $\{G_e\}_{e\in E}$; i.e., a group is attached to each edge of the graph. Moreover, $G_{e}=G_{\bar{e}}$;

•

a collection of injective morphisms $\{f_e:G_e\to G_{\tau (e)} \ | \ e\in E\}$; i.e., each edge group comes equipped with two embeddings to the incident vertex groups.

The fundamental group of a graph of groups is defined as follows.

Definition 3.3.

Let $\mathcal{G}\coloneq (G(V,E), \{G_u\}_{u\in V}, \{G_e\}_{e\in E}, \{f_e\}_{e\in E})$ be a graph of groups. Let $T$ be a maximal subtree of $G(V,E)$. Then the fundamental group, $\pi _1(\mathcal{G},T)$, of $\mathcal{G}$ with respect to $T$ is the group given by the following presentation:

$$\begin{multline*} \langle \{G_u\}_{u\in V}, \{t_e\}_{e\in E} \ | \ t^{-1}_e=t_{\bar{e}} \quad \text{for} \ e\in E, t_e=1 \quad \text{for} \ e\in T, f_e(a)=t_ef_{\bar{e}}(a)t_{\bar{e}} \\ \text{for} \ e\in E \ a\in G_e\rangle . \end{multline*}$$

Remark 3.4.

It is not hard to see that the fundamental group of a graph of groups does not depend on the choice of the maximal subtree up to isomorphism (see 19, Proposition 20, p. 44).

In order to give the main theorem of Bass-Serre theory we need the following definition.

Definition 3.5.

Let $G$ be a group acting on a simplicial tree $T$ without inversions, denote by $\Lambda$ the corresponding quotient graph, and denote by $p$ the quotient map $T \to \Lambda$. A Bass-Serre presentation for the action of $G$ on $T$ is a triple $(T^1, T^0, \{\gamma _e\}_{e\in E(T^1)\setminus E(T^0)})$ consisting of

•

a subtree $T^1$ of $T$ which contains exactly one edge of $p^{-1}(e)$ for each edge $e$ of $\Lambda$;

•

a subtree $T^0$ of $T^1$ which is mapped injectively by $p$ onto a maximal subtree of $\Lambda$;

•

a collection of elements of $G$, $\{\gamma _e\}_{e\in E(T^1)\setminus E(T^0)}$, such that if $e=(u,v)$ with $v\in T^1\setminus T^0$, then $\gamma _e\cdot v$ belongs to $T^0$.

Theorem 3.6.

Suppose $G$ acts on a simplicial tree $T$ without inversions. Let $(T^1,T^0, \{\gamma _e\})$ be a Bass-Serre presentation for the action. Let $\mathcal{G}\coloneq (G(V,E)$, $\{G_u\}_{u\in V}$, $\{G_e\}_{e\in E}$, $\{f_e\}_{e\in E})$ be the following graph of groups:

•

$G(V,E)$ is the quotient graph given by $p:T\to \Gamma$;

•

if $u$ is a vertex in $T^0$, then $G_{p(u)}=Stab_G(u)$;

•

if $e$ is an edge in $T^1$, then $G_{p(e)}=Stab_G(e)$;

•

if $e$ is an edge in $T^1$, then $f_{p(e)}:G_{p(e)}\to G_{\tau (p(e))}$ is given by the identity if $e\in T^0$ and by conjugation by $\gamma _e$ if not.

Then $G$ is isomorphic to $\pi _1(\mathcal{G})$.

Remark 3.7.

The other direction of the above theorem also holds. Whenever a group $G$ is isomorphic to the fundamental group of a graph of groups then there is a natural way to obtain a simplicial tree $T$ and an action of $G$ on $T$ (see 19, section 5.3, p. 50).

Among splittings of groups we will distinguish those with some special type vertex groups called surface type vertex groups.

Definition 3.8.

Let $G$ be a group acting on a tree $T$ without inversions and let $(T_1,T_0,\{\gamma _e\})$ be a Bass-Serre presentation for this action. Then a vertex $v\in T^0$ is called a surface type vertex if the following conditions hold:

•

$\mathrm{Stab}_G(v)=\pi _1(\Sigma )$ for a connected compact surface $\Sigma$ with nonempty boundary, such that either the Euler characteristic of $\Sigma$ is at most $-2$ or $\Sigma$ is a once punctured torus.

•

For every edge $e\in T_1$ adjacent to $v$, $\mathrm{Stab}_G(e)$ embeds onto a maximal boundary subgroup of $\pi _1(\Sigma )$, and this induces a one-to-one correspondence between the set of edges (in $T^1$) adjacent to $v$ and the set of boundary components of $\Sigma$.

We next follow 2 and define the notion of a generalized abelian decomposition (GAD).

Definition 3.9.

A GAD of a group $G$ is a graph of groups $\mathcal{G}(V,E)$ with abelian edge groups such that $V$ is partitioned as $V_S\cup V_A \cup V_R$ where:

•

each vertex in $V_S$ is a vertex of surface type for the corresponding action on a tree;

•

each vertex group for a vertex in $V_A$ is noncyclic abelian; and

•

each vertex group for a vertex $V_R$ is called rigid.

Definition 3.10 (Peripheral subgroup).

Let $A$ be a vertex group of a GAD of $G$, $(\mathcal{G}(V,E), (V_S,$ $V_A,V_R))$, whose vertex is in $V_A$. Then we denote by $P(A)$ the subgroup of $A$ generated by all incident edge groups. Moreover the subgroup of $A$ that dies under every morphism $h:A\rightarrow \mathbb{Z}$ that kills $P(A)$ is called the peripheral subgroup and is denoted by $\bar{P}(A)$.

Definition 3.11 (Dehn twists).

Let $H$ be a subgroup of $G$. Suppose $G$ splits as an:

•

amalgamated free product $A*_CB$ so that $H$ is a subgroup of $A$. Let $g$ be an element in the centralizer of $C$ in $G$. Then a Dehn twist in $g$ is the automorphism fixing $A$ pointwise and sending each element $b$ of $B$ to $gbg^{-1}$.

•

$\mathrm{HNN}$-extension $A*_C$ so that $H$ is a subgroup of $A$. Let $g$ be an element in the centralizer of $C$ in $G$. Then a Dehn twist in $g$ is the automorphism fixing $A$ and sending the Bass-Serre element $t$ to $tg$.

Definition 3.12 (Relative modular automorphisms).

Let $H$ be a subgroup of $G$. Let $\Delta \coloneq (\mathcal{G}(V,E), (V_S,V_A,V_R))$ be a GAD of $G$ in which $H$ can be conjugated into a vertex group. Then $\operatorname {Mod}_H(\Delta )$ is the subgroup of $\operatorname {Aut}_H(G)$ generated by:

•

inner automorphisms;

•

unimodular automorphisms of $G_u$ for $u\in V_A$ that fix the peripheral subgroup of $G_u$ and every other vertex group;

•

automorphisms of $G_u$ for $u\in V_S$ coming from homeomorphisms of the corresponding surface that fix all boundary components;

•

Dehn twists in elements of centralizers of edge groups, after collapsing the GAD to a one edge splitting in which $H$ is a subgroup of a vertex group.

Moreover we define the modular group of $G$ relative to $H$, $\operatorname {Mod}_H(G)$, to be the group generated by $\operatorname {Mod}_H(\Delta )$ for every $\mathrm{GAD}$ $\Delta$ of $G$.

3.2. Actions on real trees

Real trees (or $\mathbb{R}$-trees) generalize simplicial trees in the following way.

Definition 3.13.

A real tree is a geodesic metric space in which for any two points there is a unique arc that connects them.

When we say that a group $G$ acts on a real tree $T$ we will always mean an action by isometries.

Moreover, an action $G\curvearrowright T$ of a group $G$ on a real tree $T$ is called nontrivial if there is no globally fixed point and minimal if there is no proper $G$-invariant subtree. Lastly, an action is called free if for any $x\in T$ and any nontrivial $g\in G$ we have that $g\cdot x\neq x$.

We next collect some families of group actions on real trees.

Definition 3.14.

Let $G\curvearrowright ^{\lambda } T$ be a minimal action of a finitely generated group $G$ on a real tree $T$. Then we say:

(i)

$\lambda$ is of discrete (or simplicial) type if every orbit $G.x$ is discrete in $T$. In this case $T$ is simplicial, and the action can be analyzed using Bass-Serre theory.

(ii)

$\lambda$ is of axial (or toral) type if $T$ is isometric to the real line $\mathbb{R}$ and $G$ acts with dense orbits; i.e., $\overline{G.x}=T$, for every $x\in T$.

(iii)

$\lambda$ is of surface (or IET) type if $G=\pi _1(\Sigma )$ where $\Sigma$ is a surface with (possibly empty) boundary carrying an arational measured foliation and $T$ is dual to $\tilde{\Sigma }$; i.e., $T$ is the lifted leaf space in $\tilde{\Sigma }$ after identifying leaves of distance $0$ (with respect to the pseudometric induced by the measure).

We will use the notion of a graph of actions in order to glue real trees equivariantly. We follow the exposition in 3, section 1.3.

Definition 3.15 (Graph of actions).

A graph of actions $(G\curvearrowright T$, $\{Y_u\}_{u\in V(T)}$, $\{p_e\}_{e\in E(T)})$ consists of the following data:

•

a simplicial type action $G\curvearrowright T$;

•

for each vertex $u$ in $T$ a real tree $Y_u$;

•

for each edge $e$ in $T$, an attaching point $p_e$ in $Y_{\tau (e)}$.

Moreover:

(1)

$G$ acts on $R\coloneq \{\coprod Y_u : u\in V(T)\}$ so that $q:R\to V(T)$ with $q(Y_u)=u$ is $G$-equivariant;

(2)

for every $g\in G$ and $e\in E(T)$, $p_{g\cdot e}=g\cdot p_e$.

To a graph of actions $\mathcal{A}\coloneq (G\curvearrowright T,\{Y_u\}_{u\in V(T)},\{p_e\}_{e\in E(T)})$ we can assign an $\mathbb{R}$-tree $Y_{\mathcal{A}}$ endowed with a $G$-action. Roughly speaking this tree will be $\coprod _{u\in V(T)} Y_{u}/\!\sim$, where the equivalence relation $\sim$ identifies $p_e$ with $p_{\bar{e}}$ for every $e\in E(T)$. We say that a real $G$-tree $Y$ decomposes as a graph of actions $\mathcal{A}$ if there is an equivariant isometry between $Y$ and $Y_{\mathcal{A}}$.

Interesting actions on real trees can be obtained by sequences of morphisms from a finitely generated group to a free group. We explain how in the next subsection.

3.3. The Bestvina-Paulin method

The construction we are going to record is credited to Bestvina 1 and Paulin 8 independently.

We fix a finitely generated group $G$ and we consider the set of nontrivial equivariant pseudometrics $d:G\times G\to \mathbb{R}^{\geq 0}$, denoted by $\mathcal{ED}(G)$. We equip $\mathcal{ED}(G)$ with the compact-open topology (where $G$ is given the discrete topology). Note that convergence in this topology is given by

$$\begin{equation*} (d_i)_{i<\omega }\to d\quad \text{if and only if} \ \ d_i(1,g)\to d(1,g)\ \ (\text{in $\mathbb{R}$}) \quad \text{for any } g\in G. \end{equation*}$$

It is not hard to see that $\mathbb{R}^+$ acts cocompactly on $\mathcal{ED}(G)$ by rescaling; thus the space of projectivised equivariant pseudometrics on $G$ is compact.

We also note that any based $G$-space $(X,*)$ (i.e., a metric space with a distinguished point equipped with an action of $G$ by isometries) gives rise to an equivariant pseudometric on $G$ as follows: $d(g,h)=d_X(g\cdot *,h\cdot *)$.

We say that a sequence of $G$-spaces $(X_i,*_i)_{i<\omega }$ converges to a $G$-space $(X,*)$ if the corresponding pseudometrics induced by $(X_i,*_i)$ converge to the pseudometric induced by $(X,*)$ in $\mathcal{PED}(G)$.

A morphism $h:G\to H$ where $H$ is a finitely generated group induces an action of $G$ on $\mathcal{X}_H$ (the Cayley graph of $H$) in the obvious way, thus making $\mathcal{X}_H$ a $G$-space. We have:

Lemma 3.16.

Let $\mathbb{F}$ be a nonabelian free group. Let $(h_n)_{n<\omega }:G\to \mathbb{F}$ be a sequence of pairwise nonconjugate morphisms. Then for each $n<\omega$ there exists a base point $*_n$ in $\mathcal{X}_{\mathbb{F}}$ such that the sequence of $G$-spaces $(\mathcal{X}_{\mathbb{F}},*_n)_{n<\omega }$ has a convergent subsequence to a real $G$-tree $(T,*)$, where the action of $G$ on $T$ is nontrivial.

3.4. Limit groups

Definition 3.17.

Let $G$ be a group and let $(h_n)_{n<\omega }:G\rightarrow \mathbb{F}$ be a sequence of morphisms. Then the sequence $(h_n)_{n<\omega }$ is called convergent if for every $g\in G$, there exists $n_g$ such that either $h_n(g)\neq 1$ for all $n>n_g$ or $h_n(g)=1$ for all $n>n_g$.

Moreover, if $(h_n)_{n<\omega }:G\rightarrow \mathbb{F}$ is a convergent sequence, then we define its stable kernel $\mathrm{Ker}h_n\coloneq \{ g\in G \ | \ g \ \text{is eventually killed by}\ h_n \}$.

Definition 3.18.

Let $G$ be a finitely generated group. Then $G$ is a limit group if there exists a convergent sequence $(h_n)_{n<\omega }:G\rightarrow \mathbb{F}$ with trivial stable kernel.

Limit groups can be given a constructive definition. To this end we define:

Definition 3.19.

Let $\Delta \coloneq (\mathcal{G}(V,E), (V_S,V_A,V_R))$ be a $\mathrm{GAD}$ of a group $G$. For each $G_v$ with $v\in V_R$ we define its envelope $\tilde{G_v}$ in $\Delta$ in the following way: for every $a\in V_A$ we replace $G_a$ in $\mathcal{G}(V,E)$ by its peripheral subgroup. Then $\tilde{G_v}$ is the group generated by $G_v$ together with the centralizers of incident edge groups.

Definition 3.20 (Strict morphisms).

Let $\eta :G\twoheadrightarrow L$ be an epimorphism and let $\Delta \coloneq (\mathcal{G}(V,E),$ $(V_S,V_A,V_R))$ be a GAD of $G$ in which every edge group is maximal abelian in at least one vertex group of the one edged splitting induced by the edge. Then $\eta$ is strict with respect to $\Delta$ if the following hold:

•

$\eta$ is injective on each edge group;

•

$\eta$ is injective on $\tilde{G_v}$ for every $v\in V_R$;

•

$\eta$ is injective on the peripheral subgroup of each abelian vertex group;

•

$\eta (G_s)$ is not abelian for every $s\in V_S$.

Definition 3.21.

A group $L$ is a constructive limit group if it belongs to the following hierarchy of groups defined recursively.

Base step.

Level $0$ consists of finitely generated free groups.

Recursive step.

A group $G$ belongs to level $i+1$ if it is either the free product of two groups that belong to level $i$ or there exists a $\mathrm{GAD}$, $\Delta$, for $G$ and a strict map $\eta :G\twoheadrightarrow H$ with respect to $\Delta$ onto some $H$ that belongs to level $i$.

The following theorem appears as Theorem 5.12 in 15.

Theorem 3.22 (Sela).

Let $L$ be a finitely generated group. Then $L$ is a limit group if and only if it is a constructive limit group.

3.5. Graded limit groups

A graded limit group is a limit group together with a distinguished finitely generated subgroup. We will be interested in a special kind of graded limit group called solid limit groups. Solid and rigid limit groups were defined in 15, Definition 10.2, and the corresponding notion in the work of Kharlampovich-Myasnikov 6 is the notion of a group without sufficient splittings.

Definition 3.23.

Let $G$ be a limit group which is freely indecomposable with respect to a finitely generated subgroup $H$. We fix a finite generating set $\Sigma$ for $G$ and a basis $\bar{a}$ for $\mathbb{F}\coloneq \mathbb{F}(\bar{a})$. A morphism $h:G\rightarrow \mathbb{F}$ is short with respect to $\operatorname {Mod}_H(G)$ if for every $\sigma \in \operatorname {Mod}_H(G)$ and every $g\in \mathbb{F}$ that commutes with $h(H)$ we have that $\max _{s\in \Sigma }\left|h(s)\right|_{\mathbb{F}}\leq \max _{s\in \Sigma }\left|Conj(g)\circ h\circ \sigma \right|_{\mathbb{F}}$.

Definition 3.24.

Let $G$ be a limit group which is freely indecomposable with respect to a finitely generated subgroup $H$. Let $(h_n)_{n<\omega }:G\rightarrow \mathbb{F}$ be a convergent sequence of short morphisms with respect to $\operatorname {Mod}_H(G)$. Then we call $G/\mathrm{Ker}h_n$ a shortening quotient of $G$ with respect to $H$.

Definition 3.25 (Solid limit group).

Let $S$ be a limit group and let $H$ be a finitely generated subgroup of $S$. Suppose $S$ is freely indecomposable with respect to $H$. Then $S$ is solid with respect to $H$ if there exists a shortening quotient of $S$ with respect to $H$ which is isomorphic to $S$.

Example 3.26.

The surface group $\langle x_1$, $x_2$, …, $x_{2k}$, $e_1$, $e_2$, …, $e_{2m} \ | \ [x_1,x_2][x_3,x_4]\ldots [x_{2k-1}$, $x_{2k}][e_1,e_2]\cdots [e_{2m-1},e_{2m}]\rangle$ is a solid limit group with respect to the subgroup $\langle e_1$, …, $e_{2m}\rangle$.

Definition 3.27.

Let $Sld$ be a solid limit group with respect to a finitely generated subgroup $H$ with generating set $\Sigma _H$. Let $(h_n)_{n<\omega }:Sld\rightarrow \mathbb{F}$. We call $(h_n)_{n<\omega }$ a flexible sequence if for every $n$ either:

•

the morphism $h_n=h'_n\circ \eta _n$, where $\eta _n:Sld\twoheadrightarrow \mathbb{F}*\Gamma$ for some group $\Gamma$, $H$ is mapped onto $\mathbb{F}$ by $\eta _n$, $h'_n:\mathbb{F}*\Gamma \rightarrow \mathbb{F}$ stays the identity on $\mathbb{F}$, and $\eta _n$ is short with respect to $\operatorname {Mod}_H(Sld)$ or

•

the morphism $h_n$ is short with respect to $\operatorname {Mod}_H(Sld)$ and moreover$$\begin{equation*} \max \nolimits _{g\in B_n}\left|h_n(g)\right|_{\mathbb{F}}> 2^n(1+\max \nolimits _{s\in \Sigma _H}\left|h_n(s)\right|_{\mathbb{F}}), \end{equation*}$$

where $B_n$ is the ball of radius $n$ in the Cayley graph of $Sld$.

If $(h_n)_{n<\omega }:Sld\rightarrow \mathbb{F}$ is a convergent flexible sequence, then we call $Sld/\mathrm{Ker}h_n$ a flexible quotient of $Sld$.

It is not hard to see, using the shortening argument, that flexible quotients are proper. Moreover one can define a partial order and an equivalence relation on the class of flexible quotients of a solid limit group. Let $Sld$ be a solid limit group with respect to a finitely generated subgroup $H$ and let $\eta _i:Sld\twoheadrightarrow Q_i$ for $i\leq 2$ be flexible quotients with their canonical quotient maps. Then $Q_2\leq Q_1$ if $\ker \eta _1\subseteq \ker \eta _2$, and $Q_1\sim Q_2$ if there exists $\sigma \in \operatorname {Mod}_H(Sld)$ such that $\ker (\eta _1\circ \sigma )=\ker \eta _2$.

For the following theorem see the discussion before Definition 10.5 in 15.

Theorem 3.28 (Sela).

Let $Sld$ be a solid limit group with respect to a finitely generated subgroup $H$. Assume that $Sld$ admits a flexible quotient. Then there exist finitely many classes of maximal flexible quotients.

A morphism from a solid limit group to a free group that does not factor through one of the maximal flexible quotients (after precomposition by a modular automorphism) is called a solid morphism; otherwise it is called flexible (see 15, Definition 10.6).

4. Towers

In this section we are interested in limit groups that have a very special structure, namely, the structure of a tower. A tower is built recursively adding floors to a given basis, which is taken to be a free product of fundamental groups of surfaces with free abelian groups. Each floor is built by “gluing” a finite set of surface flats and abelian flats to the previous one following specific rules, to be made precise in the next subsection. The corresponding notion in the work of Kharlampovich-Myasnikov is the notion of an NTQ group, i.e., the coordinate group of a nondegenerate triangular quasiquadratic system of equations (see 4, Definition 9).

Limit groups that admit the structure of a tower play a significant role in the proof of the elementary equivalence of nonabelian free groups. This class of limit groups is connected with generalizations of Merzlyakov’s theorem as proved in 16 and 5. We will analyse and further expand this connection in section 6.

4.1. The construction of a tower

We start with defining the notion of a surface flat.

Definition 4.1 (Surface flat).

Let $G$ be a group and let $H$ be a subgroup of $G$ (see Figure 1). Then $G$ has the structure of a surface flat over $H$ if $G$ acts minimally on a tree $T$ and the action admits a Bass-Serre presentation $(T^1, T^0, \{\gamma _e\})$ such that:

•

the set of vertices of $T^1$ is partitioned into two sets, $\{v\}$ and $V$, where $v$ is a surface type vertex;

•

$T^1$ is bipartite between $v$ and $V(T^1)\setminus \{v\}$;

•

$H$ is the free product of the stabilizers of vertices in $V$;

•

either there exists a retraction $r:G\to H$ that sends $\mathrm{Stab}_G(v)$ to a nonabelian image or $H$ is cyclic and there exists a retraction $r': G * \mathbb{Z}\to H * \mathbb{Z}$ which sends $\mathrm{Stab}_G(v)$ to a nonabelian image.

Remark 4.2.

A more concise way to refer to a group $G$ that has the structure of a surface flat over a subgroup $H$ is to say that $G$ is obtained from $H$ by gluing a surface $\Sigma _{g,n}$ along its boundary onto the subgroups $\{E_i\ :\ i\leq n\}$ of $H$.

Example 4.3.

The surface group $\pi _1(\Sigma _{g})=\langle x_1,\ldots ,x_{2g} \ | \ [x_1,x_2]\cdot \ldots \cdot [x_{2g-1},x_{2g}]\rangle$ has the structure of a surface flat over $\mathbb{F}_g\coloneq \langle x_1,\ldots ,x_g\rangle$ (see Figure 2).

Concisely, $\pi _1(\Sigma _g)$ is obtained from $\mathbb{F}_g$ by gluing $\langle b,x_{g+1},\ldots ,x_{2g} \ | \ b^{-1}\cdot [x_{g+1},x_{g+2}] \cdot \ldots \cdot [x_{2g-1},x_{2g}]\rangle$ along its boundary onto $\langle [x_1,x_2]\ldots [x_{g-1},x_g]\rangle$.

We similarly define the notion of an abelian flat.

Definition 4.4.

Let $G$ be a group and let $H$ be a subgroup of $G$. Then $G$ has the structure of an abelian flat over $H$ if $G$ acts minimally on a tree $T$ with a single orbit of edges, and the action admits a Bass-Serre presentation $(T^1=T^0, T^0)$ so that if $e=\{u,v\}$ is the unique edge in $T^0$, then $\mathrm{Stab}_G(u)=H$, $\mathrm{Stab}_G(e)$ is a maximal abelian subgroup of $H$, which we call the peg of the abelian flat, and $\mathrm{Stab}_G(v)=\mathrm{Stab}_G(e)\oplus \mathbb{Z}^m$ for some $m<\omega$ (see Figure 3).

Remark 4.5.

A more concise way to refer to a group $G$ that has the structure of an abelian flat over a subgroup $H$ is to say that $G$ is obtained from $H$ by gluing a free abelian group $\mathbb{Z}^n$ along the (maximal abelian) subgroup $E$ of $H$.

We note that when we say “gluing $\mathbb{Z}^n$ along the subgroup $E$ of $H$”, the outcome will really be the amalgamated free product $(E\oplus \mathbb{Z}^n)*_EH$, but we keep this terminology as it will be convenient in what follows.

We observe that if $G$ has the structure of an abelian flat over a subgroup $H$, then it is not hard to find a retraction $r:G\rightarrow H$: one can use the projection of $E\oplus \mathbb{Z}^m$ to $E$ and extend this to a morphism from $G$ to $H$ which fixes $H$.

Example 4.6.

The group

$$\begin{equation*} G\coloneq \langle x_1,\ldots ,x_n,z_1,\ldots ,z_k \ | \ x_1^2\cdot \ldots \cdot x_n^2=1, [x_1,z_i], [z_i,z_j], i,j\leq k\rangle \end{equation*}$$

has the structure of a free abelian flat over the subgroup $\langle x_1,\ldots , x_n \ | \ x_1^2\cdot \ldots \cdot x_n^2 \rangle$.

Concisely, $G$ is obtained from $\langle x_1,\ldots , x_n \ | \ x_1^2\cdots x_n^2\rangle$ by gluing the free abelian group $\mathbb{Z}^k$ along the subgroup $\langle x_1\rangle$ of $\langle x_1,\ldots , x_n \ | \ x_1^2\cdots x_n^2\rangle$.

We can combine surface and abelian flats in order to obtain the “floors” of a tower.

Definition 4.7 (Floor).

Let $G$ be a group and let $H$ be a subgroup of $G$ (see Figure 4). Then $G$ has the structure of a floor over $H$ if $G$ acts minimally on a tree $T$ and the action admits a Bass-Serre presentation $(T^1, T^0, \{\gamma _e\})$, where the set of vertices of $T^1$ is partitioned into three subsets, $V_S, V_A,$ and $V_R$, such that:

•

each vertex in $V_S$ is a surface type vertex;

•

for each vertex $u\in V_A$, its stabilizer $G_{u}$ is a free abelian group;

•

the tree $T^1$ is bipartite between $V_S\cup V_A$ and $V_R$;

•

the subgroup $H$ of $G$ is the free product of the stabilizers of vertices in $V_R$;

•

for each $u\in V_A$, there is a unique edge $e=\{v,u\}$ connecting $u$ to a vertex $v$ in $V_R$. Moreover, $\mathrm{Stab}_G(e)$ is maximal abelian in $\mathrm{Stab}_G(v)$ and $\mathrm{Stab}_G(u)=\mathrm{Stab}_G(e)\oplus \mathbb{Z}^m$. In addition, the stabilizer $\mathrm{Stab}_{G}(e)$ of $e$ cannot be conjugated to any other stabilizer $\mathrm{Stab}_G(e')$ for $e'\neq e$ an edge connecting a vertex in $V_A$ to a vertex in $V_R$;

•

either there exists a retraction $r:G\to H$ that, for each $v\in V_S$, sends $\mathrm{Stab}_G(v)$ to a nonabelian image or $H$ is cyclic and there exists a retraction $r': G * \mathbb{Z}\to H * \mathbb{Z}$ that, for each $v\in V_S$, sends $\mathrm{Stab}_G(v)$ to a nonabelian image.

In the opposite direction a floor can be decomposed into flats in many possible ways; i.e., a floor can be seen as a sequence of surface and abelian flats, and we will often see such a sequence as giving a preferred order to the flats of the floor.

We can now bring everything together to define:

Definition 4.8.

A group $G$ has the structure of a tower (of height $m$) over a subgroup $H$ if there exists a sequence $G=G^m>G^{m-1}>\cdots >G^0=H$ such that for each $i$, $0\leq i<m$, one of the following holds:

(i)

The group $G^{i+1}$ has the structure of a floor over $G^i$, in which $H$ is contained in one of the vertex groups that generate $G_i$ in the floor decomposition of $G^{i+1}$ over $G^i$. Moreover, the pegs of the abelian flats of the floor are glued along (maximal abelian) groups that are not conjugates of each other and they cannot be conjugated into groups which correspond to abelian flats of any previous floor.

(ii)

The group $G^{i+1}$ is a free product of $G^{i}$ with a finitely generated free group.

The next lemma follows from the definition of a constructible limit group.

Lemma 4.9.

If $G$ has the structure of a tower over a limit group, then $G$ is a limit group.

If $G$ has the structure of a tower over a subgroup $H$ it will be useful to collect the information witnessing it. Thus we define:

Definition 4.10.

Suppose $G$ has the structure of a tower (of height $m$) over $H$. Then the tower corresponding to $G$, denoted by $\mathcal{T}(G,H)$, is the following collection of data:

$$\begin{equation*} ((\mathcal{G}(G^1,G^0),r_1),(\mathcal{G}(G^2,G^1),r_2),\ldots ,(\mathcal{G}(G^m,G^{m-1}),r_m)), \end{equation*}$$

where:

•

the splitting $\mathcal{G}(G^{i+1},G^i)$ is the splitting that witnesses that $G^{i+1}$ has the structure of a floor over $G^i$, respectively, the free splitting $G^{i}*\mathbb{F}_n$ for some finitely generated free group $\mathbb{F}_n$;

•

the morphism $r_{i+1}:G^{i+1}\rightarrow G^i$ (or $r_{i+1}:G^{i+1}\rightarrow G^{i}*\mathbb{Z}$) is the retraction that witnesses that $G^{i+1}$ has the structure of a floor over $G^{i}$, respectively, the retraction $r_{i+1}:G^i*\mathbb{F}_n\rightarrow G^i$.

Remark 4.11.

The notation $\mathcal{G}(G)$ will refer to a splitting of $G$ as a graph of groups. The notation $\mathcal{G}(G,H)$ will refer either to a free splitting of $G$ as $H*\mathbb{F}_n$ or to a splitting that corresponds to a floor structure of $G$ over $H$.

A tower in which no abelian flat occurs in some (any) decomposition of its floors into flats is called a hyperbolic tower (or regular NTQ group in the terminology of Kharlampovich-Myasnikov). Furthermore, if a floor consists only of abelian flats we call it an abelian floor.

For the rest of the paper we assume the following.

Convention.

Suppose $\mathcal{T}(G,\mathbb{F})$ is a tower. Let $\{E_j\}_{j\in J}$ be the collection of pegs that correspond to the abelian flats that occur along the floors of the tower. Let $\{E_j\}_{j\in J'}$, with $J'\subseteq J$, be the subcollection of the pegs that can be conjugated into a subgroup of the base floor $\mathbb{F}$; i.e., there is $\gamma _j\in G$ such that $E_j^{\gamma _j}\leq \mathbb{F}$ for every $j\in J'$.

Then, we assume that:

(1)

when the above subcollection is not empty, the first floor $\mathcal{G}(G^1,G^0)$ of the tower $\mathcal{T}(G,\mathbb{F})$ consists only of the abelian flats corresponding to the above subcollection and glued along $E_j^{\gamma _j}$ to $\mathbb{F}$, and each floor above the first (abelian) floor is either a free product or it consists of a single flat (abelian or surface);

(2)

when the above subcollection is empty, we assume that each floor is either a free product or it consists of a single flat (abelian or surface).

4.2. Twin towers

We next work towards constructing a tower by “gluing” two copies of a given tower together.

Definition 4.12.

Suppose $H$ has the structure of an abelian floor over $\mathbb{F}$ and $\mathcal{G}(H,\mathbb{F})$ is the splitting witnessing it. Let $\{\mathbb{Z}^{m_i}\}_{i\in I}$ be the collection of the free abelian groups that we glue along the corresponding pegs $\{E_i\}_{i\in I}$ in forming the abelian flats of the floor.

Then the double of $H$ with respect to $\mathcal{G}(H,\mathbb{F})$, denoted by $H_{Db}\coloneq H_{Db}(\mathcal{G}(G,\mathbb{F}))$, is the group obtained as the fundamental group of a graph of groups in which all the data is as in $\mathcal{G}(H,\mathbb{F})$ apart from replacing $\{\mathbb{Z}^{m_i}\}_{i\in I}$ by their doubles $\{\mathbb{Z}^{m_i}\oplus \mathbb{Z}^{m_i}\}_{i\in I}$. The above graph of groups $Db(\mathcal{G}(H,\mathbb{F}))\coloneq \mathcal{G}(H_{Db},\mathbb{F})$ is called the floor double of $\mathcal{G}(H,\mathbb{F})$ and naturally witnesses that $H_{Db}$ is an abelian floor over $\mathbb{F}$ (see Figure 5).

Lemma 4.13.

Suppose $H$ has the structure of an abelian floor $\mathcal{G}(H,\mathbb{F})$ over $\mathbb{F}$. Then $H$ admits two natural embeddings $f_1,f_2$ into $H_{Db}$.

Moreover, for each $i\leq 2$, the group $H_{Db}$ admits an abelian floor structure over $f_i(H)$.

Proof.

The first embedding $f_1$ can be taken to be the identity since $H$ is a subgroup of $H_{Db}$, and clearly $H_{Db}$ has an abelian floor structure over $H$ with the pegs corresponding to the maximal abelian groups of $H$ that contain the pegs of $\mathcal{G}(H,\mathbb{F})$. The second embedding $f_2$ is obtained as follows:

•

it agrees with $\mathrm{Id}$ on $\mathbb{F}$, and

•

it sends each free abelian group $\mathbb{Z}^m$ that is glued along a peg $E$ in $\mathbb{F}$ in forming the abelian flats of the abelian floor $\mathcal{G}(H,\mathbb{F})$ isomorphically onto the corresponding free abelian group glued along the peg in $H$ that contains $E$ in forming the abelian flats of the abelian floor $\mathcal{G}(H_{Db},H)$.■

The following lemmata are immediate.

Lemma 4.14.

Let $H$ be a subgroup of a group $G$. Suppose $H$ has the structure of an abelian floor $\mathcal{G}(H,\mathbb{F})$ over $\mathbb{F}$. Let $G_{Db}\coloneq H_{Db}*_HG$. Let $\mathcal{G}(G)$ be a splitting of $G$ in which $H$ is a subgroup of a vertex group $G_v$. Then $G_{Db}$ is isomorphic to the fundamental group of the graph of groups that has the same data as $\mathcal{G}(G)$ apart from replacing $G_v$ by $(G_v)_{Db}\coloneq H_{Db}*_HG_v$.

Lemma 4.15.

Suppose $G$ has an abelian floor structure over a limit group $L$. Let $B_1, B_2$ be nonconjugate (in $L$) maximal abelian subgroups of $L$ that cannot be conjugated into any of the pegs of the abelian floor. Then $B_1,B_2$ are nonconjugate maximal abelian subgroups of $G$.

We now pass to proving that replacing the first (abelian) floor by its double yields a natural tower structure for the corresponding group.

Lemma 4.16.

Suppose $G$ has the structure of a tower (of height $m$) $\mathcal{T}(G,\mathbb{F})\coloneq ((\mathcal{G}(G^1,G^0),$ $r_1),(\mathcal{G}(G^2,G^1),$ $r_2),\ldots ,(\mathcal{G}(G^m,G^{m-1}),r_m))$ over $\mathbb{F}$ and that the first floor $\mathcal{G}(G^1,G^0)$ is an abelian floor. Let, for each $i\leq m$, the group $G^i_{Db}$ be the amalgamated free product $G^1_{Db}*_{G_1}G^i$, where the edge maps are $f:G_1\to G^1_{Db}$ and the embedding induced by the tower structure from $G_1$ to $G_i$.

Then $G_{Db}\coloneq G^m_{Db}$ admits a structure of a tower over $\mathbb{F}$ witnessed by $G^m_{Db}>G^{m-1}_{Db}>\cdots > G^1_{Db}>\mathbb{F}$ and splittings $\mathcal{G}(G^{i+1}_{Db},G^i_{Db})$, which are naturally inherited from the corresponding splittings in $\mathcal{T}(G,\mathbb{F})$.

Proof.

The proof is by induction on the height $m$ of the tower.

Base step.

The group $G^1_{Db}$ has a natural abelian floor structure over $\mathbb{F}$ as observed in Definition 4.12.

Inductive step.

We will assume that the result holds for any tower of height at most $i$ and we show it for towers of height $i+1$. We take cases according to whether $G^{i+1}$ is a free product over $G^i$ or has a surface flat structure over $G^i$ or has an abelian flat structure over $G^i$:

•

Assume that $G^{i+1}=G^i*\mathbb{F}_n$; then, by Lemma 4.14, $G^{i+1}_{Db}=G^i_{Db}*\mathbb{F}_n$. So, by the induction hypothesis $G^{i+1}_{Db}$ has a tower structure over $\mathbb{F}$ corresponding naturally to the tower structure of $G^{i+1}$ over $\mathbb{F}$.

•

Assume that $G^{i+1}$ has a surface flat structure over $G^i$ witnessed by $(\mathcal{G}(G^{i+1},G^i),r_{i+1})$. We consider the rigid vertex group, say $G_u$, of the above graph of groups that contains $\mathbb{F}$. Since $G^1$ is freely indecomposable with respect to $\mathbb{F}$, the vertex group $G_u$ must contain $G^1$. We consider the graph of groups with the same data as $(\mathcal{G}(G^{i+1},G^i),r_{i+1})$ apart from replacing $G_u$ by $G_u*_{G^1}G^1_{Db}$. Then by Lemma 4.14 the fundamental group of this latter graph of groups is isomorphic to $G^{i+1}_{Db}$, and together with the retraction $r_{i+1}': G^{i+1}_{Db}\rightarrow G^i_{Db}$ that agrees with $r_{i+1}$ on $G^{i+1}$ and stays the identity on $G^1_{Db}$ it witnesses that $G^{i+1}_{Db}$ has a surface flat structure over $G^i_{Db}$.

•

Assume that $G^{i+1}$ has an abelian flat structure over $G^i$; i.e., $G^{i+1}=G^i*_A(A\oplus \mathbb{Z}^n)$. We consider the splitting $(G^i*_{G^1}G^1_{Db})*_A(A\oplus \mathbb{Z}^n)$. By Lemma 4.14 this is a splitting of $G^{i+1}_{Db}$. It is not hard to see that the group $A$ is maximal abelian in $G^i*_{G^1}G^1_{Db}$: indeed since $A$ cannot be conjugated to any of the pegs of the first abelian floor $\mathcal{G}(G^1,\mathbb{F})$ and $A$ is maximal abelian in $G^i$, it must be maximal abelian in $G^i_{Db}$. Moreover, by Lemma 4.15, it cannot be conjugated to any other peg in $G^i_{Db}$. Thus, together with the retraction $r_{i+1}':G^{i+1}_{Db}\rightarrow G^i_{Db}$ that agrees with $r_{i+1}$ on $G^{i+1}$ and stays the identity on $G^1_{Db}$ it witnesses that $G^{i+1}$ has an abelian flat structure over $G^i_{Db}$.■

Changing slightly the hypothesis of the previous lemma yields the following remark.

Remark 4.17.

Suppose $G$ has the structure of a tower (of height $m$) $\mathcal{T}(G,\mathbb{F})\coloneq ((\mathcal{G}(G^1,G^0),$ $r_1),(\mathcal{G}(G^2,G^1),$ $r_2),\ldots ,(\mathcal{G}(G^m,G^{m-1}),r_m))$ over $\mathbb{F}$ and that the first floor $\mathcal{G}(G^1,G^0)$ is an abelian floor. Let $f:G^1\hookrightarrow G^1_{Db}$ be the nonidentity natural map from $G^1$ to its double (see Lemma 4.13). Let, for each $i\leq m$, the group $_fG^i_{Db}$ be the amalgamated free product $G^1_{Db}*_{f(G^1)}G^i$.

Then $_fG_{Db}\coloneq {_fG^m_{Db}}$ admits a structure of a tower over $\mathbb{F}$ witnessed by $_fG^m_{Db}>{_fG^{m-1}_{Db}}>\cdots >{_fG^1_{Db}}>\mathbb{F}$ and splittings $\mathcal{G}({_fG^{i+1}_{Db}},{_fG^i_{Db}})$, as in Lemma 4.16.

Before moving to the definition of a twin tower we record some easy lemmata that will help us prove that our construction of a twin tower is indeed a tower.

Lemma 4.18.

Suppose $G$ has the structure of a tower over a limit group $L$. Let $E$ be a maximal abelian subgroup of $G$ and suppose $E\cap L$ is not trivial. Let $B$ be the maximal abelian group in $L$ that contains $E\cap L$. Then either $E$ is $B$ or $E$ is the free abelian group $B\oplus \mathbb{Z}^n$ that corresponds to an abelian flat of some floor of the tower glued along $B$ to $L$.

The following lemma is an easy exercise in normal forms.

Lemma 4.19.

Let $G\coloneq A*_CB$ be a limit group and let $E$ be a maximal abelian group in $A$. Suppose that no nontrivial element of $E$ commutes with a nontrivial element of $C$. Then $E$ is maximal abelian in $G$.

We define the notion of a twin tower, first in a case which is free of some technical complexity, in the following proposition.

Proposition 4.20 (Twin tower - nonabelian case).

Suppose $G$ has the structure of a tower $\mathcal{T}(G,\mathbb{F})$ over $\mathbb{F}$. Assume that the first floor $\mathcal{G}(G^1,G^0)$ is not an abelian floor. Then the amalgamated free product $G*_{\mathbb{F}}G$ admits a natural tower structure over $\mathbb{F}$ which we call the twin tower of $G$ with respect to $\mathcal{T}(G,\mathbb{F})$ (see Figure 6).

Proof.

Let $((\mathcal{G}(G^1,G^0),r_1),\ldots ,(\mathcal{G}(G^m,G^{m-1}),r_m))$ be the sequence witnessing that $G$ is a tower over $\mathbb{F}$. Let $G^{m+i}\coloneq G^i*_{\mathbb{F}}G$ be the amalgamated free product of $G^i$ with $G$ over $\mathbb{F}$. We claim that there exists a sequence

$$\begin{multline*} ((\mathcal{G}(G^1,G^0),r_1),\ldots ,(\mathcal{G}(G^m,G^{m-1}),r_m), (\mathcal{G}(G^{m+1},G^m),r_{m+1}), \\ \ldots , (\mathcal{G}(G^{2m},G^{2m-1}),r_{2m})), \end{multline*}$$

where the splitting $\mathcal{G}(G^{m+i+1},G^{m+i})$ has the same data as the splitting $\mathcal{G}(G^{i+1},G^i)$, apart from replacing the vertex group $G_u$ that contains $\mathbb{F}$ with $G_u*_{\mathbb{F}}G$, and moreover it witnesses that $G*_{\mathbb{F}}G$ has the structure of a tower over $\mathbb{F}$. We proceed by induction:

Base step.

We show that $G^{m+1}$ is a free product or has a surface flat structure or has an abelian flat structure over $G^m=G$, according to whether $G^1$ is a free product or has a surface flat structure $\mathbb{F}$. In addition, we show that it respects the requirements of being a floor of a tower together with the already given sequence of floors. We take cases:

•

Assume that $G^1=\mathbb{F}*\mathbb{F}_n$; then $G^{m+1}=(\mathbb{F}*\mathbb{F}_n)*_{\mathbb{F}}G=G*\mathbb{F}_n$. Thus, $G^{m+1}$ has a free product structure over $G$.

•

Assume that $G^1$ has a surface flat structure over $\mathbb{F}$. We consider the graph of groups with the same data as in $\mathcal{G}(G^1,\mathbb{F})$ apart from replacing the vertex group $\mathbb{F}$ by the amalgamated free product $G*_{\mathbb{F}}\mathbb{F}$. Then the fundamental group of this graph of groups is isomorphic to $G^{m+1}$, and together with the retraction $r_{m+1}:G^{m+1}\rightarrow G^m$ that agrees with $r_1$ on $G^1$ and stays the identity on $G$, it witnesses that $G^{m+1}$ has a surface flat structure over $G^m$.

Inductive step.

Assume that the result holds for all $G^{m+1},\ldots , G^{m+i}$. We will show that it holds for $G^{m+i+1}$. We take cases according to whether $G^{i+1}$ is a free product or has a surface flat structure or an abelian flat structure over $G^i$:

•

Assume that $G^{i+1}=G^i*\mathbb{F}_n$; then $G^{m+i+1}=G^{i+1}*_{\mathbb{F}}G=G^i*_{\mathbb{F}}G*\mathbb{F}_n=G^{m+i}*\mathbb{F}_n$. Thus, $G^{m+i+1}$ is a free product of $G^m$ with $\mathbb{F}_n$ and it satisfies the conditions of being part of a tower with the already given sequence of floors.

•

Assume that $G^{i+1}$ has a surface flat structure $\mathcal{G}(G^{i+1},G^i)$ over $G^i$. Consider the graph of groups decomposition with the same data as in $\mathcal{G}(G^{i+1},G^i)$, apart from replacing the vertex group $G_u$ that contains $\mathbb{F}$ with the amalgamated free product $G_u*_{\mathbb{F}}G$. The fundamental group of this graph of groups is isomorphic to $G^{m+i+1}$, and together with the retraction $r_{m+i+1}:G^{m+i+1}\rightarrow G^{m+i}$ that agrees with $r_{i+1}$ on $G^{i+1}$ and stays the identity on $G$, it witnesses that $G^{m+i+1}$ has a surface flat structure over $G^{m+i}$.

•

Assume that $G^{i+1}$ has an abelian flat structure $G^i*_A(A\oplus \mathbb{Z}^n)$ over $G^i$. Consider the amalgamated free product $(G^i*_{\mathbb{F}}G)*_A(A\oplus \mathbb{Z}^n)$. This is a splitting of $G^{m+i+1}$, and moreover, by Lemma 4.19, $A$ is maximal abelian in $G^i*_{\mathbb{F}}G$. A maximal abelian group of $G^{m+i}$ that contains a peg of a previous abelian flat must live either in $G^i$ or in $G$. Now it is enough to observe that $A$ cannot be conjugated to a maximal abelian group of $G$ and if a conjugate $A^{g}$ of $A$ intersects nontrivially another maximal abelian group of $G^i$, then $g\in G^i$. In addition, we define the map $r_{m+i+1}:G^{m+i+1}\twoheadrightarrow G^{m+i}$ to agree with $r_{i+1}$ on $G^{i+1}$ and stay the identity on $G$.■

Example 4.21.

Let $G\coloneq \langle x_1,x_2,e_1,e_2,z_1,z_2 \ | \ [x_1,x_2]\cdot [e_1,e_2], [x_1^3x_2^4,z_1], [x_1^3x_2^4,z_1], {} [z_1,z_2] \rangle$. We can give $G$ a tower structure over $\mathbb{F}_2\coloneq \langle e_1,e_2\rangle$ as follows. The tower consists of two floors:

•

The first floor is just a surface flat that is obtained by gluing $\Sigma _{1,1}$, whose fundamental group $\pi _1(\Sigma _{1,1})$ is $\langle x_1,x_2, s \ | \ s^{-1}\cdot [x_1,x_2]\rangle$, along its boundary to the subgroup $\langle [e_1,e_2]\rangle$ of the free group $\mathbb{F}_2$. In group theoretic terms the first floor is the amalgamated free product $G^1\coloneq \mathbb{F}_2*_{[e_1,e_2]=s}\pi _1(\Sigma _{1,1})$. The retraction $r_1$ sends $x_i$ to $e_i$ and stays the identity on $e_i$.

•

The second floor is just an abelian flat that is obtained by gluing $\mathbb{Z}^2$ along the (maximal) abelian subgroup $\langle x_1^3x_2^4\rangle$ of $G^1$. In group theoretic terms the second floor is the amalgamated free product $G^2\coloneq G^1*_{x_1^3x_2^4=z}(\langle z\rangle \oplus \mathbb{Z}^2)$. The retraction $r_2$ sends $z_1$ and $z_2$ to $z$ and stays the identity on $G^1$.

The group $G*_{\mathbb{F}_2}G$ can be given a twin tower structure as follows. This tower has four floors, which we describe:

•

The first two floors are identical to the floors of the tower structure of $G$.

•

The third floor is just a surface flat that is obtained by gluing $\Sigma _{1,1}$, whose fundamental group $\pi _1(\Sigma _{1,1})$ is $\langle y_1,y_2, s \ | \ s^{-1}\cdot [y_1,y_2]\rangle$, along its boundary to the subgroup $\langle [e_1,e_2]\rangle$ of the free group $\mathbb{F}_2$. In group theoretic terms the third floor is the amalgamated free product $G^3\coloneq G^2*_{[e_1,e_2]=s}\pi _1(\Sigma _{1,1})$. The retraction $r_3$ sends $y_i$ to $e_i$ and stays the identity on $G^2$.

•

The fourth floor is just an abelian flat that is obtained by gluing $\mathbb{Z}^2$ along the (maximal) abelian subgroup $\langle y_1^3y_2^4\rangle$ of $G^3$. In group theoretic terms the fourth floor is the amalgamated free product $G^4\coloneq G^3*_{y_1^3y_2^4=z}(\langle z\rangle \oplus \mathbb{Z}^2)$. The retraction $r_4$ sends $z'_1$ and $z'_2$ to $z$ and stays the identity on $G^3$.

Proposition 4.22 (Twin tower - abelian case).

Suppose $G$ has the structure of a tower $\mathcal{T}(G,\mathbb{F})$ (of height $m$) over $\mathbb{F}$. Assume that the first floor $\mathcal{G}(G^1,G^0)$ is an abelian floor (see Figure 7).

Let $G^1_{Db}$ be the double of $G^1$ with respect to $\mathcal{G}(G^1,G^0)$ and let $f:G^1\hookrightarrow G^1_{Db}$ be the nonidentity natural embedding of $G^1$ into $G^1_{Db}$.

Let $G_{Db}$ be the double of $G$ with respect to $\mathcal{G}(G^1,G^0)$, and let $_fG_{Db}$ be the amalgamated free product $G^1_{Db}*_{G^1}G$, where $f_{\bar{e}}:G^1\rightarrow G^1_{Db}$ is $f$ and $f_e:G^1\rightarrow G$ is the identity map. Then the amalgamated free product $G_{Db}*_{G^1_{Db}}(_fG_{Db})$ admits a natural tower structure over $\mathbb{F}$.

Proof.

Suppose that $((\mathcal{G}(G^1,G^0),r_1),(\mathcal{G}(G^2,G^1),r_2),\ldots ,(\mathcal{G}(G^m,G^{m-1}),r_m))$ witnesses that $G$ has the structure of a tower over $\mathbb{F}$.

Let

$$\begin{equation*} \mathcal{T}(G_{Db},\mathbb{F})\coloneq ((\mathcal{G}(G^1_{Db},G^0),r^{Db}_1),(\mathcal{G}(G^2_{Db},G^1_{Db}),r^{Db}_2),\ldots ,(\mathcal{G}(G^m_{Db},G^{m-1}_{Db}),r^{Db}_m)) \end{equation*}$$

be the natural tower structure (see Lemma 4.16) of $G_{Db}$ over $\mathbb{F}$, and let

$$\begin{equation*} \mathcal{T}(_fG_{Db},G^1_{Db})\coloneq ((\mathcal{G}(_fG^2_{Db},G^1_{Db}),r^{f}_2),\ldots ,(\mathcal{G}(_fG^m_{Db}, _fG^{m-1}_{Db}),r^{f}_m)) \end{equation*}$$

be the natural tower structure (see Remark 4.17) of $_fG_{Db}$ over $G^1_{Db}$.

For each $i< m$, let $G^{m+i}_{Db}=G_{Db}*_{G^1_{Db}}(_fG^{i+1}_{Db})$. We claim that there exists a natural sequence of floors

$$\begin{multline*} ((\mathcal{G}(G^1_{Db},G^0),r^{Db}_1),(\mathcal{G}(G^2_{Db},G^1_{Db}),r^{Db}_2),\ldots ,(\mathcal{G}(G^m_{Db},G^{m-1}_{Db}),r^{Db}_m), \\ (\mathcal{G}(G^{m+1}_{Db},G^m_{Db}), r^{Db}_{m+1}),\ldots , (\mathcal{G}(G^{2m},G^{2m-1}),r^{Db}_{2m})) \end{multline*}$$

that witnesses that $G_{Db}*_{G^1_{Db}}(_fG_{Db})$ has a tower structure over $\mathbb{F}$. We construct this sequence starting with the $m$ floors of the tower $\mathcal{T}(G_{Db},\mathbb{F})$ and we proceed recursively as follows:

Base step.

We show that $G_{Db}^{m+1}$ is a free product or has a surface flat structure or has an abelian flat structure over $G_{Db}^m$ according to whether $_fG^2_{Db}$ is a free product or has a surface flat structure or has an abelian flat structure over $G^1_{Db}$:

•

Assume that $_fG^2_{Db}=G^1_{Db}*\mathbb{F}_n$; then $G^{m+1}_{Db}=G_{Db}*_{G^1_{Db}}(G^1_{Db}*\mathbb{F}_n)=G_{Db}*\mathbb{F}_n$.

•

Assume that $_fG^2_{Db}$ has a surface flat structure $(\mathcal{G}(_fG^2_{Db},G^1_{Db}),r^{f}_2)$ over $G^1_{Db}$. We consider the graph of groups with the same data as $\mathcal{G}(_fG^2_{Db},G^1_{Db})$ apart from replacing the vertex group $G^1_{Db}$ with the group $G_{Db}$. The fundamental group of this latter graph of groups is $G^{m+1}_{Db}$, and together with the retraction $r_{m+1}^{Db}:G^{m+1}_{Db}\rightarrow G_{Db}$ that agrees with $r_2^f$ on $_fG^2_{Db}$ and stays the identity on $G_{Db}$, it witnesses that $G^{m+1}_{Db}$ has a surface flat structure over $G_{Db}$.

•

Assume that $_fG^2_{Db}$ has an abelian flat structure $G^1_{Db}*_A (A\oplus \mathbb{Z}^n)$ over $G^1_{Db}$. Consider the amalgamated free product $(G_{Db}*_{G^1_{Db}}G^1_{Db})*_A(A\oplus \mathbb{Z}^n)$; this is a splitting of $G^{m+1}_{Db}$. By definition $A$ cannot be conjugated to any other peg of some abelian flat of $\mathcal{T}(G_{Db},\mathbb{F})$; thus it is maximal abelian in $G_{Db}$ and satisfies the properties that make $G_{Db}*_A(A\oplus \mathbb{Z}^n)$ the $m+1$-th floor of our tower.

Recursive step.

Suppose we have constructed the $m+i$-th floor of the tower. We show that $G_{Db}^{m+i+1}$ is a free product or has a surface flat structure or an abelian flat structure over $G_{Db}^{m+i}$ according to whether $_fG^{i+2}_{Db}$ is a free product or has a surface flat structure or an abelian flat structure over $_fG^{i+1}_{Db}$:

•

Assume that $_fG^{i+2}_{Db}=\ _fG^{i+1}_{Db}*\mathbb{F}_n$; then $G^{m+1}_{Db}=G_{Db}*_{G^1_{Db}}(G^{i+1}_{Db}*\mathbb{F}_n)=G^{m+i}_{Db}*\mathbb{F}_n$.

•

Assume that $_fG^{i+2}_{Db}$ has a surface flat structure $(\mathcal{G}(_fG^{i+2}_{Db},\ _fG^{i+1}_{Db}),r^{f}_{i+2})$ over $_fG^{i+1}_{Db}$. We consider the graph of groups with the same data as $\mathcal{G}(_fG^{i+2}_{Db},G^{i+1}_{Db})$, apart from replacing the vertex group $G_u$ that contains $G^1_{Db}$ with the group $G_{Db}*_{G^1_{Db}}G_u$. The fundamental group of this latter graph of groups is $G^{m+i+1}_{Db}$, and together with the retraction $r_{m+i+1}^{Db}:G^{m+i+1}_{Db}\rightarrow G^{m+i}_{Db}$ that agrees with $r_{i+2}^f$ on $_fG^{i+2}_{Db}$ and stays the identity on $G_{Db}$, it witnesses that $G^{m+i+1}_{Db}$ has a surface flat structure over $G^{m+i}_{Db}$.

•

Assume that $_fG^{i+2}_{Db}$ has an abelian flat structure $_fG^{i+1}_{Db}*_A(A\oplus \mathbb{Z}^n)$ over $_fG^{i+1}_{Db}$. We consider the amalgamated free product $(G_{Db}*_{G^1_{Db}}(_fG^{i+1}_{Db}))*_A(A\oplus \mathbb{Z}^n)$; this is a splitting of $G^{m+i+1}_{Db}$. Since $A$ cannot be conjugated to any other previous peg of the tower $\mathcal{T}(G_{Db},\mathbb{F})$ and the tower $\mathcal{T}(_fG_{Db},G^1_{Db})$, we see that $A$ is maximal abelian in $G_{Db}*_{G^1_{Db}}(_fG^{i+1}_{Db})$ and it satisfies the properties that make $G^{m+i}_{Db}*_A(A\oplus \mathbb{Z}^n)$ the $m+i+1$-th floor of our tower.■

Example 4.23.

Let $G\coloneq \langle e_1,e_2,z_1,z_2,x_1,x_2 \ | \ [z_1,z_2], [z_1,e_1^2e_2^2],[z_2,e_1^2e_2^2], [x_1,x_2]\cdot [z_1,e_1]\rangle$. We can give $G$ a tower structure over $\mathbb{F}_2\coloneq \langle e_1,e_2\rangle$ as follows. The tower $\mathcal{T}(G,\mathbb{F}_2)$ consists of two floors:

•

The first floor is just an abelian flat obtained by gluing $\mathbb{Z}^2$ along the maximal abelian group $\langle e_1^2e_2^2 \rangle$ of $\mathbb{F}_2$. In group theoretic terms, the group corresponding to the first floor is the amalgamated free product $G^1\coloneq \mathbb{F}_2*_{e_1^2e_2^2=z}(\langle z\rangle \oplus \mathbb{Z}^2)$. The retraction $r_1:G^1\twoheadrightarrow \mathbb{F}_2$ sends $z_1$ and $z_2$ to $z$ and stays the identity on $\mathbb{F}_2$.

•

The second floor is just a surface flat obtained by gluing $\Sigma _{1,1}$, whose fundamental group is $\langle x_1,x_2,s \ | \ s^{-1}[x_1,x_2]\rangle$, along its boundary to the subgroup $\langle [z_1,e_1]\rangle$ of $G^1$. In group theoretic terms the group corresponding to the second floor is the amalgamated free product $G^2\coloneq G^1*_{[z_1,e_1]=s}\pi _1(\Sigma _{1,1})$. The retraction $r_2:G^2\twoheadrightarrow G^1$ sends $x_1$ to $z_1$, $x_2$ to $e_1$, and stays the identity on $G^1$.

We now consider the double of $G^1$ with respect to the splitting $\mathcal{G}(G^1,\mathbb{F})$ of the first point above. As a group $G^1_{Db}$ has the following presentation:

$$\begin{equation*} \langle e_1,e_2,z_1,z_2,y_1,y_2 \ | \ [z_i,y_j] \ i,j\leq 2, [z_1,e_1^2e_2^2],[z_2,e_1^2e_2^2], [y_1,e_1^2e_2^2],[y_2,e_1^2e_2^2] \rangle . \end{equation*}$$

It can be seen as the amalgamated free product $G^1_{Db}\coloneq \mathbb{F}_2*_{e_1^2e_2^2=z}(\langle z\rangle \oplus \mathbb{Z}^4)$.

The twin tower that corresponds to $\mathcal{T}(G,\mathbb{F}_2)$ consists of three floors as follows:

•

The first floor is the floor double of $\mathcal{G}(G^1,\mathbb{F}_2)$, and the group corresponding to this floor is $G^1_{Db}$. The retraction $r_1^{Db}$ sends each $z_1,z_2,y_1,y_2$ to $z$ and stays the identity on $\mathbb{F}_2$.

•

The second floor is just a surface flat obtained by gluing $\Sigma _{1,1}$, whose fundamental group is $\langle x_1,x_2,s \ | \ s^{-1}[x_1,x_2]\rangle$, along its boundary to the subgroup $\langle [z_1,e_1]\rangle$ of $G^1_{Db}$. In group theoretic terms the group corresponding to the second floor is the amalgamated free product $G^2_{Db}\coloneq G^1_{Db}*_{[z_1,e_1]=s}\pi _1(\Sigma _{1,1})$. The retraction $r_2^{Db}:G^2_{Db}\twoheadrightarrow G^1_{Db}$ sends $x_1$ to $z_1$, $x_2$ to $e_1$, and stays the identity on $G^1_{Db}$.

•

The third floor is again a surface flat obtained by gluing $\Sigma _{1,1}$, whose fundamental group is $\langle p_1,p_2,s \ | \ s^{-1}[p_1,p_2]\rangle$, along its boundary to the subgroup $\langle [y_1,e_1]\rangle$ of $G^2_{Db}$. In group theoretic terms the group corresponding to the second floor is the amalgamated free product $G^3_{Db}\coloneq G^2_{Db}*_{[y_1,e_1]=s}\pi _1(\Sigma _{1,1})$. The retraction $r_3^{Db}:G^3_{Db}\twoheadrightarrow G^2_{Db}$ sends $p_1$ to $y_1$, $p_2$ to $e_1$, and stays the identity on $G^2_{Db}$.

The group corresponding to the twin tower has presentation

$$\begin{multline*} \langle e_1,e_2,z_1,z_2,y_1,y_2,x_1,x_2,p_1,p_2 \ | \ [z_i,y_j] \ i,j\leq 2, [z_1,e_1^2e_2^2],[z_2,e_1^2e_2^2], [y_1,e_1^2e_2^2], \\ [y_2,e_1^2e_2^2], [x_1,x_2]\cdot [z_1,e_1],[p_1,p_2]\cdot [y_1,e_1] \rangle . \end{multline*}$$

4.3. Closures of towers

We pass to the notion of a tower closure. We first define the notion of an abelian floor closure.

Definition 4.24 (Abelian floor closure).

Suppose $G$ has the structure of an abelian floor over a limit group $L$ and $\mathcal{G}(G,L)$ is the splitting witnessing it. Let $\{\mathbb{Z}^{m_i}\}_{i\in I}$ be the collection of the free abelian groups that we glue along the corresponding pegs $\{E_i\}_{i\in I}$ in forming the abelian flats of the floor.

Let $\{A^{m_i}\}_{i\in I}$ be free abelian groups and let $f_i:E_i\oplus \mathbb{Z}^{m_i}\rightarrow E_i\oplus A^{m_i}$ be an embedding with $f_i\upharpoonright E_i=\mathrm{Id}$ and such that $f_i(E_i\oplus \mathbb{Z}^{m_i})$ is a finite index subgroup of $E_i\oplus A^{m_i}$ for every $i\in I$. We call $\{f_i\}_{i\in I}$ a family of closure embeddings.

We denote by $cl(G)$ the group that is the fundamental group of a graph of groups in which all the data is as in $\mathcal{G}(G,L)$ apart from replacing $\{\mathbb{Z}^{m_i}\}_{i\in I}$ by $\{\mathbb{Z}^{m_i}\oplus A^{m_i}\}_{i\in I}$ and adding to the vertex groups $\{E_i\oplus \mathbb{Z}^{m_i}\oplus A^{m_i}\}_{i\in I}$ the relations corresponding to the family of closure embeddings; i.e., $z=f_i(z)$ for every $z\in E\oplus \mathbb{Z}^{m_i}$. Moreover, we call the latter graph of groups, $\mathcal{G}(cl(G),L)$, the floor closure of $\mathcal{G}(G,L)$ with respect to $\{f_i\}_{i\in I}$.

In the following paragraph we identify closure embeddings with finite-index subgroups of a “formal” free abelian group $\mathbb{A}^m\coloneq \langle (1,0,\ldots ,0),\ldots ,(0,0,\ldots ,1)\rangle$. Such an identification will be of use when we wish to understand when a homomorphism from a tower $h:G\to \mathbb{F}$ extends to $cl(G)$. For simplicity and in accordance with our convention on tower structures, we phrase it for an abelian flat over $\mathbb{F}$, but it holds for an arbitrary tower.

Remark 4.25.

Suppose $G$ has the structure of an abelian flat over $\mathbb{F}$, so $G$ is isomorphic to the amalgamated product of $\mathbb{F}$ with $\left< c\right> \oplus \mathbb{Z}^m$ over $c$. Let $\gamma$ generate the centraliser of (the image of) $c$ in $\mathbb{F}$, so that $c=\gamma ^l$ for some $l$, and let $f:\mathbb{Z}^m \to A^m$ be a closure embedding. For each $i\leq m$, $f(z_i)=c^{k_{i,0}}a_1^{k_{i,1}}\cdots a_m^{k_{i,m}}$ for some $k_{i,j}\in \mathbb{Z}$. Denote by $K = (k_{i,j})_{1\leq i,j\leq m}$ the coefficient matrix. Since the image of $f$ is of finite index in $\mathbb{A}^m$, $\det K \neq 0$, and so the image of $K$ (considered as a linear transformation on $\mathbb{A} ^m$) is of finite index in $\mathbb{A}^m$.

Now let $h:G\to \mathbb{F}$ be a morphism which restricts to the identity on $\mathbb{F}$. Then $h$ sends each $z_i$ to some power $x_i$ of $\gamma$, and we can assign to $h$ the vector $(x_1,\dots ,x_m)$. If $h$ extends to a morphism $H:cl(G)\to \mathbb{F}$, then there are integers $(y_1,\dots ,y_m)$ such that $H(a_i)=\gamma ^{y_i}$ and $x_i = k_{i,0}l+k_{i,1}y_1+\cdots + k_{i,m}y_m$. Clearly, the existence of such $y_i$’s is also a sufficient condition for $h$ to extend, so $h$ extends to $cl(G)$ if and only if $(x_i-l\cdot k_{i,0})_{i=1}^m$ belongs to $\operatorname {Im}(K)$ or $(x_i,\ldots ,x_m)$ belongs to $(l\cdot k_{1,0},\ldots ,l\cdot k_{m,0})+\operatorname {Im}(K)$.

Example 4.26.

Consider the amalgamated product $G=\mathbb{F}_2*_{e_1^5=c}(\langle c\rangle \oplus \mathbb{Z}^2)$. This group admits the structure of an abelian flat over $\mathbb{F}_2\coloneq \langle e_1,e_2\rangle$.

Let $f:\langle c\rangle \oplus \mathbb{Z}^2 \rightarrow \langle c\rangle \oplus A^2$ be the morphism that restricts to the identity on $c$ and sends $z_1$ to $c^2a_1^3a_2$ and $z_2$ to $a_1a_2$. This is a closure embedding, and $cl(G)$ is the amalgamated product $cl(G)=\mathbb{F}_2*_{e_1^5=c}(\langle c\rangle \oplus \mathbb{Z}^2 \oplus A^2 /\langle z_1=c^2a_1^3a_2,z_2=a_1a_2\rangle )$.

A morphism $h:G\rightarrow \mathbb{F}_2$ (that restricts to the identity on $\mathbb{F}_2$) satisfies $h(z_i)=e_1^{x_i}$ for $i=1,2$. It extends to a morphism $H:cl(G)\rightarrow \mathbb{F}_2$ if and only if $x_1=5l+3y_1+y_2$ and $x_2=y_1+y_2$ for some $y_1,y_2\in \mathbb{Z}$. Such $y_1,y_2$ exist if and only if $(x_1-5l,x_2)=y_1(3,1)+y_2(1,1)$; that is, $(x_1,x_2)$ belongs to the coset $[5l,0]$ of the image of $K=\left(\begin{smallmatrix} 3 & 1 \\ 1 & 1 \end{smallmatrix}\right)$.

Remark 4.27.

Suppose $G$ has the structure of a floor over a limit group $L$. Suppose $\{\mathbb{Z}^{m_i}\}_{i\in I}$ is the collection of the free abelian groups that we glue along the corresponding pegs $\{E_i\}_{i\in I}$ in forming the abelian flats of the floor. Let $(\mathcal{G}(cl(G),L), \{f_i\}_i\in I)$ be the closure of $G$ with respect to some family of closure embeddings. Let $G=G^1*\cdots *G^m$ be a free splitting of $G$. Then for each $i\in I$ there exists $\gamma _i\in G$ so that $\gamma _i E_i\oplus \mathbb{Z}^{m_i}\gamma _i^{-1}$ is a subgroup of some $G_j$ for $j\leq m$.

Thus, there exists a free splitting, $H^1*\cdots *H^m$, of $cl(G)$ such that each $H^i$ is the group obtained by gluing $\gamma _iA^{m_i}\gamma _i^{-1}$ in $G^i$ along each maximal abelian group of the form $\gamma _i E_i\oplus \mathbb{Z}^{m_i}\gamma _i^{-1}$ that is contained in $G^i$ and adding the relations in this new vertex group according to the family of the closure embeddings; i.e., $\gamma _i z\gamma _i^{-1}=\gamma _if_i(z)\gamma _i^{-1}$ for every $z\in E_i\oplus \mathbb{Z}^{m_i}$.

Definition 4.28 (Tower closure).

Let $\mathcal{T}(G,\mathbb{F})$ be a tower of height $m$. Let, for $i<m$, $\mathcal{G}(G^{i+1},G^i)$ be the $i$-th floor, and let $\{f^i_j\}_{j\in J}$ be a family of closure embeddings (in the case where the $i$-th floor is a free product or a hyperbolic floor we take the family to be empty). Then the tower closure, $cl(\mathcal{T}(G,\mathbb{F}))$, of the tower $\mathcal{T}(G,\mathbb{F})$ with respect to the previous families of closure embeddings for each floor of the tower is defined recursively as follows.

Base step.

The first floor consists of the floor closure $\mathcal{G}(cl(G^1),\mathbb{F})$ with respect to $\{f^1_j\}_{j\in J}$.

Recursive step.

Let $G^i_{cl}$ be the group that corresponds to the $i$-th floor of the tower closure.Then $G^{i+1}_{cl}$ is the fundamental group of the graph of groups in which:

•

the underlying graph is the same as the underlying graph of $\mathcal{G}(G^{i+1},G^{i})$;

•

the vertex groups $G^i_1,\ldots ,G^i_m$ whose free product is $G^i$ are replaced by the corresponding groups in $G^i_{cl}$;

•

the abelian flats $\mathbb{Z}^{m_j}$ are replaced by $A^{m_j}\oplus \mathbb{Z}^{m_j}$; and

•

in the new abelian vertex groups $E_j\oplus A^{m_j}\oplus \mathbb{Z}^{m_j}$ we add relations according to the family of closure embeddings; i.e., $z=f^{i+1}_j(z)$ for every $z\in E_j\oplus \mathbb{Z}^{m_j}$.

It is not hard to see the following.

Lemma 4.29.

Let $\mathcal{T}(G,\mathbb{F})$ be a tower over $\mathbb{F}$. Then $cl(\mathcal{T}(G,\mathbb{F}))$ with respect to any families of closure embeddings is a tower over $\mathbb{F}$.

4.4. Symmetrizing closures of twin towers

When moving to the closure of a twin tower it could be the case that “twin” abelian flats that appear in the floors of the twin tower embed in different ways in the ambient free abelian groups. For example, if $E\oplus \mathbb{Z}^2$ is an abelian flat generated by $z_1,z_2$ and $\hat{E}\oplus \hat{\mathbb{Z}^2}$ is its twin generated by $\hat{z_1},\hat{z_2}$, then we could have the closure embeddings $f:E\oplus \mathbb{Z}^2\rightarrow E\oplus \mathbb{A} ^2$ which correspond to $(1,2)+\operatorname {Im}(K),K=\left(\begin{smallmatrix} 3 & 1\\ 1 & 1 \end{smallmatrix}\right)$ and $\hat{f}:\hat{E}\oplus \hat{\mathbb{Z}^2}\rightarrow \hat{E}\oplus \hat{\mathbb{A}^2}$ which correspond to $(3,4)+\operatorname {Im}(\hat{K}),\hat{K}=\left(\begin{smallmatrix} 4 & 5\\ 1 & 3 \end{smallmatrix}\right)$. We would like the images of $z_i, \hat{z_i}$ under $f,\hat{f}$ to correspond to the same words in the generators of a closure. This is achieved by closure-embedding $\mathbb{A}^2, \hat{\mathbb{A}^2}$ in $\mathbb{B}^2, \hat{\mathbb{B}^2}$ with corresponding closure-embeddings $g, \hat{g}$ with coefficient matrices $L, \hat{L}$ such that $KL=\hat{K}\hat{L}$. In this case, the closure embedding $g \circ f$ corresponds to $(1,2)+\operatorname {Im}(KL)$, and $\hat{g}\circ \hat{f}$ corresponds to $(3,4)+\operatorname {Im}(KL)$. Both columns of $KL=\hat{K}\hat{L}$ are in the image of $K$, so finding $L,\hat{L}$ comes down to finding two vectors in the images of both $K$ and $\hat{K}$, which are linearly independent. This can be done, since $\operatorname {Im}(K), \operatorname {Im}(\hat{K})$ are of finite index in $\mathbb{A}^2$, and so is their intersection.

We define the symmetric closure of a closure of a twin tower as follows.

Definition 4.30.

Suppose $G$ has the structure of a tower (of height $m$) $\mathcal{T}(G,\mathbb{F})$ and that the first floor $\mathcal{G}(G^1,G^0)$ is an abelian floor. Enumerate the abelian flats in $G$: $E_1\oplus \mathbb{Z}_1 ^{m_1},\ldots ,E_s\oplus \mathbb{Z}_s ^{m_s}$. Let $\mathcal{T}\#\mathcal{T}(G,\mathbb{F})$ be the twin tower of $\mathcal{T}(G,\mathbb{F})$. For each $1< i\leq s$, let $\hat{E_i}\oplus \hat{\mathbb{Z}^{m_i}}$ be the “twin” of the $i$-th flat.

Let $cl(\mathcal{T}\#\mathcal{T}(G,\mathbb{F}))$ be a closure with respect to some closure embeddings $\{(f_i,\hat{f}_i)_{1\leq i\leq s}\}$, where $f_i,\hat{f_i}$ are defined on $E_1\oplus \mathbb{Z}_i ^{m_i},\hat{E_i}\oplus \hat{\mathbb{Z}^{m_i}}$, respectively, and correspond to $p_i+\operatorname {Im}(K_i), \hat{p_i}+\operatorname {Im}(\hat{K_i})$. Let $L_i,\hat{L_i} \in M_{m_i}(\mathbb{Z})$ be such that $K_iL_i=\hat{K_i}\hat{L_i}$ for each $i$. Let $\{g_i,\hat{g_i}\}_{1\leq i\leq s}$ be the closure embedding corresponding to $p_i+\operatorname {Im}(K_iL_i),\hat{p_i}+\operatorname {Im}(K_iL_i)$ for each $i$. Then $\{g_i,\hat{g_i}\}_{1\leq i\leq s}$ is called the symmetric closure embedding of $\{f_i,\hat{f_i}\}_{1\leq i\leq s}$.

Lemma 4.31.

The symmetric closure of the closure of a twin tower is a closure of its closure. Moreover, a homomorphism $h:G \to \mathbb{F}$ of a twin tower extends to the symmetric closure if and only if for each $i$ the vector $(x_1,\ldots ,x_{m_i})$ corresponding to $h \restriction \mathbb{Z}^{m_i}$ is in $p_i+\operatorname {Im}(K_iL_i)$ (respectively, $h \restriction \hat{\mathbb{Z}^{m_i}}$ is in $\hat{p_i}+\operatorname {Im}(K_iL_i)$). As a consequence, if $h:G\to \mathbb{F}$ extends to a closure of $G$, it also extends to its symmetric closure.

5. Solid limit groups and strictly solid morphisms

In this section we record the definitions of a strictly solid morphism and a family of such morphism as given by Sela. A strictly solid morphism is a morphism from a solid limit group to a free group that satisfies certain conditions. These morphisms are of fundamental importance in the work of Sela in answering Tarski’s question: first because of a boundedness result (see 17, Theorem 2.9) and second because in contrast to solid morphisms they are first order definable.

The definition of a strictly solid morphism requires a technical construction, called the completion of a strict map. The next subsection explains this construction.

In subsection 5.2 we define the above special class of morphisms and their families.

5.1. Completions

We start by modifying a $\mathrm{GAD}$ for a limit group $G$ in order to simplify the conditions for a map $\eta :G\twoheadrightarrow L$ to be strict with respect to it. The goal is to transform the $\mathrm{GAD}$ in a way that the rigid vertex groups will be enlarged to their envelopes, every edge group connecting two rigid vertex groups will be maximal abelian in both vertex groups of the one edged splitting induced by its edge (and after replacing all abelian vertex groups with their peripheral subgroups), and abelian vertex groups will be leaves connected through a rigid vertex to the rest of the graph.

The following lemma of Sela will be helpful.

Lemma 5.1 (Sela).

Let $L$ be a limit group and let $M$ be a noncyclic maximal abelian subgroup. Then:

•

if $L$ admits an amalgamated free product splitting with abelian edge group, then $M$ can be conjugated into one of the factors;

•

if $L$ admits an $HNN$ extension splitting $A*_C$ where $C$ is abelian, then either $M$ can be conjugated into $A$ or $M$ can be conjugated in $\langle C,t\rangle$ and $L=A*_C\langle C,t\rangle$.

Lemma 5.2.

Let $G$ be a limit group and let $\Delta \coloneq (\mathcal{G}(V,E),(V_S,V_A,V_R))$ be a $GAD$ for $G$, where the image of each edge group is maximal abelian in at least one vertex group of the one edged splitting induced by the edge. Assume moreover that $V_A$ is empty. Then there exists a $GAD$ $\hat{\Delta }\coloneq (\hat{\mathcal{G}}(\hat{V},\hat{E}),(\hat{V}_S, \hat{V}_A, \hat{V}_R))$ for $G$ satisfying the following properties:

(1)

the underlying graph $\hat{\mathcal{G}}$ is the same as $\mathcal{G}$ up to some sliding of edges;

(2)

the set of rigid vertices and the set of surface type vertices are the same;

(3)

every rigid vertex group of the graph of groups $\hat{\mathcal{G}}$ coincides with the envelope of the corresponding rigid vertex group in $\mathcal{G}$;

(4)

the image of every edge group connecting two rigid vertices of $\hat{V}_R$ is maximal abelian in both vertex groups in the splitting induced by the edge.

Remark 5.3.

Suppose $\eta :G\rightarrow L$ is a strict map with respect to the $\mathrm{GAD}$ $\Delta$ for $G$. We consider the following modification of $\Delta$:

Step 1.

We replace every vertex group $G_u$ with $u\in V_A$ by its peripheral subgroup and we place $u$ in the set $V_R$ (i.e., we consider it rigid); we call this $\mathrm{GAD}$ $\Delta _0$.

Step 2.

We modify $\Delta _0$ according to Lemma 5.2 in order to obtain $\hat{\Delta }_0$.

Step 3.

To every rigid vertex in $\hat{\Delta }_0$ whose vertex group was a peripheral subgroup in $\Delta _0$ we attach an edge whose edge group is the peripheral subgroup itself, and the vertex group on its other end is the abelian group that contained the peripheral subgroup in $\Delta$. These new vertex groups will be abelian type vertex groups. We denote by $\hat{\Delta }$ this $\mathrm{GAD}$ for $G$.

We will either explicitly or implicitly use the above modification for the rest of the paper.

Definition 5.4.

Let $G$ be a group and let $\mathcal{G}(G)$ be a $\mathrm{GAD}$ for $G$ with $m$ edges and at least one rigid vertex. Let $\eta :G\twoheadrightarrow H$ and let $(\mathcal{G}(G),\eta )$ be strict.

Let $\mathcal{G}_0\subseteq \mathcal{G}_1\subseteq \cdots \subseteq \mathcal{G}_m\coloneq \mathcal{G}(G)$ be a sequence of subgraphs of groups such that $\mathcal{G}_i$ has $i$ edges. We define the group $Comp_i$ together with its splitting $\mathcal{G}(Comp_i)$ by the following recursion:

Base step.

The subgraph of groups $\mathcal{G}_0$ consists of a single vertex $V$ which we may assume is rigid. Then $Comp_0$ is the group $H$ and $\mathcal{G}(Comp_0)$ is the trivial splitting.

Recursive step.

Let $e_{i+1}$ be the edge in $\mathcal{G}_{i+1}\setminus \mathcal{G}_i$. We take cases:

Case 1.

Suppose $e_{i+1}$ connects two rigid vertices. We further take cases:

1A.

Assume that the centralizer of $\eta (G_{e_{i+1}})$ in $Comp_i$ cannot be conjugated either to the centralizer of $\eta (G_{e})$ for some edge $e$ in $\mathcal{G}_i$ that connects two rigid vertex groups or to the centralizer of the image of the peripheral subgroup of some abelian vertex group in $\mathcal{G}_i$. Then $Comp_{i+1}$ is the fundamental group of the graph of groups obtained by gluing to $\mathcal{G}(Comp_i)$ a free abelian flat of rank $1$ along the centralizer of $\eta (G_{e_{i+1}})$ in $Comp_i$. The latter graph of groups is $\mathcal{G}(Comp_{i+1})$.

1B.

Assume that the centralizer of $\eta (G_{e_{i+1}})$ in $Comp_i$ can be conjugated to the centralizer $C$ (in $Comp_i$) either of $\eta (G_{e})$ for some edge $e$ in $\mathcal{G}_i$ that connects two rigid vertex groups or of the image of the peripheral group of some abelian vertex group (i.e., a vertex group whose vertex belongs to $V_A$) in $\mathcal{G}_i$. Then $Comp_i$ is the fundamental group of the graph of groups obtained by gluing to $\mathcal{G}(Comp_i)$ a free abelian flat of rank $1$ along $C$.

Case 2.

Suppose $e_{i+1}$ connects a rigid vertex with a free abelian vertex group of rank $n$ and let $P$ be its peripheral subgroup. We may assume that the free abelian vertex group is not in $\mathcal{G}_i$ and we further take cases:

2A.

Assume that $\eta (P)$ cannot be conjugated either to the centralizer of $\eta (G_{e})$ for some edge $e$ in $\mathcal{G}_i$ that connects two rigid vertex groups or to the centralizer of the image of the peripheral subgroup of some abelian vertex group in $\mathcal{G}_i$. Then $Comp_{i+1}$ is the fundamental group of the graph of groups obtained by gluing to $\mathcal{G}(Comp_i)$ a free abelian flat of rank $n$ along the centralizer of $\eta (P)$ in $Comp_i$, and moreover in the new abelian vertex group we add the relations identifying the peripheral subgroup in its centralizer. The latter graph of groups is $\mathcal{G}(Comp_{i+1})$.

2B.

Assume that $\eta (P)$ can be conjugated to the centralizer $C$ (in $Comp_i$) either of $\eta (G_{e})$ for some edge $e$ in $\mathcal{G}_i$ that connects two rigid vertex groups or of the image of the peripheral group of some abelian vertex group (i.e., a vertex group whose vertex belongs to $V_A$) in $\mathcal{G}_i$. Then $Comp_{i+1}$ is the group corresponding to the graph of groups obtained by gluing to $\mathcal{G}(Comp_i)$ a free abelian flat of rank $n$ along $C$.

Case 3.

Suppose $e_{i+1}\coloneq (u,v)$ connects a surface type vertex with a rigid vertex (i.e., a vertex that belongs to $V_R$). We further take cases according to whether the surface type vertex belongs to $\mathcal{G}_i$ or not:

3A.

Assume that the surface vertex group $G_v$ does not belong to $\mathcal{G}_i$. Then $Comp_{i+1}$ is the amalgamated free product $Comp_i*_{G_{e_{i+1}}}\tilde{G}_v$ where $\tilde{G}_v$ is an isomorphic copy of $G_v$ witnessed by the isomorphism $f:G_v\rightarrow \tilde{G}_v$, and the edge group embeddings $\tilde{f}_{e_{i+1}}, \tilde{f}_{\bar{e}_{i+1}}$ are defined as follows: $\tilde{f}_{e_{i+1}}=f\circ f_{e_{i+1}}$ and $\tilde{f}_{\bar{e}_{i+1}}=\eta \circ f_{\bar{e}_{i+1}}$, where $f_{e_i+1}, f_{\bar{e}_{i+1}}$ are the injective morphisms that correspond to the edge group of the splitting $\pi _1(\mathcal{G}_i)*_{G_{e_{i+1}}}G_v$.

3B.

Assume that the surface vertex group $G_u$ belongs to $\mathcal{G}_i$. Then by our recursive hypothesis there exists an isomorphic copy of $G_u$, say $f:G_u\rightarrow \tilde{G}_u$, in $Comp_i$. We define $Comp_{i+1}$ to be the $\mathrm{HNN}$ extension $Comp_i*_{G_{e_{i+1}}}$, where the edge group embeddings $\tilde{f}_{e_{i+1}}, \tilde{f}_{\bar{e}_{i+1}}$ are defined as follows: $\tilde{f}_{e_{i+1}}=\eta \circ f_{e_{i+1}}$ and $\tilde{f}_{\bar{e}_{i+1}}=f\circ f_{\bar{e}_{i+1}}$, where $f_{e_{i+1}},f_{\bar{e}_{i+1}}$ are the injective morphisms that correspond to the edge group of the splitting $\pi _1(\mathcal{G}_i)*_{G_{e_{i+1}}}G_v$ (in the case $v$ is not in $\mathcal{G}_i$) or of the splitting $\pi _1(\mathcal{G}_i)*_{G_{e_{i+1}}}$ (in the case $v$ is in $\mathcal{G}_i$).

Finally the group $Comp(\mathcal{G}(G),\eta )\coloneq Comp_m$ is called the completion of $G$ with respect to $\mathcal{G}(G)$, the sequence $\mathcal{G}_0\subset \cdots \subset \mathcal{G}_m$, and $\eta$.

The completion of a group $G$ with respect to a strict map $\eta :G\rightarrow L$ has a natural structure of a floor over $L$. Moreover, Sela has proved 16, Lemma 1.13 that $G$ admits a natural embedding into its completion.

Lemma 5.5.

Let $G$ be a group and let $(\mathcal{G}(G),(V_S,V_A,V_R))$ be a $\mathrm{GAD}$ for $G$. Let $L$ be a limit group and let $\eta :G\twoheadrightarrow L$ be such that $(\mathcal{G}(G),\eta )$ is strict. Let $Comp(\mathcal{G}(G),\eta )$ be the completion of $G$ with respect to $\mathcal{G}(G)$ and $\eta$. Then $G$ admits a natural embedding to $Comp(\mathcal{G}(G),\eta )$.

Proof.

Let $\mathcal{G}_0\subset \mathcal{G}_1\subset \cdots \subset \mathcal{G}_m\coloneq \mathcal{G}(G)$ be the sequence of subgraphs of groups that covers the graph of groups $\mathcal{G}(G)$ and with respect to which we have constructed the completion $Comp(\mathcal{G}(G),\eta )$. We will prove by induction that for each $i\leq m$, there exists an injective map $f_i:\pi _1(\mathcal{G}_i)\rightarrow Comp_i$ such that $f_{i+1}\supset f_i$ and $\bigcup f_i\coloneq f:G\rightarrow Comp(\mathcal{G}(G),\eta )$ agrees with $\eta$ up to conjugation, by an element that is either trivial or does not live in $L$, in the vertex groups whose vertices belong to $V_R$ in the $\mathrm{GAD}$ for $G$.

Base step.

We take $f_0$ to be $\eta \upharpoonright \pi _1(\mathcal{G}_0)$. Since we have assumed that the unique vertex in $\mathcal{G}_0$ belongs $V_R$ and $Comp_0$ is $L$, one sees that $f_0:\pi _1(\mathcal{G}_0)\rightarrow Comp_0$ is injective and respects our hypothesis on vertex groups whose vertices belong to $V_R$.

Inductive step.

Let $f_i:\pi _1(\mathcal{G}_i)\rightarrow Comp_i$ be the morphism that satisfies our induction hypothesis. We find an injective morphism $f_{i+1}:\pi _1(\mathcal{G}_{i+1})\rightarrow Comp_{i+1}$ that extends $f_i$ and satisfies the hypothesis on vertex groups whose vertices belong to $V_R$. We take cases according to the initial and terminal vertices of the edge $e\coloneq (u,v)\in \mathcal{G}_{i+1}\setminus \mathcal{G}_i$.

•

Suppose we are in Case 1A of Definition 5.4. Then $\pi _1(\mathcal{G}_{i+1})$ is either the amalgamated free product $\pi _1(\mathcal{G}_i)*_{G_e}G_v$ (if $v\notin \mathcal{G}_i$) or the $\mathrm{HNN}$ extension $\pi _1(\mathcal{G}_i)*_{G_e}$ (if $v\in \mathcal{G}_i$) and $Comp_{i+1}$ is the amalgamated free product $Comp_i*_C(C\oplus \langle z\rangle )$ where $C$ is the centralizer of $\eta (G_e)$ in $Comp_i$.

Suppose that $\pi _1(\mathcal{G}_{i+1})$ is an amalgamated free product. By the induction hypothesis $f_i\upharpoonright G_u=Conj(\gamma _u)\circ \eta \upharpoonright G_u$. We define $f_{i+1}$ to agree with $f_i$ on $\pi _1(\mathcal{G}_i)$ and $f_{i+1}(g)=\gamma _u z \eta (g)z^{-1}\gamma _u^{-1}$ for $g\in G_v$. Note that obviously $\gamma _uz$ does not belong to $Comp_i$; thus it does not belong to $L$. The map $f_{i+1}$ is indeed a morphism since for any $g\in G_e$ we have that $f_{\bar{e}}(g)$ is an element that lives in $G_u$; thus $f_{i+1}(f_{\bar{e}}(g))=f_i(f_{\bar{e}}(g))=\gamma _u\eta (f_{\bar{e}}(g))\gamma _u^{-1}$. On the other hand $f_{i+1}(f_{e}(g))=\gamma _uz\eta (f_{e}(g))z^{-1}\gamma _u^{-1}$, and since $\eta (f_e(g))$ is in $C$ we get that $f_{i+1}(f_{e}(g))=\gamma _u\eta (f_e(g))\gamma _u^{-1}$. Therefore, since $\eta (f_{\bar{e}}(g))=\eta (f_e(g))$, we see that $f_{i+1}(f_e(g))=f_{i+1}(f_{\bar{e}}(g))$. We continue by proving that $f_{i+1}$ is injective. Let $g\coloneq a_1b_1\cdots a_nb_n$ be an element of $\pi _1(\mathcal{G}_i)*_{G_e}G_v$ in reduced form. Then $f_{i+1}(g)= f_i(a_1)\gamma _u z \eta (b_1)z^{-1}\gamma _u^{-1}\cdots f_i(a_n) \gamma _u z \eta (b_n)z^{-1}\gamma _u^{-1}$. We show that this form is reduced with respect to $Comp_i*_C(C\oplus \langle z\rangle )$. Suppose not. Then either $\gamma _u^{-1}f_i(a_j)\gamma _u$, for some $2\leq j\leq n$, or $\eta (b_j)$ for some $j\leq n$ is in $C$. In the first case this means that $\gamma _u^{-1}f_i(a_j)\gamma _u$ commutes with some (any) nontrivial element, say $\gamma$, of $\eta (G_e)$. Thus $f_i(a_j)$ commutes with $\gamma _u\gamma \gamma _u^{-1}$, but $\gamma _u\gamma \gamma _u^{-1}$ is the image of an element in $G_e$ under $f_i$, and since $f_i$ is injective we have that $a_j$ commutes with an element of $G_{e}$ by the maximality condition of $G_{e}$. This shows that $a_j$ belongs to $G_e$, a contradiction. In the second case, this means that $\eta (b_j)$ commutes with some (any) nontrivial element, say $\gamma$, of $\eta (G_e)$. Thus since $\eta$ is injective on $G_v$, we see that $b_j$ commutes with an element of $G_e$. By the maximality condition of $G_e$, $b_j$ must belong to $G_e$, a contradiction.

Suppose that $\pi _1(\mathcal{G}_{i+1})$ is an $\mathrm{HNN}$ extension. By the induction hypothesis $f_i\upharpoonright G_u=Conj(\gamma _u)\circ \eta \upharpoonright G_u$, $f_i\upharpoonright G_v=Conj(\gamma _v)\circ \eta \upharpoonright G_v$. We define $f_{i+1}$ to agree with $f_i$ on $\pi _1(\mathcal{G}_i)$ and $f_{i+1}(t)=\gamma _v\eta (t)z\gamma _u^{-1}$, where $t$ is the stable letter of the $\mathrm{HNN}$ extension. The map $f_{i+1}$ is indeed a morphism, since for any $g\in G_e$ we have that $f_{e}(g)$ is an element that lives in $G_v$; thus $f_{i+1}(f_{e}(g))=f_i(f_e(g))=\gamma _v\eta (f_e(g))\gamma _v^{-1}$. On the other hand $f_{\bar{e}}(g)$ is an element that lives in $G_u$. Thus $f_{i+1}(tf_{\bar{e}}(g)t^{-1})= \gamma _v\eta (t)z\gamma _u^{-1}\gamma _u\eta (f_{\bar{e}}(g))\gamma _u^{-1}\gamma _u z^{-1}\eta (t)^{-1} \gamma _v^{-1}=\gamma _v\eta (tf_{\bar{e}}(g)t^{-1})\gamma _v^{-1}$, and it follows that $f_{i+1}$ is a morphism. We continue by proving that $f_{i+1}$ is injective. Let $g\coloneq g_0t^{\epsilon _1}g_1t^{\epsilon _2}\cdots t^{\epsilon _n}g_n$ with $\epsilon _i\in \{-1,1\}$ be an element in reduced form with respect to the $\mathrm{HNN}$ extension. Then $f_{i+1}(g)=f_i(g_0)(\gamma _v\eta (t)z\gamma _u^{-1})^{\epsilon _1}f_i(g_1)(\gamma _v$ $\eta (t)z\gamma _u^{-1})^{\epsilon _2}\cdots$ $(\gamma _v\eta (t)z\gamma _u^{-1})^{\epsilon _n}f_i(g_n)$. We will show by induction that for every $n\geq 1$, if $g$ is an element of $\pi _1(\mathcal{G}_{i+1})$ of length $n$ (in reduced form) with respect to the $\mathrm{HNN}$ extension that $\pi _1(\mathcal{G}_{i+1})$ admits, then $f_{i+1}(g)$ can be put in reduced form of length at least one with respect to the amalgamated free product $Comp_i*_C(C\oplus \langle z\rangle )$ that $Comp_{i+1}$ admits. Moreover $f_{i+1}(g)$ ends with either $z\gamma _u^{-1}f_i(g_n)$ or $z^{-1}\eta (t)^{-1}\gamma _v^{-1}f_i(g_n)$ depending on whether $\epsilon _n$ is positive or negative. For the base step ($n=1$), the result is obvious. Suppose it is true for every $k<n$; we show it is true for $n$. We take cases with respect to whether $\epsilon _{n-1},\epsilon _{n}$ are positive or negative. Since the cases when both are negative or both are positive are symmetric we assume that both are positive and we leave the symmetric case as an exercise. Thus, $f_{i+1}(g)=f_{i+1}(g_0t^{\epsilon _1}g_1t^{\epsilon _2}\cdots tg_{n-1})\cdot f_{i+1}(t g_n)$, and by the induction hypothesis $f_{i+1}(g_0t^{\epsilon _1}g_1t^{\epsilon _2}\cdots t^{\epsilon _{n-1}}g_{n-1})$ can be put in reduced form of length at least one that ends with $z\gamma _u^{-1}f_i(g_{n-1})$. Therefore $f_{i+1}(g)=\gamma z\gamma _u^{-1}f_i(g_{n-1})\gamma _v\eta (t)z\gamma _u^{-1}f_i(g_n)$. This latter element has the desired properties, since if $\gamma _u^{-1}f_i(g_{n-1})\gamma _v\eta (t)$ belongs to $C$, then we consider $z\gamma _u^{-1}f_i(g_{n-1})\gamma _v\eta (t)z$ as an element of $C\oplus \langle z\rangle \setminus C$, and if not, then already the element is in reduced form ending with $z\gamma _u^{-1}f_i(g_n)$. We now treat the case where $\epsilon _{n-1}=1$ and $\epsilon _n=-1$. In this case $f_{i+1}(g)=f_{i+1}(g_0t^{\epsilon _1}g_1t^{\epsilon _2}\cdots tg_{n-1})\cdot f_{i+1}(t^{-1} g_n)$, and by the induction hypothesis$$\begin{equation*} f_{i+1}(g_0t^{\epsilon _1}g_1t^{\epsilon _2}\ldots tg_{n-1}) \end{equation*}$$

can be put in reduced form of length at least one that ends with $z\gamma _u^{-1}f_i(g_{n-1})$. Thus, $f_{i+1}(g)=\gamma z\gamma _u^{-1}f_i(g_{n-1})\gamma _uz^{-1}\eta (t)^{-1}\gamma _v^{-1}f_i(g_n)$. It is enough to show that $\gamma _u^{-1}f_i(g_{n-1})\gamma _u$ does not belong to the centralizer $C$ of $\eta (G_e)$. Suppose, for a contradiction, that it does. Then $\gamma _u^{-1}f_i(g_{n-1})\gamma _u$ commutes with some (any) element of $\eta (G_e)$; such an element can be written as $\gamma _u^{-1}f_i(\beta )\gamma _u$ for some $\beta \in G_u$. Therefore, $f_i(g_{n-1})$ commutes with $f_i(b)$, and since $f_i$ is injective, we see that $g_{n-1}$ commutes with $b$. We can now use the maximality of $f_{\bar{e}}(G_e)$ to conclude that $g_{n-1}$ belongs to it, contradicting the reduced form for $g$. The case where $\epsilon _{n-1}=-1$ and $\epsilon _n=1$ is symmetric to the previous case, and we leave it to the reader.

•

Suppose we are in case 1B of Definition 5.4. Suppose the centralizer of $\eta (G_{e})$ (in $Comp_i$) can be conjugated by the element $\gamma$ into $C$, where $C$ satisfies the hypothesis of case 1B. Then $\pi _1(\mathcal{G}_{i+1})$ is either the amalgamated free product $\pi _1(\mathcal{G}_i)*_{G_e}G_v$ (if $v\notin \mathcal{G}_i$) or the $\mathrm{HNN}$ extension $\pi _1(\mathcal{G}_i)*_{G_e}$ (if $v\in \mathcal{G}_i$), and $Comp_{i+1}$ is the amalgamated free product $Comp_i*_C(C\oplus \langle z\rangle )$.

Suppose that $\pi _1(\mathcal{G}_{i+1})$ is an amalgamated free product. By the induction hypothesis $f_i\upharpoonright G_u=Conj(\gamma _u)\circ \eta \upharpoonright G_u$. We define $f_{i+1}$ to agree with $f_i$ on $\pi _1(\mathcal{G}_i)$ and $f_{i+1}(g)=\gamma _u\gamma ^{-1} z \gamma \eta (g)\gamma ^{-1} z^{-1}\gamma \gamma _u^{-1}$ for $g\in G_v$. It is not hard to check that $f_{i+1}$ is a morphism, the reason being that for any element $g$ of $G_e$, since $\gamma \eta (f_{e}(g)) \gamma ^{-1}$ belongs to $C$, it commutes with $z$; thus $f_{i+1}(f_e(g))=\gamma _u\eta (f_e(g))\gamma _u^{-1}$ and this is enough. We next prove that $f_{i+1}$ is injective. Let $g\coloneq a_1b_1\cdots a_nb_n$ be an element of $\pi _1(\mathcal{G}_i)*_{G_e}G_v$ in reduced form. Then$$\begin{equation*} f_{i+1}(g) =f_i(a_1)\gamma _u\gamma ^{-1} z\gamma \eta (b_1)\gamma ^{-1}z^{-1}\gamma \gamma _u^{-1}\cdots f_i(a_n) \gamma _u\gamm