# Towers and the first-order theory of hyperbolic groups

Vincent Guirardel, Gilbert Levitt, Rizos Sklinos V.G. conducted this work
within the framework of the Henri Lebesgue Center ANR-11-LABX-0020-01. R.S.
was supported by the National Science Foundation under Grant No. 1953784.

###### Abstract

This paper is devoted to the first-order theory of torsion-free hyperbolic
groups. One of its purposes is to review some results and to provide precise
and correct statements and definitions, as well as some proofs and new
results.

A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-
Myasnikov). We discuss them thoroughly.

We state and prove a new general theorem which unifies several results in the
literature: elementarily equivalent torsion-free hyperbolic groups have
isomorphic cores (Sela); if $H$ is elementarily embedded in a torsion-free
hyperbolic group $G$, then $G$ is a tower over $H$ relative to $H$ (Perin);
free groups (Perin-Sklinos, Ould-Houcine), and more generally free products of
prototypes and free groups, are homogeneous.

The converse to Sela and Perin’s results just mentioned is true. This follows
from the solution to Tarski’s problem on elementary equivalence of free
groups, due independently to Sela and Kharlampovich-Myasnikov, which we treat
as a black box throughout the paper.

We present many examples and counterexamples, and we prove some new model-
theoretic results. We characterize prime models among torsion-free hyperbolic
groups, and minimal models among elementarily free groups. Using Fraïssé’s
method, we associate to every torsion-free hyperbolic group $H$ a unique
homogeneous countable group ${\mathcal{M}}$ in which any hyperbolic group
$H^{\prime}$ elementarily equivalent to $H$ has an elementary embedding.

In an appendix we give a complete proof of the fact, due to Sela, that towers
over a torsion-free hyperbolic group $H$ are $H$-limit groups.

##  1 Introduction

Two groups $G,G^{\prime}$ are _elementarily equivalent_ (denoted $G\equiv
G^{\prime}$) if they satisfy the same first-order sentences in the language of
groups $\mathcal{L}:=\\{\cdot,^{-1},{\bf 1}\\}$. In 1946 Tarski asked whether
all non-abelian free groups are elementarily equivalent. Although the class of
free groups had been extensively studied by group theorists and geometers, the
existing tools at the time seemed to be inadequate and the question proved
very hard to tackle. It was only after more than fifty years, in 2001, that
Sela and separately Kharlampovich-Myasnikov answered the question in the
positive. Both works required profound tools to be discovered and culminated
in many voluminous papers that take in total more than a thousand pages. In
addition, the techniques developed allowed Sela to classify torsion-free
hyperbolic groups up to elementary equivalence [Sel09].

One should note that Tarski’s question was answered in the following strong
way: the chain of non-abelian free groups under the natural embeddings is
elementary. This means that not only do non-abelian free groups share the same
first-order theory, but also elements (or rather finite tuples) of a non-
abelian free group $F$ do not change first-order properties when seen as
elements of a larger free group $F*F^{\prime}$: the natural embedding
$F\hookrightarrow F*F^{\prime}$ is elementary. As in the case of elementary
equivalence, the techniques generalize to certain embeddings of torsion-free
hyperbolic groups.

One purpose of the current paper is to review some of the above results and to
provide precise and correct statements and definitions, as well as some proofs
and further results. A key concept of this work is that of a tower. This
notion has been introduced by Sela and is the counterpart of Kharlampovich-
Myasnikov’s NTQ systems (for Non-degenerate Triangular Quasi-quadratic
systems).

It has been used for both the characterization of elementary embeddings and
the characterization of elementary equivalence. For instance, combining
results by Sela [Sel09] and Perin [Per11], one can almost state that, given a
torsion-free hyperbolic group $G$ and a non-abelian subgroup $H$, the
embedding $H\hookrightarrow G$ is elementary if and only if $G$ is a tower
over $H$ (precisely: if and only if $G$ is an extended tower over $H$ relative
to $H$, see below).

For this to be true, one must be careful with the definition of a “tower” and
make additional requirements on the tower involved in this statement (see
Theorem 2.2 below); in particular, the original definition of hyperbolic
$\omega$-residually free towers of [Sel01] does not exactly characterize
elementarily free groups (and neither do regular NTQ groups), as first found
out by the third named author while working on [PS12]. As we shall see, these
problems come from low-complexity surfaces and are the source of faulty
results in the literature.

We quickly mention some of the main ideas of the paper, with a detailed
overview containing precise statements appearing in the next section. All
groups $G$ considered in the paper are assumed to be finitely generated and
torsion-free.

Following Sela and Perin, we distinguish two kinds of towers (Section 4):
simple towers ([Sel01, Sel09]), and more general extended towers ([Per11,
Per13]) where the base may be “disconnected”. The two notions are closely
related, as one can transform an extended tower into a simple one by taking a
free product with a free group (Proposition 5.2).

Associated to the notion of a tower is a partial ordering on isomorphism
classes of groups: $G\mathrel{\geqslant_{t}}H$ if $G$ is an extended tower
over $H$. Following Sela, minimal elements for this ordering are called
_prototypes_ , and a _core_ of $G$ is a prototype $C$ such that
$G\mathrel{\geqslant_{t}}C$. In Section 6 we prove the uniqueness of the core
(up to isomorphism) for any finitely generated torsion-free CSA group $G$.
This is due to Sela [Sel09] for hyperbolic groups; we give a different proof.

In order to relate towers to model theory, we isolate a result by Sela
(Theorem 2.1 below) which implies the solution to Tarski’s problem, and which
we unfortunately have to take as a black box. We refer to it as “Tarski” in
the paper. It says that, if $G$ and $H$ are non-abelian torsion-free
hyperbolic groups such that $G$ is a simple tower over $H$, then $H$ is
elementarily embedded in $G$.

In Section 9 we state and prove a new general result going in the converse
direction (Theorem 2.5 below). It is stated in terms of partial elementary
maps, which can be interpreted as elementary equivalence of two groups in a
language with constants. It says that a partial elementary maps between
torsion-free hyperbolic groups extends to an isomorphism between their
relative cores. This allows us to unify several results in the literature:

  * •

elementarily equivalent torsion-free hyperbolic groups have isomorphic cores
(Sela [Sel09], the converse follows from “Tarski”)

  * •

if $H$ is elementarily embedded in a torsion-free hyperbolic group $G$, then
$G$ is an extended tower over $H$ relative to $H$ (Perin [Per11], here also
the converse follows from “Tarski”)

  * •

the homogeneity of free groups (due to Perin and Sklinos [PS12] and Ould-
Houcine [OH11]), and more generally of free products of prototypes and free
groups (Theorem 2.7 below).

We devote the full Section 10 to working out examples and counterexamples.

In the last section we prove some new model-theoretic results. We characterize
prime models among torsion-free hyperbolic groups, and minimal models among
elementarily free groups. Using Theorem 2.5 and Fraïssé’s method, we associate
to every torsion-free hyperbolic group $H$ a unique homogeneous countable
group ${\mathcal{M}}$ in which any hyperbolic group $H^{\prime}$ elementarily
equivalent to $H$ has an elementary embedding (see Theorem 2.10). Finally,
Theorem 2.5 allows us to characterize tuples of elements of torsion-free
hyperbolic groups with the same first-order properties in terms of towers
(Corollary 9.11).

In an appendix we give a complete proof of the fact, due to Sela, that towers
over a torsion-free hyperbolic group $H$ are $H$-limit groups.

###### Contents

  1. 1 Introduction
  2. 2 Overview of the paper
  3. 3 Preliminaries
    1. 3.1 Elementary equivalence and elementarily embedded subgroups
    2. 3.2 Surfaces
    3. 3.3 Splittings and trees
    4. 3.4 Grushko decompositions.
    5. 3.5 Surface-type vertices
    6. 3.6 The cyclic JSJ decomposition of hyperbolic groups and CSA groups
    7. 3.7 Two lemmas about splittings
  4. 4 Towers
    1. 4.1 Centered splittings
    2. 4.2 Boundary-preserving maps, pinched quotient
    3. 4.3 A key lemma
    4. 4.4 Simple towers
    5. 4.5 Extended towers and retractable splittings
    6. 4.6 Relative extended towers
    7. 4.7 Relation with other definitions
  5. 5 Some properties of towers
    1. 5.1 Making an étage simple
    2. 5.2 Narrowing the range
    3. 5.3 Retracting a Grushko factor
    4. 5.4 Reading an étage in the JSJ
  6. 6 Towers as an ordering: core and prototypes
    1. 6.1 Defining an order
    2. 6.2 Proof of the Minimal Base Lemma 6.14
  7. 7 Elementary equivalence
  8. 8 Elementary embeddings
    1. 8.1 Characterizing elementarily embedded subgroups
    2. 8.2 Prototypes
    3. 8.3 Lattice-like properties
  9. 9 Partial elementary maps extend to the relative core
    1. 9.1 Isomorphic cores yield partial elementary maps
    2. 9.2 $A$-groups and plain groups
    3. 9.3 Preliminary choices
    4. 9.4 Using the shortening argument
    5. 9.5 A first-order criterion for isomorphism
    6. 9.6 Proof of Theorem 9.16 and Corollary 9.9
    7. 9.7 Preretractions
  10. 10 Examples
    1. 10.1 Simple towers vs extended towers
    2. 10.2 A construction
    3. 10.3 $\omega$-residually free towers and NTQ groups
    4. 10.4 The core
    5. 10.5 Elementary equivalence and quasi-isometry
    6. 10.6 Parachutes
  11. 11 Minimal and maximal models
    1. 11.1 Primality
    2. 11.2 Minimality
    3. 11.3 Universality
  12. A Appendix: Towers are limit groups
    1. A.1 Applying Baumslag’s lemma
    2. A.2 Surfaces with sockets
    3. A.3 A reduction
    4. A.4 Decomposing into pieces with non-abelian image
    5. A.5 Proof of Theorem A.5
    6. A.6 Ping-pong lemmas

##  2 Overview of the paper

Figure 1: Two simple towers and an extended tower, with the underlying graphs
of groups.

##### Simple towers

To introduce towers, let us start with an example (see Figure 1). Let
$G=B_{1}*_{g=[x,y]}F(x,y)$ be the cyclic amalgam of a group $B_{1}$ with a
free group of rank 2, which we view as the fundamental group of a punctured
torus; we identify the boundary element $[x,y]$ with some non-trivial $g\in
B_{1}$ (all groups are assumed to be torsion-free, so $g$ has infinite order).

If $g$ is not a commutator in $B_{1}$, then clearly the embedding of $B_{1}$
into $G$ is not elementary: the sentence saying that $g$ is a commutator is
true or false, depending on whether we interpret it in $G$ or in $B_{1}$.

Now suppose that $g$ is a commutator $[a,b]$ in $B_{1}$. In this case we say
that $G$ is a tower over $B_{1}$; it follows from “Tarski” (Theorem 2.1) that
the embedding of $B_{1}$ into $G$ is elementary if $G$ is hyperbolic.

This example may be generalized, leading to the definition of a simple tower.
Instead of gluing a once-punctured torus to $B_{1}$, we may glue any compact
surface $\Sigma$ (possibly non-orientable, with several boundary components).
In topological terms, we are attaching every boundary component $C_{i}$ of
$\Sigma$ to a space $X_{1}$ with $\pi_{1}(X_{1})=B_{1}$ along curves
representing non-trivial elements of $B_{1}$ (see Figure 1). Van Kampen’s
theorem expresses the resulting group $G$ as the fundamental group of a graph
of groups $\Gamma$ having two vertices: a central vertex $v$ carrying
$\pi_{1}(\Sigma)$ and a bottom vertex $v_{1}$ carrying $B_{1}$; there is one
edge between $v$ and $v_{1}$ per boundary component of $\Sigma$.

We shall say that $G$ is a _simple étage_ over $B_{1}$ if $B_{1}$ is non-
abelian and two conditions are satisfied. First, $\Sigma$ must be _non-
exceptional_ : it must carry a pseudo-Anosov homeomorphism; equivalently,
$\Sigma$ is a once-punctured torus or its Euler characteristic satisfies
$\chi(\Sigma)\leq-2$.

The second condition is a generalization of $g$ being a commutator in $B_{1}$
in the example studied above, but it is expressed without referring to a
specific element. There are two ways of stating it.

One may require the existence of a _retraction_ $\rho:G\to B_{1}$ such that
$\pi_{1}(\Sigma)$ has non-abelian image (in the example, $\rho$ sends $x$ to
$a$ and $y$ to $b$ if $g=[a,b]$). One may also require the existence of a map
$r$ from $\pi_{1}(\Sigma)$ to $G$ (such as the restriction of $\rho$ to
$\pi_{1}(\Sigma)$) which is a _non-degenerate boundary-preserving map_ (called
local preretraction in [GLS19]): it must agree with a conjugation on each
boundary subgroup $\pi_{1}(C_{i})$, have non-abelian image, not be an
isomorphism between $\pi_{1}(\Sigma)$ and some conjugate.

It is shown in Proposition 4.14 that the two requirements are equivalent if
$G$ is a torsion-free CSA group and $B_{1}$ is not abelian.

If $\Sigma$ is non-exceptional and $\rho$ (equivalently, $r$) exists, with
$B_{1}$ non-abelian, we say that $G$ (assumed to be CSA) is a _simple étage of
surface type_ over $B_{1}$ (we use the French word étage rather than the
English word floor to emphasize the fact that there are two levels: a single-
story house has no étage).

There is another type of étage: we say that $G$ is a _simple étage of free
product type_ over $B_{1}$ if $G=B_{1}*{\mathbb{Z}}$. We mention for
completeness another type of étage, although it will not be used in this
paper: $G$ is a simple étage of abelian type over $B_{1}$ if $G$ is an amalgam
$G=B_{1}*_{A}(A\oplus{\mathbb{Z}})$, with ${A}$ a maximal abelian subgroup of
$B_{1}$. Abelian étages play a role in obtaining formal solutions/implicit
function theorems (see [Sel03, KM05]), but they are not used in characterizing
elementary equivalence and embeddings.

A tower is obtained by stacking étages on top of each other: $G$ is a _simple
tower_ over a subgroup $H$ if there exists a chain of subgroups
$G=G_{0}>G_{1}>\dots>G_{k}=H$ such that $G_{i}$ is a simple étage over
$G_{i+1}$; we allow the case $k=0$, so $G$ is always a simple tower over
itself. We note that $\omega$-residually free towers, as defined in [Sel01],
are precisely simple towers over a (non-abelian) subgroup which is a free
product of cyclic groups and fundamental groups of closed surfaces of Euler
characteristic $\leq-2$ (see Remark 4.31).

We can now state the _main elementary embedding theorem_ , which implies the
solution to Tarski’s problem (we shall refer to it simply as “Tarski” and
treat it as a black box):

######  Theorem 2.1 (“Tarski”, [Sel09, Theorem 7.6]).

Let $G,H$ be non-abelian torsion-free hyperbolic groups. If $G$ is a _simple_
tower over $H$, then the inclusion $H\hookrightarrow G$ is an elementary
embedding. In particular, $G$ and $H$ are elementarily equivalent.

This statement does not explicitly appear in [Sel09] but, as pointed out in
[Per11], it follows from the arguments used in the proof of Theorem 7.6 of
[Sel09].

The exact converse to Theorem 2.1 is not true: in Subsection 10.3 we construct
a group which is not a simple étage but contains a proper elementarily
embedded subgroup. The opposite direction in Theorem 2.1, which is due to
Perin [Per11], thus has to be formulated differently.

We shall rephrase Perin’s result as follows (see Theorem 2.2 below): if
$H\hookrightarrow G$ is an elementary embedding, there exists $r\geq 0$ such
that the free product $G*{\mathbb{F}}_{r}$ of $G$ with the free group of rank
$r$ is a simple tower over $H$. If we want to avoid using an auxiliary free
group, we have to use extended towers.

##### Extended towers.

These towers were introduced by Perin in [Per11, Per13].

We defined a simple étage of surface type by attaching a surface $\Sigma$ to a
single connected space $X_{1}$ (see the first two pictures on Figure 1). We
now consider connected spaces $X_{1},\dots,X_{n}$ with $n\geq 1$, and we
attach each boundary component of $\Sigma$ to one of these spaces so as to get
a connected space.

The associated graph of groups $\Gamma$, which we call a centered splitting
(see Definition 4.1), has bottom vertices $v_{i}$ for $i=1,\dots,n$, each
carrying a group $B_{i}=\pi_{1}(X_{i})$; each $v_{i}$ is joined to the central
vertex $v$ carrying $\pi_{1}(\Sigma)$ by one or more edges (see the last
picture on Figure 1).

If the surface $\Sigma$ is not exceptional and there exists a non-degenerate
boundary-preserving map $p:\pi_{1}(\Sigma)\to G$, or equivalently a retraction
$\rho:G\to\tilde{B}_{1}*\dots*\tilde{B}_{n}$ with $\rho(\pi_{1}(\Sigma))$ non-
abelian, for some choices of conjugates $\tilde{B}_{i}$ of $B_{i}$, we say
that $G$ is an _extended étage of surface type_ over
$H=\tilde{B}_{1}*\dots*\tilde{B}_{n}$ (as pointed out in [Per13], the
requirements on the retraction $\rho$ must be modified when $n=1$ and $B_{1}$
is abelian). We then define _extended towers_ by stacking extended étages of
surface type and étages of free product type.

If no surface of low complexity is involved, any extended tower may be
upgraded to a simple tower: we prove (Proposition 5.1) that, if $G$ is an
extended étage over $H$ with the associated surface $\Sigma$ having Euler
characteristic $\chi(\Sigma)\leq-3$, then $G$ is a simple étage over some
subgroup $H^{\prime}$ isomorphic to $H$. We also point out (Proposition 5.2)
that, regardless of $\Sigma$, one obtains a simple étage after stabilizing: if
$G$ is an extended étage over $H$, some $G*{\mathbb{F}}_{r}$ is a simple étage
over a subgroup $\tilde{H}\subset G*{\mathbb{F}}_{r}$ isomorphic to $H$.

Figure 2: The group $K(B_{1},B_{2},b_{1},b_{2})$ constructed by attaching a
twice-punctured Klein bottle. This is an extended étage over $B_{1}*B_{2}$ if
$b_{1}$ and $b_{2}$ are squares in $B_{1}$ and $B_{2}$ respectively.

On the other hand, consider as in Figure 2 and Example 10.2 a double cyclic
amalgam

$G=K(B_{1},B_{2},b_{1},b_{2})=B_{1}*_{b_{1}={g_{1}}}K*_{{g_{2}}=b_{2}}B_{2}$

with $K=\langle g_{1},g_{2},x,y\mid g_{1}g_{2}=x^{2}y^{2}\rangle$ the
fundamental group of a twice-punctured Klein bottle (note that it is a non-
exceptional surface since its Euler characteristic is $-2$). It defines an
extended étage over $B_{1}*B_{2}$ if each $b_{i}$ is a square in $B_{i}$. If
$B_{1}$ and $B_{2}$ have no cyclic splitting, then $G$ is not a simple étage
(see Lemma 10.5). Moreover, $G$ has no proper elementarily embedded subgroup
(see Proposition 10.8): it is _minimal_. Similar examples may be constructed
using other surfaces of Euler characteristic $-2$.

##### Classifying elementarily embedded subgroups.

Because of such examples, Theorem 2.1 is not true for extended towers. But it
will be true if we add an extra condition. One says that an étage is
_relative_ to a subgroup $H<G$ if $H$ is conjugate into one of the $B_{i}$’s
(étage of surface type), into $B_{1}$ if $G=B_{1}*{\mathbb{Z}}$. A tower is
relative to $H$ if all its étages are (see Definition 4.36). We can now state
the following characterization of elementarily embedded subgroups.

######  Theorem 2.2 ([Per11], see Theorem 8.1).

Let $G$ be a non-abelian torsion-free hyperbolic group, and let $H<G$ be a
non-abelian subgroup. The following are equivalent:

  1. 1.

$H$ is elementarily embedded in $G$;

  2. 2.

$G$ is an extended tower over $H$ relative to $H$ (in the sense of Definition
4.36);

  3. 3.

there exists a finitely generated free group $F$ such that $G*F$ is a simple
tower over $H$ (viewed as a subgroup of $G*F$ in the obvious way).

$1\Rightarrow 2$ is Perin’s original statement. Deducing 1 from 2 or 3
requires “Tarski” (Theorem 2.1).

##### Classifying hyperbolic groups up to elementary equivalence.

Given torsion-free CSA groups $G$ and $H$, we write
$G\mathrel{\geqslant_{t}}H$ if $G$ is an extended tower over a subgroup
isomorphic to $H$. This defines a _partial order_ on the set of isomorphism
classes, and minimal elements are called _prototypes._ In Section 6 we prove
that, given a CSA group $G$, there is a unique prototype $\mathrm{c}(G)$ such
that $G\mathrel{\geqslant_{t}}\mathrm{c}(G)$; it is called the _core_ of $G$.

The difficulty here is to prove uniqueness. In the case of hyperbolic groups,
Sela deduces it from “Tarski”, while our arguments in Section 6 are purely
group-theoretical. We associate a _minimal base_ $M$ to any 2-acylindrical
cyclic splitting of a group $G$, and we use it to show confluence: if
$G_{0}\mathrel{\geqslant_{t}}G_{1}$ and $G_{0}\mathrel{\geqslant_{t}}G_{2}$,
with $G_{0}$ a CSA group, there exists $G_{3}$ such that
$G_{1}\mathrel{\geqslant_{t}}G_{3}$ and $G_{2}\mathrel{\geqslant_{t}}G_{3}$.

The core behaves nicely with respect to free products: $c(G_{1}*G_{2})\simeq
c(G_{1})*c(G_{2})$, and a group is a prototype if and only if it is a free
product of one-ended prototypes.

In Section 7 we use extended towers to present Sela’s classification of
torsion-free hyperbolic groups up to elementary equivalence. Note that any
finitely generated group which is elementarily equivalent to a hyperbolic
group is itself hyperbolic ([Sel09, And18a]).

We shall deduce from Theorem 2.5 below that, if two hyperbolic groups $G$ and
$G^{\prime}$ are elementarily equivalent (denoted $G\equiv G^{\prime}$), their
cores are isomorphic. Our arguments do not use “Tarski”, but “Tarski” is
needed to prove that $G$ is elementarily equivalent to its core, thus
obtaining Sela’s complete classification.

######  Theorem 2.3 ([Sel09], see Theorem 7.1).

Two non-abelian torsion-free hyperbolic groups are elementarily equivalent if
and only if their cores are isomorphic.

To be precise, it is not quite true that $G$ is always elementarily equivalent
to its core: if $G\equiv{\mathbb{F}}_{n}$, its core is the trivial group,
which is not equivalent to $G$. To remedy this, we also define the _elementary
core_ $\mathrm{\widetilde{c}}(G)$, which is equal to $\mathrm{c}(G)$ if
$\mathrm{c}(G)\neq\\{1\\}$ and to ${\mathbb{F}}_{2}$ if
$\mathrm{c}(G)=\\{1\\}$, so that $G\equiv\mathrm{\widetilde{c}}(G)$ always
holds. Note however that $\tilde{c}(G)$ does not behave as well as $c(G)$ with
respect to free products.

Corollary 7.10 gives many characterizations of groups which are elementarily
free ($G\equiv{\mathbb{F}}_{2}$), correcting [Sel06, Theorem 7] and [KM06,
Theorem 41].

The elementary core $\mathrm{\widetilde{c}}(G)$ being isomorphic to a subgroup
of $G$ and elementarily equivalent to $G$, one may wonder whether it embeds
elementarily into $G$. This is the case if $\mathrm{\widetilde{c}}(G)$ is one-
ended (Corollary 8.8) but the following result, based on the groups
$G=K(B_{1},B_{2},b_{1},b_{2})$ introduced above, shows that this is not true
in general, thus correcting Theorem 7.6 of [Sel09].

######  Theorem 2.4 (See Theorem 10.13).

If $P$ is a prototype with more than one end, there is a torsion-free
hyperbolic group $G$ such that $G$ is an extended étage over $P$ (so that
$P\simeq\mathrm{\widetilde{c}}(G)$ is the core of $G$ and $G\equiv P$), but
$P$ does not elementarily embed into $G$.

One can get an elementary embedding of the core, however, if one is ready to
stabilize (Lemma 8.7): some $G*{\mathbb{F}}_{r}$ with $r\geq 0$ has an
elementarily embedded subgroup which is isomorphic to
$\mathrm{\widetilde{c}}(G)$.

For example, if $G=K(B_{1},B_{2},b_{1},b_{2})$ with $B_{i},b_{i}$ as above
($b_{i}$ is a square and $B_{i}$ has no cyclic splitting), and
$P=B_{1}*B_{2}$, then $P\simeq\mathrm{\widetilde{c}}(G)$ has no elementary
embedding in $G$, but one can show that $B_{1}*tB_{2}t^{-1}$ is an
elementarily embedded subgroup of $G*\langle t\rangle$.

We also discuss in Section 10 to what extent $\mathrm{c}(G)$ is unique as a
subgroup of $G$, and we study _parachutes_. These are groups having an
arbitrary centered splitting with $n=1$ and $B_{1}\simeq{\mathbb{Z}}$. Some of
these groups admit non-degenerate boundary-preserving maps, but no retraction
from $G$ to $B_{1}$ may have non-abelian image, so these groups are an
exception in the theory. We determine which of these groups are simple étages,
and which have a proper elementarily embedded subgroup.

##### Extending partial elementary maps.

One of the main results of the paper is Theorem 2.5 below. This result may be
viewed as a common generalization of results by Sela and Perin mentioned
above, and it also implies the homogeneity of free groups [PS12, OH11], and
more generally of their free products with prototypes (see Theorem 2.7 below).

We consider hyperbolic groups $G,G^{\prime}$ with a subgroup $A\subset G$, and
$f:A\to G^{\prime}$ sending $A$ isomorphically to a subgroup
$A^{\prime}\subset G^{\prime}$. We say that $f$ is a _partial elementary map_
if, for any formula $\varphi(x_{1},\dots,x_{n})$ and any $a_{1},\dots,a_{n}$
in $A$, one has $G\models\varphi(a_{1},\dots,a_{n})$ if and only if
$G^{\prime}\models\varphi(f(a_{1}),\dots,f(a_{n}))$.

The notion of partial elementary map somehow interpolates between elementary
equivalence and elementary embedding: if $A=\\{1\\}$, the trivial map is
partial elementary if and only if $G\equiv G^{\prime}$; if $A=G$, a partial
elementary map $f$ defines an elementary embedding of $G$ into $G^{\prime}$.

A _core of $G$ relative to $A$_ is a subgroup $C\subset G$ containing $A$,
such that $G$ is a tower over $C$ relative to $A$, and $C$ is not an étage
relative to $A$. Thus, if $A=\\{1\\}$, the core of $G$ relative to $A$ is just
the core $c(G)$.

######  Theorem 2.5 (See Theorem 9.7).

Given $A\subset G$ and $A^{\prime}\subset G^{\prime}$, with $G,G^{\prime}$
torsion-free hyperbolic groups, let $C$ (resp. $C^{\prime}$) be a core of $G$
relative to $A$ (resp. of $G^{\prime}$ relative to $A^{\prime}$).

Then any partial elementary map $f:A\to G^{\prime}$ with $f(A)=A^{\prime}$
extends to an isomorphism $F$ between $C$ and $C^{\prime}$.

######  Remark 2.6.

Using “Tarski”, one can see that the converse is also true: if $f$ extends, it
is partial elementary (see Section 9.1).

This theorem is proved in Section 9, building on ideas due to Sela, Perin and
Sklinos.

The argument goes as follows (see Subsection 9.6). Using $f$, one can view $G$
and $G^{\prime}$ as $A$-groups, i.e. groups with a specified embedding of $A$;
morphisms between $A$-groups must commute with these embeddings. The goal is
then to prove that $C$ and $C^{\prime}$ are isomorphic as $A$-groups. We
introduce a notion of _(strong) engulfing_ between two $A$-groups, which can
be translated into a first-order statement and satisfies some kind of a co-
Hopf property: if two $A$-groups engulf in each other, they are isomorphic (to
be precise, we have to define both engulfing and strong engulfing, see details
in Subsection 9.5). The fact that $C$ is the core of $G$ implies that $C$
strongly engulfs in $G$. Since strong engulfing is first-order and $f$ is a
partial elementary map, $C$ strongly engulfs in $G^{\prime}$. Using the fact
that $G^{\prime}$ is a tower over $C^{\prime}$, we deduce that $C$ engulfs in
$C^{\prime}$. By symmetry of the argument, $C$ and $C^{\prime}$ engulf in each
other and are therefore isomorphic.

An important step in this proof consists in proving that the inclusion of $C$
into $G$ is a strong engulfing. This step is based on the notion of
preretraction introduced by Perin in [Per11]. By way of contradiction, assume
that $C\hookrightarrow G$ is not a strong engulfing. This directly translates
into the existence of a non-injective preretraction $r:C\rightarrow G$. In
Subsection 9.7 we give a rather short proof of Propositions 5.11 and 5.12 from
[Per11], which use $r$ to show that $C$ is an étage, thus contradicting the
fact that $C$ is a prototype.

##### Homogeneity.

Theorem 2.5 can be used to prove the homogeneity of groups having very
restricted tower structures. Recall that a countable group $G$ is
_homogeneous_ if, for every pair of tuples $\bar{a}=(a_{1},\dots,a_{n})$ and
$\bar{b}=(b_{1},\dots,b_{n})$ in $G^{n}$ satisfying the same first-order
formulas, there exists an automorphism of $G$ that sends $\bar{a}$ to
$\bar{b}$. Equivalently, given $\bar{a}$ and $\bar{b}$ such that the map
$a_{i}\mapsto b_{i}$ extends to a partial elementary map
$f:\langle\bar{a}\rangle\rightarrow G$, there is an extension of $f$ to an
automorphism of $G$.

Given such an $f$, Theorem 2.5 says that $f$ extends to an isomorphism between
the cores $C$ and $C^{\prime}$ of $G$ relative to $\langle\bar{a}\rangle$ and
$\langle\bar{b}\rangle$ respectively. In particular, if $G$ is a prototype,
then $C=C^{\prime}=G$ and it immediately follows that $G$ is homogeneous. If
$G$ is free, it cannot be written as an étage of surface type ([Per11, PS12],
see also Corollary 5.16), so the only possibilities for the relative cores $C$
and $C^{\prime}$ are free factors. Homogeneity of free groups immediately
follows. This argument generalizes naturally to free products of free groups
and prototypes:

######  Theorem 2.7 (see Corollary 9.10, extending [PS12, OH11]).

If a torsion-free hyperbolic group $G=P_{1}*\dots*P_{p}*{\mathbb{F}}_{r}$ is a
free product of one-ended prototypes $P_{i}$ with a free group, then $G$ is
homogeneous.

See [DBP19] for a characterization of homogeneous hyperbolic groups.

##### Primality, minimality, universality.

A group is _prime_ if it elementarily embeds into any other group elementarily
equivalent to it. We show:

######  Theorem 2.8 (see Theorem 11.1).

A non-abelian torsion-free hyperbolic group is prime if and only if it is a
one-ended prototype.

A group is _minimal_ if it has no proper elementarily embedded subgroup. The
classification of minimal torsion-free hyperbolic groups depends on the nature
of their core.

If $G$ is a one-ended prototype, it is minimal and any minimal finitely
generated group elementarily equivalent to $G$ is isomorphic to $G$. On the
other hand, given two one-ended prototypes $B_{1},B_{2}$ (so that
$B_{1}*B_{2}$ is also a prototype), different choices of $b_{1},b_{2}$ in the
groups $K(B_{1},B_{2},b_{1},b_{2})$ constructed above provide different groups
with core $B_{1}*B_{2}$ which are minimal and elementarily equivalent to
$B_{1}*B_{2}$. More generally:

######  Theorem 2.9 (see Theorem 11.4).

Let $G$ be a non-abelian torsion-free hyperbolic group.

  * •

If the core of $G$ is one-ended, then $G$ is minimal if and only if $G$ is a
prototype (i.e. $G$ is equal to its core).

  * •

If the core of $G$ is infinitely-ended, there are infinitely many minimal
hyperbolic groups elementarily equivalent to $G$.

Theorem 11.4 also gives a classification of finitely generated minimal
elementarily free groups.

We use Theorem 2.5 to show that the class of all finitely generated groups
elementarily equivalent to a given torsion-free hyperbolic group is an
elementary Fraïssé class (see [KMS18] for this notion and [Fra54] for the
original notion of a Fraïssé class). Using standard arguments, this directly
implies:

######  Theorem 2.10 (see Theorem 11.5).

Given any non-abelian torsion-free hyperbolic group $H$, there exists a unique
countable (infinitely generated) group ${\mathcal{M}}_{H}$ elementarily
equivalent to $H$ with the following properties:

  1. 1.

${\mathcal{M}}_{H}$ is homogeneous;

  2. 2.

there exists an elementary chain of finitely generated subgroups $G_{1}\prec
G_{2}\prec\dots\prec G_{n}\prec\cdots$ such that
${\mathcal{M}}_{H}=\bigcup_{n\geq 1}G_{n}$;

  3. 3.

all torsion-free hyperbolic groups elementarily equivalent to $H$ elementarily
embed into ${\mathcal{M}}_{H}$.

##### Towers are limit groups

One of the purposes of this paper is to fill gaps in the literature. In an
appendix we give a complete proof of the fact, due to Sela [Sel09], that, if
$G$ is a tower over a non-abelian torsion-free hyperbolic group $H$, then $G$
is an $H$-limit group.

The key step (Theorem A.5) is to prove that, given a homomorphism $f$ from the
fundamental group of a non-exceptional surface $\Sigma$ to $H$ which is
injective on each boundary subgroup and has non-abelian image, one may cut
$\Sigma$ into subsurfaces such that the restriction of $f$ to the fundamental
group of each subsurface is injective. One then uses a standard argument based
on Baumslag’s lemma [Bau67] to conclude.

##### Acknowledgments

We thank Pablo Cubides, François Dahmani, Olga Kharlampovich, Chloé Perin,
Zlil Sela, Karen Vogtmann for useful conversations.

##  3 Preliminaries

All groups $G$ considered in this paper will be finitely generated and
torsion-free. From Subsection 4.4 on, $G$ will be assumed to be CSA. In
Section 7 we will start discussing model theory, and $G$ will be hyperbolic.

###  3.1 Elementary equivalence and elementarily embedded subgroups

We refer to [TZ12, CK90] for basic notions of model theory. Two groups
$G,G^{\prime}$ are _elementarily equivalent_ , denoted $G\equiv G^{\prime}$,
if they satisfy the same first-order sentences in the language of groups
without constants: such a sentence (in prenex form) consists of a string of
quantifiers and variables followed by a boolean combination of word equations
and inequations in these variables and their inverses.

For finite groups, and for finitely generated abelian groups (but not for
finitely generated nilpotent groups), elementary equivalence is the same as
isomorphism. On the other hand, all non-abelian free groups are elementarily
equivalent (“Tarski’s problem”).

A subgroup $H\subset G$ is _elementarily embedded_ , denoted $H\preceq_{e}G$,
if, for any first-order sentence $\varphi$ in the language of groups with
constants in $H$, one has $H\models\varphi$ if and only if $G\models\varphi$.
Equivalently, this means that, for every first-order formula
$\varphi(x_{1},\ldots,x_{n})$ (in the language of groups, without constants)
with free variables $x_{1},\dots,x_{n}$, and every $n$-uple
$\bar{a}=(a_{1},\dots,a_{n})$ from $H$, we have $H\models\varphi(\bar{a})$ if
and only if $G\models\varphi(\bar{a})$. In model theory, one usually says that
$H$ is an elementary subgroup of $G$, but we will avoid this terminology as it
may be confusing in the context of hyperbolic groups.

Intuitively, a subgroup of a group is elementarily embedded if the first-order
properties of its tuples do not change if we see them as tuples of the ambient
group rather than of the subgroup. Note that, if $H$ is an elementarily
embedded subgroup of $G$, then $G$ and $H$ are elementarily equivalent, as
sentences without constants are in particular sentences with constants in $H$.

######  Example 3.1.

The following are examples of subgroups $H\subset G$ which are elementarily
equivalent to $G$ but are not elementarily embedded in $G$.

  * •

The subgroup of even integers $2{\mathbb{Z}}$ of the group of integers
${\mathbb{Z}}$ is not elementarily embedded: the element $2$ has a root in
${\mathbb{Z}}$, but not in $2{\mathbb{Z}}$.

  * •

Let $\pi_{1}(S_{g})=\langle
e_{1},\ldots,e_{2g}\mid[e_{1},e_{2}]\cdots[e_{2g-1},e_{2g}]=1\rangle$ be the
fundamental group of the closed orientable surface of genus $g$, with $g\geq
2$. Let $F=\langle e_{1},e_{2},e_{3},e_{4}\rangle$. It is a free group of rank
$4$ if $g\geq 3$. If $g=3$, then $F$ is not elementarily embedded because
$[e_{1},e_{2}][e_{3},e_{4}]$ is a single commutator $[e_{6},e_{5}]$ in
$\pi_{1}(S_{g})$ but not in $F$. On the other hand, $F$ is elementarily
embedded if $g\geq 4$ (see [Per11]).

The following facts are easy consequences of the definition.

######  Remark 3.2.

If $H\preceq_{e}G$ and $K$ is conjugate to $H$, then $K\preceq_{e}G$.

If $K\preceq_{e}H$ and $H\preceq_{e}G$, then $K\preceq_{e}G$.

If $K\preceq_{e}G$ and $H\preceq_{e}G$ with $K\subset H$, then
$K\preceq_{e}H$.

###  3.2 Surfaces

We denote by $S_{g,b}$ and $N_{g,b}$ respectively the orientable and non-
orientable compact surfaces of genus $g$ with $b$ boundary components (we just
write $S_{g}$ or $N_{g}$ when there is no boundary).

We will denote by $Q=\pi_{1}(\Sigma)$ the fundamental group of a compact
(possibly non-orientable) surface $\Sigma$. Subgroups of $\pi_{1}(\Sigma)$
contained up to conjugacy in the fundamental group of a boundary component of
$\Sigma$ will be called _boundary subgroups_. The maximal boundary subgroups
are the fundamental groups of boundary components.

Recall that the Euler characteristic $\chi(\Sigma)$ equals $2-2g-b$ if
$\Sigma$ is orientable, $2-g-b$ if it is not orientable (with $g$ the genus
and $b$ the number of boundary components).

We always assume that $Q$ is not virtually abelian (equivalently,
$\chi(\Sigma)$ is negative). This rules out the non-hyperbolic surfaces: disc,
annulus, Möbius band, sphere, projective plane, torus, Klein bottle.

######  Definition 3.3 (Exceptional surfaces).

Four hyperbolic surfaces with $\chi(\Sigma)=-1$ are considered _exceptional_
because their mapping class group is “too small” (they do not carry pseudo-
Anosov diffeomorphisms): the pair of pants (thrice-punctured sphere)
$S_{0,3}$, the twice-punctured projective plane $N_{1,2}$, the once-punctured
Klein bottle $N_{2,1}$, and the closed non-orientable surface $N_{3}$ of genus
3 (the once-punctured torus $S_{1,1}$ is contained in $N_{3}$ but is not
exceptional).

Figure 3: The exceptional surfaces with infinite mapping class group, and the
corresponding special curves

Two exceptional surfaces $\Sigma$ have infinite mapping class group: the once-
punctured Klein bottle $N_{2,1}$ and $N_{3}$, the closed non-orientable
surface of genus 3. Both of them have a _special simple closed curve_
$c_{\Sigma}$ whose isotopy class is invariant under the mapping class group,
see Figure 3.

The punctured Klein bottle $N_{2,1}$ has a unique isotopy class $c_{N_{2,1}}$
of non-separating two-sided simple closed curves whose complement is
orientable (hence a pair of pants) [Stu06, Prop. A.3].

In the case of $N_{3}$, the special curve $c_{N_{3}}$ is the unique (up to
isotopy) one-sided simple closed curve whose complement is orientable
[GAnMB06, Prop. 2.1]. Cutting $N_{3}$ along the boundary of a regular
neighbourhood of $c_{N_{3}}$ decomposes $N_{3}$ into a once-punctured torus
and a Möbius band, and represents $\pi_{1}(N_{3})$ as an amalgam over $\langle
g_{N_{3}}^{2}\rangle$ (we denote by $g_{N_{3}}$ a generator of
$\pi_{1}(c_{N_{3}})$, it is well defined up to inversion and conjugacy).

######  Definition 3.4 (Pinched curve, pinching map).

Let $p$ be a homomorphism from $\pi_{1}(\Sigma)$ to a group.

A _pinched curve_ is a 2-sided simple closed curve which is not null-homotopic
and whose fundamental group is contained in $\ker p$. A family of pinched
curves is a family of disjoint, pairwise non-parallel, pinched curves.

The map $p$ is _pinching_ if there is a pinched curve, _non-pinching_ if there
is none.

###  3.3 Splittings and trees

Graphs of groups are a generalization of free products with amalgamation and
HNN-extensions. Recall that a graph of groups consists of a graph $\Gamma$,
the assignment of a group $G_{v}$ or $G_{e}$ to each vertex $v$ or non-
oriented edge $e$, and injections $G_{e}\to G_{v}$ whenever $v$ is the origin
of an oriented edge (see [Ser77] for precise definitions); the image of
$G_{e}$ is called an incident edge group at $v$. This data yields a group, the
fundamental group of the graph of groups, with an action on a simplicial tree
$T$ (the Bass-Serre tree). We usually call the graph of groups $\Gamma$, with
the extra data implicit.

A graph of groups decomposition, or splitting, of a group $G$ is an
isomorphism of $G$ with the fundamental group of a graph of groups $\Gamma$.
The vertex and edge groups $G_{v}$ and $G_{e}$ are then naturally viewed as
subgroups of $G$. We always assume that $G$ is finitely generated and that the
graph of groups is minimal (no proper connected subgraph of $\Gamma$ carries
the whole of $G$); it follows that $\Gamma$ is a finite graph. We allow the
trivial splitting, when $\Gamma$ is a single vertex $v$ with $G_{v}=G$.

A splitting of $G$ yields an action of $G$ on the Bass-Serre tree $T$.
Conversely, an action of $G$ on a tree $T$ defines a quotient graph of groups
$\Gamma=T/G$.

We denote by $G_{e}$ (resp. $G_{v}$) the stabilizer of an edge $e$ of $T$
(resp. a vertex $v$); it is conjugate to an edge (resp. vertex) group of
$\Gamma$. We say that a vertex $v$ of $\Gamma$ or $T$ is _cyclic_ if $G_{v}$
is cyclic.

All splittings considered in this paper will be cyclic splittings: edge groups
are cyclic (possibly trivial). We allow redundant vertices: $\Gamma$ may have
a vertex of valence 2 with both incident edge groups equal to the vertex
group; this corresponds to vertices of valence 2 in $T$.

A subgroup of $G$ is _elliptic_ if it fixes a point in $T$ (equivalently, it
is contained in a conjugate of a vertex group of $\Gamma$). The splitting
$\Gamma$ and the tree $T$ are _relative to subgroups_ $H_{1},\dots,H_{p}$ if
each $H_{i}$ is elliptic.

If $T,T^{\prime}$ are trees with an action of $G$, we say that $T$ _dominates_
$T^{\prime}$ (or that $\Gamma=T/G$ dominates $\Gamma^{\prime}=T^{\prime}/G$)
if there is a $G$-equivariant map from $T$ to $T^{\prime}$, equivalently if
every subgroup which is elliptic in $T$ is also elliptic in $T^{\prime}$. Two
trees (or two splittings) belong to the same _deformation space_ if each one
dominates the other (equivalently, if they have the same elliptic subgroups).

In particular, $T$ dominates all trees $T^{\prime}$ obtained by collapsing
each edge of $T$ belonging to a given $G$-invariant set to a point. Such a
$T^{\prime}$ is called a _collapse_ of $T$; one passes from $\Gamma=T/G$ to
$\Gamma^{\prime}=T^{\prime}/G$ by collapsing certain edges and redefining the
vertex groups. Conversely, we say that $T$ is a refinement of $T^{\prime}$.
One obtains $\Gamma$ from $\Gamma^{\prime}$ by blowing up certain vertices $v$
of $\Gamma^{\prime}$ using a splitting of $G_{v}$ relative to the incident
edge groups.

The (closed) _star_ of a vertex $v$ in a graph or a tree is the union of all
edges containing $v$. The open star of $v$ is obtained by removing all
vertices $w\neq v$ from the closed star.

Given an action of $G$ on a tree $T$, we say that $T$ is _1-acylindrical near
a vertex_ $v$ if $G_{e}\cap G_{e^{\prime}}=\\{1\\}$ whenever $e,e^{\prime}$
are distinct edges containing $v$. It is _2-acylindrical_ if all segments of
length 3 have trivial stabilizer.

###  3.4 Grushko decompositions.

A finitely generated group $G$ is either trivial, or one-ended, or isomorphic
to ${\mathbb{Z}}$, or it has infinitely many ends (equivalently, it is a non-
trivial free product); recall that all groups are assumed to be torsion-free.

Any finitely generated group $G$ has a _Grushko decomposition_ : it is
isomorphic to a free product $A_{1}*\cdots*A_{n}*\mathbb{F}_{m}$, where each
$A_{i}$ is one-ended and $\mathbb{F}_{m}$ is a free group of rank $m$.
Moreover, the numbers $n$ and $m$ are unique, as well as the $A_{i}$’s up to
conjugacy (in $G$) and permutation. We call the $A_{i}$’s the _Grushko
factors_ of $G$.

Decompositions as free products correspond to graph of groups decompositions
with trivial edge groups (splittings over the trivial group). If $G$ has
infinitely many ends, it has a Grushko decomposition
$G=A_{1}*\cdots*A_{n}*\mathbb{F}_{m}$ as above. It may be viewed as the
fundamental group of a graph of groups with one central vertex $v$ carrying a
trivial group, joined to vertices $v_{1},\dots,v_{n}$ carrying
$A_{1},\dots,A_{n}$ by a single edge, with $m$ loops attached to $v$.

If ${\mathcal{H}}=\\{H_{1},\dots,H_{p}\\}$ is a family of non-trivial
subgroups of $G$, we say that $G$ is one-ended relative to ${\mathcal{H}}$ if
it cannot be written as a non-trivial free product with each $H_{i}$ contained
in a conjugate of one of the factors. If $G$ is not one-ended relative to
${\mathcal{H}}$, it has a non-trivial Grushko decomposition relative to
${\mathcal{H}}$; it is of the form $G=A_{1}*\cdots*A_{n}*\mathbb{F}_{m}$, with
each $H_{i}$ contained in some $A_{j}$ (up to conjugacy), and $A_{j}$ one-
ended relative to the conjugates of the $H_{i}$’s which it contains ($A_{j}$
free is allowed only if it contains a conjugate of an $H_{i}$).

###  3.5 Surface-type vertices

Let $\Gamma$ be a splitting of $G$, with Bass-Serre tree $T$.

######  Definition 3.5 (Surface-type vertex).

A vertex $v$ of $\Gamma$ is called a _surface-type vertex_ if the following
conditions hold:

  * •

the group $G_{v}$ carried by $v$ is the fundamental group of a compact surface
$\Sigma$ (usually with boundary, possibly non-orientable), with
$\chi(\Sigma)<0$;

  * •

incident edge groups are maximal boundary subgroups of $\pi_{1}(\Sigma)$, and
this induces a bijection between the set of incident edges and the set of
boundary components of $\Sigma$.

The vertex $v$ is _exceptional_ if $\Sigma$ is exceptional (see Definition
3.3).

We say that $G_{v}=\pi_{1}(\Sigma)$, which we often call $Q$, is a surface-
type vertex group, and that $\Sigma$ is a surface of $\Gamma$. If $u$ is any
lift of $v$ to $T$, we say that $u$ is a surface-type vertex, and its
stabilizer $G_{u}$ (which is conjugate to $Q$) is a surface-type vertex
stabilizer.

If $G$ is the fundamental group of a closed hyperbolic surface, it is a
surface-type vertex group (with $\Gamma$ the trivial splitting).

Surface-type vertices are QH (quadratically hanging) vertices, see [GL17]. The
tree $T$ is 1-acylindrical near any surface-type vertex $u$: if $e$ and
$e^{\prime}$ are distinct edges containing $u$, then $G_{e}\cap
G_{e^{\prime}}=\\{1\\}$.

We will need two standard lemmas about surface groups.

######  Lemma 3.6 (compare Lemma 1.4 of [Sel04] and Lemma 3.11 of [Per11]).

Let $Q=\pi_{1}(\Sigma)$ be a surface-type vertex group of a splitting of $G$.
Let $J\subset G$ be a one-ended finitely generated group. If $J\cap Q$ is non-
trivial and has infinite index in $Q$, then $J\cap Q$ is a boundary subgroup
of $Q=\pi_{1}(\Sigma)$, in particular $J\cap Q\simeq{\mathbb{Z}}$.

###### Sketch of proof.

Assume that $J\cap Q$ has infinite index in $Q$, is non-trivial, not a
boundary subgroup. By Scott’s Theorem [Sco78], there is a finite covering
$\Sigma_{0}$ of $\Sigma$ such that $J\cap Q$ is the fundamental group of a
proper subsurface $\Sigma_{A}\subset\Sigma_{0}$. This subsurface contains a
boundary component of $\Sigma_{0}$ because $J$, being one-ended, must have
non-trivial intersection with some boundary subgroup of $Q$. It also has a
boundary component $C$ which is not parallel to a component of
$\partial\Sigma_{0}$. An essential arc going from $C$ to $C$ within
$\Sigma_{A}$ yields a free splitting of $J$, contradicting one-endedness. ∎

######  Lemma 3.7.

Let $\Sigma$ be a compact surface. Let $p:\pi_{1}(\Sigma)\to G_{1}*G_{2}$ be a
homomorphism to a non-trivial free product such that the image of every
boundary subgroup of $\pi_{1}(\Sigma)$ is contained in a conjugate of $G_{1}$
or $G_{2}$, with at least one image contained in a conjugate of $G_{1}$. If
$p$ is non-pinching (Definition 3.4), its image is contained in a conjugate of
$G_{1}$.

###### Sketch of proof.

If the conclusion is false, consider a $p$-equivariant map $f$ from the
universal covering $\tilde{\Sigma}$ of $\Sigma$ to the Bass-Serre tree of the
amalgam $G_{1}*G_{2}$, with $f$ mapping each boundary component of
$\tilde{\Sigma}$ to a vertex of the tree. The projection to $\Sigma$ of the
preimage of midpoints of edges contains a pinched curve. ∎

###  3.6 The cyclic JSJ decomposition of hyperbolic groups and CSA groups

Recall that a finitely generated group $G$ is hyperbolic if its Cayley graph
(with respect to some, hence any, finite generating set) is a hyperbolic
metric space: there exists $\delta>0$ such that any point on one side of any
geodesic triangle is $\delta$-close to one of the other two sides (see [BH99,
GdlH90]). Small cancellation groups and fundamental groups of negatively
curved closed manifolds are hyperbolic. In particular free groups, fundamental
groups of closed surfaces with negative Euler characteristic, and free
products of such groups, are hyperbolic.

Any torsion-free hyperbolic group is Hopfian [Sel99]: it cannot be isomorphic
to a proper quotient. Any one-ended torsion-free hyperbolic group is co-
Hopfian [Sel97]: it cannot be isomorphic to a proper subgroup.

A one-ended torsion-free hyperbolic group $G$ does not split over a trivial
group, but it may have splittings with edge groups isomorphic to
${\mathbb{Z}}$ and it has a canonical JSJ decomposition
$\Gamma_{\\!\mathrm{can}}$ over cyclic groups [Sel97, Bow98, GL17]. We mention
the features that will be important for this paper. Similar considerations
apply to relative JSJ decompositions, see [GL17].

A subgroup $H\subset G$ is _universally elliptic_ if it is elliptic in every
cyclic splitting of $G$. A tree is universally elliptic if its edge
stabilizers are. A JSJ tree is a universally elliptic tree which dominates
every universally elliptic tree. In general, only the deformation space of JSJ
trees is well-defined, but in the case of one-ended hyperbolic groups there is
a canonical JSJ tree $T_{\mathrm{can}}$ (see [GL17]); it is the tree of
cylinders of any JSJ tree. It may also be constructed from the topology of the
Gromov boundary $\partial G$ of $G$ [Bow98]. It is invariant under
automorphisms of $G$.

All edge groups of the graph of groups
$\Gamma_{\\!\mathrm{can}}=T_{\mathrm{can}}/G$ are infinite cyclic. The graph
$\Gamma_{\\!\mathrm{can}}$ is bipartite, with every edge joining a cyclic
vertex (i.e. a vertex carrying a cyclic group) to a non-cyclic vertex. Non-
cyclic vertex groups are either rigid (universally elliptic) or surface-type
(in the sense of Definition 3.5). The action of $G$ on the associated Bass-
Serre tree $T_{\mathrm{can}}$ is 1-acylindrical near every non-cyclic vertex.

The tree $T_{\mathrm{can}}$ is universally compatible: if $T$ is any tree with
cyclic stabilizers on which $G$ acts, $T_{\mathrm{can}}$ and $T$ have a common
refinement $\hat{T}$ (their lcm, see [GL17] Section A.5): one may obtain each
of the trees $T_{\mathrm{can}}$ and $T$ from $\hat{T}$ by collapsing each edge
in some $G$-invariant set to a point.

More generally, recall that a group is CSA if maximal abelian subgroups are
malnormal. Equivalently, a group is CSA if it is commutative transitive and,
whenever $[tat^{-1},a]=1$ for some $a\neq 1$, then $t$ commutes with $a$. In
particular, torsion-free hyperbolic groups are CSA.

A one-ended torsion-free CSA group has a canonical cyclic JSJ decomposition
relative to non-cyclic abelian subgroups (see Theorem 9.5 of [GL17]).

It has properties similar to the hyperbolic case, but since we consider the
cyclic decomposition rather than the abelian one it is not bipartite. Each
edge joins an abelian vertex (i.e. a vertex carrying an abelian group) to a
non-abelian vertex, or two non-abelian vertices. Non-abelian vertex groups are
rigid or surface-type, and no edge joins two surface-type vertices. The Bass-
Serre tree is 2-acylindrical.

Figure 4: The modified JSJ decomposition $\tilde{\Gamma}_{\\!\mathrm{can}}$
(with $R_{1},R_{2}$ rigid vertex groups)

##### The modified JSJ decomposition

In Section 9 we will need to modify $\Gamma_{\\!\mathrm{can}}$ so that all
exceptional surfaces appearing at surface-type vertices have finite mapping
class group, while preserving the invariance under automorphisms (see the
proof of Lemma 9.20).

Recall that the exceptional surfaces with infinite mapping class group are
$N_{2,1}$ (the once-punctured Klein bottle) and the closed surface $N_{3}$
(see Subsection 3.2). As explained in [GL17, Remark 9.32], if
$\Gamma_{\\!\mathrm{can}}$ has vertices which are surface-type with associated
surface $N_{2,1}$, one may cut the surfaces along their special curve
$c_{N_{2,1}}$ and thus obtain a splitting $\tilde{\Gamma}_{\\!\mathrm{can}}$
refining $\Gamma_{\\!\mathrm{can}}$ (see Figure 4). Since the homotopy class
of $c_{N_{2,1}}$ is invariant under the mapping class group of $N_{2,1}$, the
splitting $\tilde{\Gamma}_{\\!\mathrm{can}}$ is still invariant under the
group of automorphisms of $G$. Similarly, if $G\simeq\pi_{1}(N_{3})$, we
replace $\Gamma_{\\!\mathrm{can}}$ (which is the trivial splitting) by the
splitting dual to the decomposition of $N_{3}$ into a once-punctured torus and
a Möbius band.

###  3.7 Two lemmas about splittings

The following lemma is folklore (see the proof of Proposition 4.7 in [GLS19]).

######  Lemma 3.8.

Let $A\subset G$, and let $G_{v}$ be a vertex group of a splitting $\Gamma$ of
$A$. Let $p:G_{v}\to G$ be a homomorphism such that the restriction of $p$ to
any incident edge group $G_{e}\subset G_{v}$ agrees with conjugation by some
element of $G$ (depending on $e$). Then $p$ may be extended “by the identity”
to $\hat{p}:A\to G$. The extension $\hat{p}$ agrees with a conjugation on each
vertex group of $\Gamma$ other than $G_{v}$. ∎

The next lemma will be used in Subsection 9.7. It generalizes the injectivity
part of Proposition 6.1 of [Per11]. The argument is the same as in [Per11], we
give it for completeness.

######  Lemma 3.9.

Let $A$ and $G$ be groups acting on trees $T_{A}$, $T_{G}$ with all edge
stabilizers isomorphic to ${\mathbb{Z}}$. Assume that these trees are
bipartite, with vertices of type 0 or 1, and 1-acylindricity holds near
vertices of type 1.

Let $f:A\to G$ be a homomorphism sending any vertex stabilizer of type 0 of
$T_{A}$ injectively into a vertex stabilizer of type 0 of $T_{G}$, and any
vertex stabilizer of type 1 bijectively to a vertex stabilizer of type 1.

If $f$ is not injective, there exist two non-conjugate vertex stabilizers of
type 1 of $T_{A}$ with the same image.

###### Proof.

By acylindricity, vertex groups of type 1 are not ${\mathbb{Z}}$ so fix a
unique point. Vertex groups of type 0 fix a unique vertex of type 0 (and
possibly vertices of type 1). We denote by $A_{v}$ the stabilizer of a vertex
$v$ of $T_{A}$.

Given a vertex $v$ of $T_{A}$, let $\varphi(v)$ be the unique vertex of
$T_{G}$ which has the same type as $v$ and is fixed by $f(A_{v})$. By
1-acylindricity, $\varphi$ maps adjacent vertices to adjacent vertices. Assume
that $f$ is not injective. There must then be distinct edges $vu_{1}$ and
$vu_{2}$ of $T_{A}$ with $\varphi(u_{1})=\varphi(u_{2})$. By $1$-acylindricity
and injectivity of $f$ on vertex groups, $v$ is of type 0 (and thus
$u_{1},u_{2}$ are of type 1). The lemma is proved if $u_{1}$ and $u_{2}$ are
in different orbits, so assume $u_{2}=au_{1}$ with $a\in A$.

Since $f(a)$ fixes $\varphi(u_{1})$, which is of type 1, there exists
$a^{\prime}\in A_{u_{1}}$ such that $f(a^{\prime})=f(a)$. Replacing $a$ by
$aa^{\prime}{{}^{-1}}$ lets us assume $u_{2}=au_{1}$ with $a\in\ker f$. This
implies $av\neq v$.

Let $a_{1},a_{2}$ be non-trivial elements of $A$ fixing the edges $vu_{1}$ and
$vu_{2}$ respectively. The element $aa_{1}a^{-1}$ fixes the edge joining $av$
and $u_{2}$, so $\langle aa_{1}a^{-1},a_{2}\rangle$ is not cyclic by
1-acylindricity near $u_{2}$. On the other hand, $f(aa_{1}a^{-1})=f(a_{1})$
and $f(a_{2})$ both fix the edge joining $\varphi(v)$ to
$\varphi(u_{1})=\varphi(u_{2})$, so generate a cyclic subgroup of $G$. This
contradicts the fact that $f$ is injective on $A_{u_{2}}$. ∎

######  Corollary 3.10.

Let $G$ be a non-cyclic torsion-free hyperbolic group acting on a tree $T$
with all edge stabilizers ${\mathbb{Z}}$.

Let $f:G\to G$ be a homomorphism sending any non-cyclic vertex stabilizer and
any edge stabilizer bijectively to a conjugate. Then $f$ is injective.

###### Proof.

We apply the lemma with $A=G$ and $T_{A},T_{G}$ both equal to the tree of
cylinders of $T$ (see [GL11] or [GL17, Section 7]). It is bipartite:
stabilizers of vertices of type 0 are maximal cyclic and commensurable to an
edge stabilizer of $T$, stabilizers of vertices of type 1 are the non-cyclic
vertex stabilizers of $T$. The hypotheses of Lemma 3.9 are satisfied. ∎

##  4 Towers

In the beginning of this section, $G$ will be an arbitrary torsion-free
finitely generated group. In Proposition 4.14, we will see that subgroups
$B<G$ having non-trivial elements $x,y$ such that $x$ commutes with all
conjugates of $y$ play a special role, and from then on we shall assume that
$G$ is CSA to ensure that such subgroups $B$ are abelian. This is mostly a
matter of convenience, as it makes statements simpler. In Subsections 5.4 and
6.2.2, however, we will use the canonical cyclic JSJ decomposition of $G$, and
this really requires $G$ to be CSA.

Figure 5: A centered splitting of $G$ with three bottom groups
$B_{1},B_{2},B_{3}$. Its base is the free product $B_{1}*B_{2}*B_{3}$.

###  4.1 Centered splittings

######  Definition 4.1 (Centered splitting).

A _centered splitting_ of $G$ is a graph of groups decomposition
$G=\pi_{1}(\Gamma)$ such that the vertices of $\Gamma$ are
$v,v_{1},\dots,v_{n}$, with $n\geq 1$, where $v$ is surface-type and every
edge joins $v$ to some $v_{i}$ (see Figure 5). Note that all edge groups are
isomorphic to ${\mathbb{Z}}$.

The vertex $v$ is called the _central vertex_ of $\Gamma$, the group carried
by $v$ is denoted by $Q=\pi_{1}(\Sigma)$, and $\Sigma$ is the surface
associated to $\Gamma$. The vertices $v_{1},\dots,v_{n}$ are the _bottom
vertices_ , and we denote by $B_{i}$ the group carried by $v_{i}$. We call
$B_{i}$ a bottom group, and $Q$ is the central group.

The _base_ of $\Gamma$ is the abstract free product
$B_{\Gamma}=B_{1}*\dots*B_{n}$.

The centered splitting $\Gamma$ is _simple_ if $n=1$, _non-exceptional_ if the
surface $\Sigma$ is non-exceptional.

If $\Gamma$ is simple and $B_{1}\simeq{\mathbb{Z}}$, we say that $G$ is a
_parachute_ (parachutes are studied in Propositions 6.7 and 6.8 of [GLS19] and
in Subsection 10.6).

######  Remark 4.2.

Each $B_{i}$ is well-defined up to conjugacy in $G$, but if $n\geq 2$ the
subgroup $\langle B_{1},\dots,B_{n}\rangle$ itself is not well-defined up to
conjugacy (it depends on choices of individual conjugates), and it does not
have to be isomorphic to $B_{\Gamma}$ in general. See Proposition 4.21, where
particular choices of conjugates are made.

######  Remark 4.3.

If $G$ is hyperbolic, the groups $B_{i}$ are quasiconvex, hence hyperbolic.

Like all splittings, $\Gamma$ is assumed to be minimal: if $v_{i}$ belongs to
only one edge, the group carried by this edge is a proper subgroup of $B_{i}$.
As in every surface-type vertex, each conjugacy class of boundary subgroups of
$Q$ is carried by exactly one incident edge (all boundary components are
used). We allow redundant vertices: $\Gamma$ may have a cyclic vertex $v_{i}$
of valence 2 with both incident edge groups equal to the vertex group.

Given any non-trivial splitting with a surface-type vertex $v$, collapsing all
edges not containing $v$ (and subdividing if there is a loop at $v$) yields a
centered splitting.

######  Remark 4.4.

By 1-acylindricity near $v$, the family $(B_{1},\dots,B_{n})$ is malnormal: if
$gB_{i}g^{-1}\cap B_{j}\neq\\{1\\}$, then $i=j$ and $g\in B_{i}$. In
particular, each $B_{i}$ is malnormal and, if two elements of $B_{i}$ are
conjugate in $G$, then they are conjugate in $B_{i}$.

See Subsection 5.3 for a few general facts about centered splittings.

###  4.2 Boundary-preserving maps, pinched quotient

######  Definition 4.5 (Boundary-preserving map).

Let $G$ be a group, and $Q\simeq\pi_{1}(\Sigma)$ a subgroup identified with
the fundamental group of a compact surface $\Sigma$.

A _boundary-preserving map_ (with values in $G$) is a homomorphism $p:Q\to G$
such that, on each boundary subgroup of $\pi_{1}(\Sigma)$, the map $p$ agrees
with the conjugation by some element of $G$.

The boundary-preserving map is _non-degenerate_ if additionally:

  * •

the image of $p$ is non-abelian;

  * •

$p$ is not an isomorphism of $Q$ onto some conjugate of $Q$;

  * •

the surface $\Sigma$ is non-exceptional.

If $Q$ is the group carried by a surface-type vertex $v$ of a splitting
$G_{1}=\pi_{1}(\Gamma)$ of a subgroup $G_{1}<G$, we say that the boundary-
preserving map is _associated to_ $\Gamma$, or to $v$ (most often, $G_{1}$
will be $G$ or a free factor of $G$).

######  Remark 4.6.

The first two additional conditions ensure that $p$ carries meaningful
information; in particular, it is not the identity and does not factor through
an abelian group. Non-degenerate boundary-preserving maps were called local
preretractions in [GLS19].

The non-exceptionality assumption is not needed until Section 7, but it is
essential for applications to model theory: in particular, “Tarski” (Theorem
2.1) would not hold without it.

Figure 6: A once-punctured torus glued on some non-trivial element $g\in
B_{1}$.

######  Example 4.7 (See Figure 6).

Let $G=B_{1}*_{g=[x,y]}F(x,y)$ be a cyclic amalgam of a group $B_{1}$ with the
free group $F(x,y)$, with $g$ some non-trivial element of $B_{1}$. The amalgam
defines a simple centered splitting $\Gamma$ of $G$, with $\Sigma$ a punctured
torus.

If $g$ is a commutator $[b,b^{\prime}]$ with $b,b^{\prime}\in B_{1}$, sending
$x$ to $b$ and $y$ to $b^{\prime}$ defines a non-degenerate boundary-
preserving map $p$ associated to $\Gamma$ (Lemma 4.9 below implies that there
is no such $p$ if $g$ is not a commutator in $B_{1}$).

Figure 7: A pinched quotient.

Recall (Definition 3.4) that a curve $C$ is pinched by $p$ if
$\pi_{1}(C)\subset\ker p$. In the following definition, we will often take
$G_{1}=G$, or $G_{1}$ a Grushko factor of $G$.

######  Definition 4.8 (Pinched quotient, see Figure 7 and Definition 5.26 of
[Per11]).

Given a boundary-preserving map $p:\pi_{1}(\Sigma)\to G$, let ${\mathcal{C}}$
be a family of pinched curves on $\Sigma$. If $G_{1}$ is a subgroup of $G$
containing $\pi_{1}(\Sigma)$, the _pinched quotient_ of $G_{1}$ is the
quotient $(G_{1})_{{\mathcal{C}}}$ of $G_{1}$ obtained by killing all curves
in ${\mathcal{C}}$. It can be described as follows.

For each curve $C\in{\mathcal{C}}$, we cut $\Sigma$ along $C$, glue a disc to
each of the two boundary curves thus created, and join these two discs by an
arc. We get a space $\Sigma_{\mathcal{C}}$ consisting of surfaces
$\Sigma_{\mathcal{C}}^{j}$ and arcs; these surfaces are closed or bounded by
components of $\partial\Sigma$.

The surfaces/arcs of $\Sigma_{\mathcal{C}}$ correspond to vertices/edges of a
graph of groups decomposition $\Delta_{\mathcal{C}}$ of
$\pi_{1}(\Sigma_{\mathcal{C}})$ with trivial edge groups. There is a quotient
map $\pi_{\mathcal{C}}:\pi_{1}(\Sigma)\to\pi_{1}(\Sigma_{\mathcal{C}})$. The
map $p$ factors through $\pi_{\mathcal{C}}$ and induces a map
$p_{\mathcal{C}}:\pi_{1}(\Sigma_{\mathcal{C}})\to G$.

Now suppose that $p$ is associated to a splitting $G_{1}=\pi_{1}(\Gamma)$ of a
subgroup $G_{1}\subset G$. Since $p$ is boundary-preserving, no component of
$\partial\Sigma$ is pinched, and we can define a graph of groups
$\Gamma_{\mathcal{C}}$ by blowing up the vertex $v$ of $\Gamma$ carrying
$\Sigma$ using $\Delta_{\mathcal{C}}$.

The fundamental group of $\Gamma_{\mathcal{C}}$ is isomorphic to the pinched
quotient $(G_{1})_{\mathcal{C}}$. The quotient map from $G_{1}$ to
$(G_{1})_{\mathcal{C}}$ is an extension of
$\pi_{\mathcal{C}}:\pi_{1}(\Sigma)\to\pi_{1}(\Sigma_{\mathcal{C}})$ and we
also denote it by $\pi_{\mathcal{C}}$. It is injective on vertex groups of
$\Gamma$ other than $\pi_{1}(\Sigma)$.

###  4.3 A key lemma

The following key lemma will be used extensively to study boundary-preserving
maps.

######  Lemma 4.9 ([GLS19, Lemma 3.3]).

Let $p:\pi_{1}(\Sigma)\to G$ be a non-degenerate boundary-preserving map
associated to a centered splitting $\Gamma$ of $G$. Let ${\mathcal{C}}$ be a
maximal family of pinched curves on $\Sigma$, and let $\Sigma^{\prime}$ be a
component of the surface obtained by cutting $\Sigma$ along ${\mathcal{C}}$.

Viewing $\pi_{1}(\Sigma^{\prime})$ as a subgroup of $\pi_{1}(\Sigma)$, the
image of $\pi_{1}(\Sigma^{\prime})$ by $p$ is contained in a conjugate of a
bottom vertex group $B_{i}$ of $\Gamma$. ∎

######  Remark 4.10.

By 1-acylindricity, the lemma implies that all edges of $\Gamma$ associated to
boundary components of $\Sigma$ contained in $\Sigma^{\prime}$ join the
central vertex to the same bottom vertex, namely the vertex $v_{i}$ carrying
$B_{i}$.

The following strengthening of Lemma 4.9 will be used later. Recall that a
subsurface is incompressible if the inclusion induces an injection on
fundamental groups.

######  Lemma 4.11 ([GLS19, Remark 3.4]).

Let $\Gamma$ be a centered splitting of $G$. We denote by $\Sigma$ its
surface. Let $S$ be another compact surface, and let $p:\pi_{1}(S)\to G$ be a
homomorphism such that the image of each boundary subgroup of $\pi_{1}(S)$ is
contained in a bottom group of $\Gamma$ (up to conjugacy).

Let ${\mathcal{C}}$ be a maximal family of pinched curves on $S$, and let
$S^{\prime}$ be a component of the surface obtained by cutting $S$ along
${\mathcal{C}}$. Then, up to conjugacy in $G$, the image of
$\pi_{1}(S^{\prime})$ by $p$ is contained in a bottom group of $\Gamma$, or
there is an incompressible subsurface $Z\subset S^{\prime}$ such that $p$ maps
boundary subgroups of $\pi_{1}(Z)$ into boundary subgroups of
$\pi_{1}(\Sigma)$, and $p(\pi_{1}(Z))$ is a finite index subgroup of
$\pi_{1}(\Sigma)$. ∎

The next lemma will be used in Subsection 9.3. Recall that the closed non-
orientable surface of genus 3 contains a one-sided simple closed curve
$c_{N_{3}}$ whose complement is a once-punctured torus (see Section 3.2); it
is unique up to isotopy. We denote by $g_{N_{3}}$ a generator of
$\pi_{1}(c_{N_{3}})$ (well-defined up to inversion and conjugacy).

######  Lemma 4.12.

Let $\Gamma$ and $p$ be as in Lemma 4.11. Assume furthermore that $S$ is
exceptional and $\Sigma$ is not. If the image of $p$ is not contained in a
conjugate of a bottom group of $\Gamma$, then $S$ is a closed non-orientable
surface of genus 3, $\Sigma$ is a once-punctured torus, and $p(g_{N_{3}}^{2})$
belongs to the edge group of $\Gamma$ (up to conjugacy).

###### Proof.

Assume that the image of $p$ is not contained in a conjugate of a bottom group
of $\Gamma$ and apply Lemma 4.11.

We claim that there is no pinching. Assume there is. Since $S$ is exceptional,
$p$ factors through a group isomorphic to
${\mathbb{Z}}*{\mathbb{Z}}/2{\mathbb{Z}}$,
${\mathbb{Z}}^{2}*{\mathbb{Z}}/2{\mathbb{Z}}$, or
$\pi_{1}(N_{2})*{\mathbb{Z}}/2{\mathbb{Z}}$. Since $G$ is torsion-free, the
image of $p$ is abelian or isomorphic to the Klein bottle group
$\pi_{1}(N_{2})$. This implies that the image of $p$ is conjugate into a
bottom group of $\Gamma$, a contradiction.

The exceptional surface $S$ has Euler characteristic $-1$ and fundamental
group ${\mathbb{F}}_{2}$, so there is an incompressible subsurface $Z\subset
S$ such that $p$ induces an isomorphism between $\pi_{1}(Z)$ and
$\pi_{1}(\Sigma)$, with $\Sigma$ a once-punctured torus. Up to conjugacy, $p$
maps the boundary subgroups of $Z$ into the boundary subgroups of $\Sigma$. In
particular, boundary subgroups of $\pi_{1}(Z)$ are contained in the derived
subgroup of $Z$, and this is possible only if $Z$ is a once-punctured torus.
It follows that $S$ is the exceptional closed surface, the union of the
punctured torus $Z$ and a Möbius band. ∎

We will also need:

######  Lemma 4.13.

If $p:Q\to G$ is a non-pinching non-degenerate boundary-preserving map
associated to a centered splitting $\Gamma$ of $G$, there is a unique bottom
group $B_{1}$ (i.e. $\Gamma$ is simple), and the image of $Q$ is contained in
a conjugate of $B_{1}$.

If $\hat{p}:G\rightarrow G$ is an extension of $p$ which maps $B_{1}$ onto a
conjugate $B_{1}^{g}$, then $B_{1}^{g}$ contains the whole image of $\hat{p}$.
In particular, $\hat{p}$ is not injective.

Such extensions $\hat{p}$ exist by Lemma 3.8.

###### Proof.

The first assertion follows from Lemma 4.9 and Remark 4.10.

By 1-acylindricity, any two distinct conjugates of $B_{1}$ have trivial
intersection. Now consider two adjacent vertices in the Bass-Serre tree $T$ of
$\Gamma$. Their stabilizers are conjugate to $Q$ or $B_{1}$, and their
intersection is a cyclic group on which $\hat{p}$ is injective, so their
images by $\hat{p}$ are contained in the same conjugate of $B_{1}$. Thus all
vertex stabilizers are mapped into the same conjugate. This conjugate is
invariant under conjugation by every element of $\hat{p}(A)$, and malnormal,
so the lemma follows. ∎

###  4.4 Simple towers

Recall that a centered splitting is simple if it has a single bottom group
(Definition 4.1).

######  Proposition 4.14.

Let $G$ be any torsion-free finitely generated group. Let $\Gamma$ be a non-
exceptional _simple_ centered splitting of $G$, with bottom group $B_{1}$ and
surface group $Q=\pi_{1}(\Sigma)$. Assume that any two non-trivial elements of
$B_{1}$ have conjugates which do not commute (this holds in particular if
$B_{1}$ is CSA and non-abelian). Then the following are equivalent:

  1. 1.

There exists a non-degenerate boundary-preserving map $p:Q\to G$.

  2. 2.

There exists a retraction $\rho:G\to B_{1}$ with $\rho(Q)$ non-abelian.

###### Proof.

For $2\Rightarrow 1$, we simply let $p$ be the restriction of $\rho$ to $Q$.
We now prove $1\Rightarrow 2$. Denoting by $b$ the number of components of
$\partial\Sigma$, we may present $G$ as

$G=\langle\pi_{1}(\Sigma),B_{1},t_{2},\dots,t_{b}\mid
c_{1}=a_{1},t_{i}c_{i}t_{i}^{-1}=a_{i}\ \text{for}\ i=2,\dots,b\rangle$

where $a_{i}\in B_{1}$ and the $c_{i}$’s are generators of maximal boundary
subgroups $C_{i}$ of $\pi_{1}(\Sigma)$, see Figure 8.

Since $p$ maps $c_{1}$ to a conjugate, we may assume after composing $p$ with
a conjugation that $p(c_{1})=c_{1}$. Now assume that there is no pinched
curve. By Lemma 4.9, the image of $\pi_{1}(\Sigma)$ by $p$ is contained in a
conjugate of $B_{1}$. This conjugate contains $p(c_{1})=a_{1}$, so must be
$B_{1}$ (by malnormality, see Remark 4.4).

We define $\rho$ as being the identity on $B_{1}$, and equal to $p$ on
$\pi_{1}(\Sigma)$. This ensures that $\pi_{1}(\Sigma)$ has non-abelian image.
We must now define $\rho(t_{i})\in B_{1}$ in a way which is compatible with
the relation $t_{i}c_{i}t_{i}^{-1}=a_{i}$. We know that $p(c_{i})$ belongs to
$B_{1}$ and is conjugate to $c_{i}$, hence to $a_{i}$, in $G$. By
malnormality, $p(c_{i})$ is conjugate to $a_{i}$ in $B_{1}$, so
$p(c_{i})=g_{i}a_{i}g_{i}^{-1}$ for some $g_{i}\in B_{1}$. We then define
$\rho(t_{i})=g_{i}^{-1}$.

Figure 8: Proof of Proposition 4.14.

We now allow pinching (see Figure 8). We consider a pinched quotient
$\pi_{\mathcal{C}}:G\to G_{\mathcal{C}}$ as in Definition 4.8 (with
$G_{1}=G$), with its splitting $\Gamma_{\mathcal{C}}$. We identify $B_{1}$
with its image in $G_{\mathcal{C}}$. We shall define a retraction
$\rho^{\prime}:G_{\mathcal{C}}\to B_{1}$, and $\rho$ will be
$\rho^{\prime}\circ\pi_{\mathcal{C}}$.

Let $G^{\prime}$ be the fundamental group of the maximal subgraph
$\Gamma^{\mathbb{Z}}_{\mathcal{C}}$ of $\Gamma_{\mathcal{C}}$ containing no
edge with trivial group (the star of the bottom vertex $v_{1}$). It is a free
factor of $G_{\mathcal{C}}$, and we can write $G_{\mathcal{C}}\simeq
G^{\prime}*G^{\prime\prime}$, with $G^{\prime\prime}$ isomorphic to the free
product of closed surface groups (corresponding to surfaces
$\Sigma_{\mathcal{C}}^{j}$ disjoint from $\partial\Sigma$) and a free group
$F$ (the topological fundamental group of the graph obtained from
$\Gamma_{\mathcal{C}}$ by collapsing $\Gamma^{\mathbb{Z}}_{\mathcal{C}}$ to a
point, of rank 2 on Figure 8). Note that $G^{\prime\prime}$ either maps onto
${\mathbb{Z}}$ or is a (possibly trivial) free product of cyclic groups of
order 2 (fundamental groups of projective planes).

First suppose that all components of $\partial\Sigma$ belong to the same
surface $\Sigma_{\mathcal{C}}^{j_{0}}$. We can define $\rho^{\prime}$ on
$G^{\prime}$ in the same way as we defined $\rho$ in the non-pinching case,
and define $\rho^{\prime}$ arbitrarily on $G^{\prime\prime}$. We just have to
make sure that the image of $\pi_{1}(\Sigma_{\mathcal{C}})$ is not abelian.

This is easy to do, using the assumption on $B_{1}$, if $G^{\prime\prime}$
maps onto ${\mathbb{Z}}$, since $\pi_{1}(\Sigma_{\mathcal{C}})$ may be
identified with $\pi_{1}(\Sigma_{\mathcal{C}}^{j_{0}})*G^{\prime\prime}$. If
$G^{\prime\prime}$ is a free product of cyclic groups of order at most 2, it
is mapped trivially by the map
$p_{\mathcal{C}}:\pi_{1}(\Sigma_{\mathcal{C}})\to G$ because $G$ is torsion-
free, and the image of $\pi_{1}(\Sigma_{\mathcal{C}})$ by $\rho^{\prime}$ is
non-abelian because it is isomorphic to the image of $\pi_{1}(\Sigma)$ by $p$.

Now suppose that there are at least two surfaces
$\Sigma_{\mathcal{C}}^{j_{1}}$ and $\Sigma_{\mathcal{C}}^{j_{2}}$ containing a
component of $\partial\Sigma$. The key difference with the previous case is
that we can no longer assume
$G^{\prime\prime}\subset\pi_{1}(\Sigma_{\mathcal{C}})$, so making the image of
$\pi_{1}(\Sigma_{\mathcal{C}})$ non-abelian is less direct.

We can still define $\rho^{\prime}$ on $G^{\prime}$: for each
$\Sigma_{\mathcal{C}}^{j}$ containing a component of $\partial\Sigma$, we
change the restriction of $p$ to $\pi_{1}(\Sigma_{\mathcal{C}}^{j})$ by a
conjugation so that it fixes some boundary subgroup of
$\pi_{1}(\Sigma_{\mathcal{C}}^{j})$, just like we ensured $p(c_{1})=c_{1}$ in
the non-pinching case, and we define $\rho^{\prime}$ on the whole of
$G^{\prime}$ as above. We now explain how to use $G^{\prime\prime}$ to make
the image of $\pi_{1}(\Sigma_{\mathcal{C}})$ non-abelian.

In defining $\rho^{\prime}$ on $G^{\prime}$, we have viewed
$\pi_{1}(\Sigma_{\mathcal{C}}^{j_{1}})$ and
$\pi_{1}(\Sigma_{\mathcal{C}}^{j_{2}})$ as subgroups of $G^{\prime}$ (vertex
groups of $\Gamma^{\mathbb{Z}}_{\mathcal{C}}$). When we view them as subgroups
of $\pi_{1}(\Sigma_{\mathcal{C}})$, hence as vertex groups of the free
splitting $\Delta_{\mathcal{C}}$ of $\pi_{1}(\Sigma_{\mathcal{C}})$, they get
replaced by conjugates. More precisely, connectedness of $\Sigma$ implies that
the free group $F$ appearing as a free factor in $G^{\prime\prime}$ is non-
trivial, and we can find a primitive element $t$ of $F$ such that a conjugate
of $\pi_{1}(\Sigma_{\mathcal{C}})$ contains
$\pi_{1}(\Sigma_{\mathcal{C}}^{j_{1}})$ and
$t\pi_{1}(\Sigma_{\mathcal{C}}^{j_{2}})t^{-1}$.

We may choose $\rho^{\prime}(t)$ arbitrarily, and we shall define it so that
the image of $\pi_{1}(\Sigma_{\mathcal{C}})$ by $\rho^{\prime}$ is non-
abelian. Since the images of $\pi_{1}(\Sigma_{\mathcal{C}}^{j_{1}})$ and
$\pi_{1}(\Sigma_{\mathcal{C}}^{j_{2}})$ by $\rho^{\prime}$ contain some non-
trivial elements $a_{1},a_{2}$, our assumption on $B_{1}$ ensures that there
exists $x\in B_{1}$ such that $\langle a_{1},xa_{2}x^{-1}\rangle$ is non-
abelian, and we can define $\rho^{\prime}(t)=x$. We conclude by extending
$\rho^{\prime}$ from $\langle t\rangle$ to $G^{\prime\prime}$ arbitrarily. ∎

If $G$ is CSA (in particular, if $G$ is hyperbolic), the hypothesis on $B_{1}$
simply asks that $B_{1}$ be non-abelian. _From now on, we assume that $G$ is a
torsion-free finitely generated CSA group._

######  Remark 4.15.

Assume that $B_{1}$ is abelian and 1 holds. Then $p$ is pinching by Lemma 4.9
and there exists an epimorphism
$\pi^{\prime\prime}:G^{\prime\prime}\twoheadrightarrow{\mathbb{Z}}$. Using
$\pi^{\prime\prime}$, we can construct a surjective map $\tilde{\rho}:G\to
B_{1}*{\mathbb{Z}}$, equal to the identity on $B_{1}$, such that
$\tilde{\rho}(Q)$ is non-abelian. If we let $t\in G$ be any preimage of a
generator of ${\mathbb{Z}}$ under $\tilde{\rho}$, then $B_{1}$ and $\langle
t\rangle$ generate their free product in $G$, and we can view $\tilde{\rho}$
as a retraction from $G$ to its subgroup $B_{1}*\langle t\rangle$. This will
be used in the proof of Proposition 4.21.

######  Definition 4.16 (Simple étage).

A CSA group $G$ is a _simple étage of surface type_ over a non-abelian
subgroup $H$ if $G$ has a non-exceptional simple centered splitting $\Gamma$,
with bottom group $B_{1}=H$, and the equivalent conditions of Proposition 4.14
are satisfied.

We say that $G$ is a _simple étage of free product type_ over an arbitrary
subgroup $H$ if $G$ splits as a free product $G=H*{\mathbb{Z}}$.

In either case, we say that $G$ is a _simple étage_ over $H$.

######  Remark 4.17.

We do not allow $H$ to be abelian in the surface-type case (in particular, the
splitting defining a parachute as in Definition 4.1 is not a simple étage,
though it is an extended étage in the sense of Definition 4.26). In a simple
étage of free product type, we do allow $H$ to be abelian, in particular $G$
may be ${\mathbb{F}}_{2}$ or ${\mathbb{Z}}$. In some statements (such as
Theorem 2.1), we specifically assume that $H$ is non-abelian also in this
case.

######  Definition 4.18 (Simple tower).

A CSA group $G$ is a simple tower over a subgroup $H$ if there exists a chain
of subgroups $G=G_{0}>G_{1}>\dots>G_{k}=H$ such that $G_{i}$ is a simple étage
over $G_{i+1}$. The tower is _trivial_ if $k=0$.

We allow the case $k=0$, so $G$ is always a simple tower over itself. On the
other hand, not every group is a simple étage.

######  Remark 4.19.

If $G$ is a simple tower over $H$, and we have an embedding $H\hookrightarrow
H^{\prime}$, then $G*_{H}H^{\prime}$ is a simple tower over $H^{\prime}$. In
particular, $G*K$ is a simple tower over $H*K$.

######  Remark 4.20.

If $G$ is a simple tower over $H$, then $H$ is malnormal in $G$ (see Remark
4.4).

###  4.5 Extended towers and retractable splittings

As mentioned earlier, we assume from now on that $G$ is CSA.

We want to extend Proposition 4.14 to non-simple splittings. The statement
given below is more complicated than Proposition 4.14. One reason is the
following. If $\Gamma$ is a simple centered splitting, its base
$B_{\Gamma}=B_{1}$ is well-defined up to conjugacy as a subgroup of $G$; if
$\Gamma$ is not simple, each bottom group $B_{i}$ is a subgroup of $G$ well-
defined up to conjugacy, but their abstract free product $B_{\Gamma}$ is not.
In terms of logic, “Tarski” (Theorem 2.1) about elementary embeddings is valid
for simple étages, but not always for extended étages (see Theorem 10.13).

######  Proposition 4.21.

Let $\Gamma$ be a non-exceptional centered splitting of $G$, with bottom
groups $B_{1},\dots,B_{n}$ and surface group $Q=\pi_{1}(\Sigma)$. The
following are equivalent:

  1. 1.

There exists a non-degenerate boundary-preserving map $p:Q\to G$.

  2. 2.

There exist:

     * •

conjugates $\tilde{B}_{1},\dots,\tilde{B}_{n}$ of the bottom groups generating
their free product $\tilde{B}_{\Gamma}=\tilde{B}_{1}*\dots*\tilde{B}_{n}$,

and:

     * •

a retraction $\rho:G\rightarrow\tilde{B}_{\Gamma}^{e}$ with $\rho(Q)$ non-
abelian, where $\tilde{B}_{\Gamma}^{e}=\tilde{B}_{\Gamma}$ if
$\tilde{B}_{\Gamma}$ is non-abelian, and
$\tilde{B}_{\Gamma}^{e}=\tilde{B}_{\Gamma}*\langle t\rangle$ for some non-
trivial extra element $t\in G$ if $n=1$ and $B_{1}$ is abelian.

Proposition 4.21 is equivalent to Proposition 4.14 when $\Gamma$ is simple and
$B_{1}$ is non-abelian, but it extends it in two ways: we allow non-simple
splittings, and simple splittings with $B_{1}$ abelian.

We shall say that $\Gamma$ is retractable if it satifies the equivalent
conditions above (see Definition 4.23). Note that, when $n=1$ and $B_{1}$ is
abelian, requiring $\rho(Q)$ to be non-abelian forces us to enlarge
$B_{\Gamma}$.

######  Remark 4.22.

The extra condition when $n=1$ and $B_{1}$ is abelian has been added in the
erratum [Per13] after the third named author, while working on [PS12],
observed that the first condition in Proposition 4.21 does not always imply
the second. For example, $\pi_{1}(N_{4})$ admits a non-exceptional centered
splitting with central group $\pi_{1}(N_{3,1})$ and a single bottom group
$B_{1}$ isomorphic to $\mathbb{Z}$, and there is an obvious non-degenerate
boundary-preserving map $p:\pi_{1}(N_{3,1})\rightarrow\pi_{1}(N_{4})$, but
since $B_{1}$ is abelian no retraction $\rho:\pi_{1}(N_{4})\rightarrow B_{1}$
with $\rho(\pi_{1}(N_{3,1}))$ non-abelian can be found.

###### Proof.

First assume that $\Gamma$ is not simple, i.e. $n\geq 2$. Let
$\pi_{\mathcal{C}}:G\to G_{\mathcal{C}}$ be a pinched quotient (Definition
4.8). We denote by $v_{1},\dots,v_{n}$ the bottom vertices of $\Gamma$ and
also the corresponding vertices of $\Gamma_{\mathcal{C}}$. Note that a vertex
of $\Gamma_{\mathcal{C}}$ carrying some $\pi_{1}(\Sigma_{\mathcal{C}}^{j})$
cannot be joined to two distinct $v_{i}$’s by Remark 4.10. We may therefore
write $G_{\mathcal{C}}$ as a free product
$G_{\mathcal{C}}=G^{\prime}_{1}*\dots*G^{\prime}_{n}*G^{\prime\prime}$, with
$G^{\prime}_{i}$ the fundamental group of the star of the vertex $v_{i}$ in
$\Gamma_{\mathcal{C}}$.

As in the proof of Proposition 4.14, we may retract each $G^{\prime}_{i}$ onto
a subgroup $\tilde{B}^{\prime}_{i}<G^{\prime}_{i}$ conjugate to the stabilizer
of $v_{i}$ in $\Gamma_{\mathcal{C}}$. These retractions extend to a retraction
$\rho^{\prime}:G_{\mathcal{C}}\to\langle\tilde{B}^{\prime}_{1},\dots,\tilde{B}^{\prime}_{n}\rangle\simeq
B_{1}*\dots*B_{n}\simeq B_{\Gamma}$. We choose a lift $\tilde{B}_{i}\subset G$
which is mapped isomorphically onto $\tilde{B}^{\prime}_{i}$ by
$\pi_{\mathcal{C}}$. Since
$\tilde{B}^{\prime}_{1},\dots\tilde{B}^{\prime}_{n}$ generate their free
product, so do $\tilde{B}_{1},\dots,\tilde{B}_{n}$, and $\pi_{\mathcal{C}}$
induces an isomorphism from $\langle\tilde{B}_{1}*\dots*\tilde{B}_{n}\rangle$
to $\langle\tilde{B}^{\prime}_{1},\dots,\tilde{B}^{\prime}_{n}\rangle$.

We define $\rho$ by postcomposing $\rho^{\prime}\circ\pi_{\mathcal{C}}$ with
the inverse of this isomorphism. The image of $Q$ is automatically non-abelian
because it contains a free group generated by the image of two boundary
subgroups of $\pi_{1}(\Sigma)$ contained in $B_{1}$ and $B_{2}$ respectively
(up to conjugacy).

If $\Gamma$ is simple and $B_{1}$ is abelian, then 2 $\Rightarrow$ 1 is clear
and 1 $\Rightarrow$ 2 follows from Remark 4.15. ∎

Proposition 4.21 explains the terminology in the following definition.

######  Definition 4.23 (Retractable surface, retractable splitting).

Let $Q$ be a subgroup of $G$ isomorphic to the fundamental group of a non-
exceptional surface $\Sigma$. We say that $\Sigma$, and $Q$, are _retractable
(in $G$)_ if there exists a non-degenerate boundary-preserving map
$p:Q\rightarrow G$.

A non-exceptional centered splitting $\Gamma$ satisfying the equivalent
conditions of Proposition 4.21 is called a _retractable_ centered splitting,
and we say that its central vertex is retractable.

More generally, if a non-degenerate boundary-preserving map $p:Q\rightarrow G$
is associated to a surface-type vertex $v$ of a splitting $\Gamma$ of a
subgroup $G_{1}<G$, we say that $\Gamma$ and $v$ are _retractable in $G$_
(when $G_{1}=G$, we just say retractable).

######  Remark 4.24.

If a cyclic splitting $\Gamma$ is retractable, so is the centered splitting
obtained by collapsing all edges not containing the vertex carrying $\Sigma$
(and subdividing if needed). In particular, $G$ has a retractable centered
splitting as soon as some cyclic splitting has a surface-type vertex group $Q$
with a non-degenerate boundary-preserving map $p:Q\to G$.

Also note that retractable centered splittings have to be non-exceptional.

######  Remark 4.25.

Recall (Corollary 3.5 of [GLS19]) that, if a centered splitting $\Gamma$ is
retractable, then $g\geq n_{1}$, with $g$ the genus of $\Sigma$ and $n_{1}$
the number of bottom vertices of $\Gamma$ having valence 1.

The following definition was introduced by Perin [Per11, Per13].

######  Definition 4.26 (Extended étage, extended tower).

If $G$ has a non-exceptional centered splitting $\Gamma$ satisfying the
equivalent conditions of Proposition 4.21, we say that $G$ is an _extended
étage of surface type_ over the subgroup
$\tilde{B}_{\Gamma}=\tilde{B}_{1}*\dots*\tilde{B}_{n}$ appearing in assertion
2 (always isomorphic to the base of $\Gamma$, even if $n=1$ and $B_{1}$ is
abelian).

An _extended étage of free product type_ over $H$ is a free product
$G=H*{\mathbb{Z}}$ (as in Definition 4.16).

A group $G$ is an _extended tower_ over a subgroup $H$ if there exists a
sequence of subgroups $G=G_{0}>G_{1}>\dots>G_{k}=H$ such that $G_{i}$ is an
extended étage over $G_{i+1}$.

As in Definitions 4.16 and 4.18, we allow $H$ to be abelian in étages of free
product type, and we allow towers with no étage, so that $G$ is always an
extended tower over itself.

######  Remark 4.27.

Beware that an extended étage of surface type with a single bottom group $H$
(i.e. a simple retractable centered splitting) does not define a simple étage
if $H$ is abelian (see Remark 4.17); in particular, a parachute is not always
a simple étage.

######  Remark 4.28.

If $G$ is an extended tower over $H$, there is a retraction from $G$ to $H$,
so one may view $H$ both as a subgroup and a quotient of $H$. In general,
however, $H$ is not canonically defined as a subgroup of $G$; for instance,
changing the choice of the conjugates $\tilde{B}_{i}$’s in Proposition 4.21
expresses $G$ as an étage over isomorphic, but non-conjugate, subgroups. What
is canonical is the family of homomorphisms $B_{\Gamma}\rightarrow G$ which
agree with a conjugation on each $B_{i}$. Depending on context, one may want
to think of $H$ either as a subgroup of $G$, or as an abstract group,
forgetting about its embedding in $G$. Roughly speaking, Sections 6 and 7 are
about abstract groups, Section 8 about subgroups.

######  Remark 4.29.

When $n=1$ and $B_{1}\simeq{\mathbb{Z}}$ ($G$ is a parachute), the existence
of the retraction $G\rightarrow\tilde{B}_{1}*\langle t\rangle$ in Proposition
4.21 is equivalent to the existence of a retraction
$\tilde{\rho}:G*{\mathbb{Z}}\to B_{1}*{\mathbb{Z}}$ such that
$\tilde{\rho}(Q)$ is not abelian. This is the way Perin defines it in [Per13].
Note in particular that $G*{\mathbb{Z}}$ is a simple étage of surface type
over $B_{1}*{\mathbb{Z}}\simeq{\mathbb{F}}_{2}$.

The following remarks apply to both simple and extended étages/towers.

######  Remark 4.30.

If $G$ is an étage over $H$, then $G*G^{\prime}$ is an étage over
$H*G^{\prime}$ for any $G^{\prime}$ (replace the bottom group $B_{1}$ of
$\Gamma$ by $B_{1}*G^{\prime}$). It follows that, if $G$ is a tower over $H$
and $G^{\prime}$ is a tower over $H^{\prime}$, then $G*G^{\prime}$ is a tower
over $H*H^{\prime}$.

######  Remark 4.31.

In a tower, one can switch étages of surface type and of free product type as
follows: if $G_{i-1}=G_{i}*{\mathbb{Z}}$ and $G_{i}$ is an étage of surface
type over $G_{i+1}$, then $G_{i-1}$ is an étage of surface type over
$G^{\prime}_{i}=G_{i+1}*{\mathbb{Z}}$, which is an étage of free product type
over $G_{i+1}$. This allows us to assume that all étages of surface type are
at the top of the tower.

Proposition 4.21 is about single étages. The following is about towers.

######  Proposition 4.32.

Assume that $G$ is an extended tower over a subgroup $H$.

Then there are subgroups $G=G_{0}>G_{1}>\dots>G_{k}=H^{\prime}$, a (possibly
trivial) free group $F\subset H^{\prime}$ such that $H^{\prime}=H*F$, and for
$i<k$ non-exceptional retractable centered splittings $\Gamma_{i}$ of $G_{i}$,
and retractions $\rho_{i}:G_{i}\rightarrow G_{i+1}$ such that:

  * •

the image under $\rho_{i}$ of the central vertex group $Q_{i}$ of $\Gamma_{i}$
is non-abelian;

  * •

for $i>0$, the group $G_{i}$ is the free product of some conjugates of the
bottom groups of $\Gamma_{i-1}$, plus an extra cyclic factor if $i=k$ and
$\Gamma_{k-1}$ is simple with an abelian base.

###### Proof.

Using Remark 4.31, starting from a sequence of étages defining the tower, one
can move the étages of free product type down to get a sequence of groups
$G=G_{0},G_{1},\dots,G_{k}=H^{\prime}$, where each $G_{i}$ is an étage of
surface type over $G_{i+1}$ and $H^{\prime}=H*F$ with $F$ free. We conclude by
applying Proposition 4.21. If $G_{k}$ is abelian, we must enlarge
$G_{k}=H^{\prime}$ and $F$ by taking their free product with $\langle
t\rangle$ as in Proposition 4.21. ∎

######  Remark 4.33.

Assume for simplicity that $G$ is hyperbolic (or more generally that all
abelian subgroups of $G$ are cyclic). The subgroups appearing in this
proposition are almost the same as those in Definition 4.26, but not exactly,
because of parachutes: a parachute is an étage over ${\mathbb{Z}}$, but in
order to have a retraction such that the surface group has non-abelian image
one must retract to a free group of rank 2. Conversely, if there are subgroups
$G_{i}$ as in the proposition, then $G$ is an extended tower over $H$, except
possibly if $H=G_{k}\simeq{\mathbb{F}}_{2}$: in this case it may happen that
$G$ is a tower over a cyclic free factor of $H$, but not over $H$ itself.

Figure 9: Top part: non-exceptional surfaces except $N_{4}$ are simple étages
over ${\mathbb{F}}_{2}$ (the top right picture is not a valid étage because
$N_{2,1}$ is exceptional).
Bottom part: $N_{4}$ is an extended étage in two different ways.

The following lemma implies that a given $G$ cannot be a tower of arbitrary
length.

######  Lemma 4.34.

Let $G$ be a torsion-free CSA group. Denote by $b_{1}^{2}(G)$ the first Betti
number modulo 2 (the dimension of $\mathrm{Hom}(G,{\mathbb{Z}}/2{\mathbb{Z}})$
over ${\mathbb{Z}}/2{\mathbb{Z}}$). If $G$ is an extended étage over $H$, then
$b_{1}^{2}(G)>b_{1}^{2}(H)$.

###### Proof.

The result is clear if the étage is of free product type. Assume it is of
surface type, associated to a surface $\Sigma$ and a centered splitting
$\Gamma$. Since $H$ is a retract of $G$, it suffices to find a homomorphism
from $G$ onto ${\mathbb{Z}}/2{\mathbb{Z}}$ vanishing on $H$. This is easy to
do if $\Sigma$ is not planar or if $\Gamma$ is not a tree of groups. In the
remaining case ($\Sigma$ is planar and $\Gamma$ is a tree), $\Gamma$ cannot be
retractable [GLS19]: a retractable $\Gamma$ with $\Gamma$ planar has no
terminal vertex (see Remark 4.25). ∎

Recall that $S_{g,b}$ and $N_{g,b}$ denote the orientable/non-orientable
surfaces of genus $g$ with $b$ boundary components, and $S_{g}=S_{g,0}$,
$N_{g}=N_{g,0}$.

######  Example 4.35 (Closed surface groups, see Figure 9).

_All non-exceptional closed surface groups except $\pi_{1}(N_{4})$ are simple
étages over ${\mathbb{F}}_{2}$, and $\pi_{1}(N_{4})$ is an extended étage over
${\mathbb{Z}}$ and ${\mathbb{F}}_{2}$. _

We sketch the proof (compare [Per11, LPS13]). Every $S_{g}$ with $g\geq 2$ is
non-exceptional and $\pi_{1}(S_{g})$ is a simple étage over ${\mathbb{F}}_{2}$
(see Figure 9). For $g\geq 5$, decomposing $N_{g}$ into $N_{g-2,1}$ and
$N_{2,1}$ makes $\pi_{1}(N_{g})$ a simple étage over ${\mathbb{F}}_{2}$. When
$g=4$, this does not yield an étage structure because $N_{2,1}$ is
exceptional.

To express $\pi_{1}(N_{4})$ as an extended étage, view $N_{4}$ as a twice-
punctured Klein bottle $N_{2,2}$ with a Möbius band glued to each boundary
component. This yields a retractable splitting of its fundamental group
$\pi_{1}(N_{4})$ with base ${\mathbb{F}}_{2}$, so $\pi_{1}(N_{4})$ is an
extended étage over ${\mathbb{F}}_{2}$ (the associated retraction maps
$\pi_{1}(N_{4})=\langle a_{1},a_{2},b_{1},b_{2}\mid
a_{1}^{2}a_{2}^{2}=b_{1}^{2}b_{2}^{2}\rangle$ to $\langle a_{1},a_{2}\rangle$
by sending $b_{i}$ to $a_{i}$). One may also decompose $N_{4}$ as the union of
$N_{3,1}$ and a Möbius band, so $\pi_{1}(N_{4})$ has a simple retractable
splitting with base ${\mathbb{Z}}$ (but is not a simple étage, see Remark
4.17).

The surface $N_{3}$ is exceptional, its fundamental group is not an extended
étage.

###  4.6 Relative extended towers

Recall that, if $G$ is an extended étage of surface type over $H$, associated
to a centered splitting $\Gamma$ of $G$, the bottom groups are by definition
elliptic in $\Gamma$ and are well-defined up to conjugacy as subgroups of $G$.
On the other hand, if the étage is not simple, there is no canonical embedding
of $H$ into $G$. This motivates the following definition, which is important
for applications.

######  Definition 4.36 (Relative extended tower [Per11]).

Let $G=G_{0}>G_{1}>\dots>G_{k}=H$ be an extended tower. The tower is _relative
to a subgroup $A\subset H$_ if each étage of surface type is relative to $A$
in the following sense: the group $A$ is contained, up to conjugacy, in a
bottom group of the centered splitting defining the étage.

######  Example 4.37.

Any simple tower over $H$ is relative to $H$, or to any subgroup $A\subset H$.
Now suppose that $G$ has a retractable centered splitting with two bottom
groups $B_{1},B_{2}$ and base $G_{1}=B_{1}*B_{2}$. Also suppose that $G_{1}$
is a simple étage over $H$. The tower $G>G_{1}>H$ is relative to $H$ if and
only if $H$ is contained in a conjugate of $B_{1}$ or $B_{2}$.

######  Remark 4.38.

If $G$ is an étage of free product type over $H<G$, i.e. $G=H*{\mathbb{Z}}$,
then the étage is automatically relative to $A$ for any subgroup $A<H$.

######  Remark 4.39.

Proposition 4.32 holds if $G$ is an extended tower over $H$ relative to $A$,
with $A$ contained in a conjugate of a bottom group of $\Gamma_{i}$ for
$i=0,\dots,k-1$.

######  Remark 4.40.

If $G_{i}$ as in Definition 4.36 is an extended étage of surface type over
$G_{i+1}$, then $G_{i+1}$ is the free product of the bottom vertex groups of
$\Gamma_{i}$. Thus, if $A<H$ is a one-ended subgroup, the tower is
automatically relative to $A$. More generally, if $A$ is one-ended relative to
a subgroup $A^{\prime}$, and the tower is relative to $A^{\prime}$, then the
tower is also relative to $A$.

######  Remark 4.41.

An extended étage of surface type over $H$ is relative to $H$ if and only if
it is simple. If $G$ is an extended tower over $H$ relative to $H$, then the
last étage (between $G_{k-1}$ and $G_{k}$) has to be an étage of free product
type or a simple étage.

###  4.7 Relation with other definitions

Sela introduced hyperbolic $\omega$-residually free towers in [Sel01]. They
are defined as simple towers (involving only étages of surface type) over a
(non-abelian) free product of (maybe 0) non-exceptional closed surface groups
and a (possibly trivial) free group. Because étages of free product type may
be moved to the bottom by Remark 4.31, one does not change the definition if
one allows étages of free product type in this definition. Moreover, since
fundamental groups of non-exceptional surfaces except $N_{4}$ are simple
towers over ${\mathbb{F}}_{2}$ (Example 4.35), and since
$\pi_{1}(N_{4})*\pi_{1}(N_{4})$ and $\pi_{1}(N_{4})*{\mathbb{Z}}$ are simple
towers over ${\mathbb{F}}_{2}$, $\omega$-residually free towers coincide with
simple towers over ${\mathbb{F}}_{2}$ or over $\pi_{1}(N_{4})$.

Our extended towers are almost identical to the extended hyperbolic towers
defined by Perin in [Per13].

Perin defined hyperbolic floors/étages in [Per11] allowing the possibility of
a non-simple centered splitting, i.e. a splitting having several bottom
vertices. In [Per13, Section 4] Perin alludes to the possibility that Sela
also thinks of non-simple floors/étages, but this definitely does not appear
in print.

As mentioned in Remark 4.22, Perin’s definition needed a fine tuning, hence
the introduction of extended hyperbolic floors in [Per13]. We do not use the
same retraction as Perin in the case of parachutes, but this makes no
difference (see Remark 4.29).

Unlike Perin, we insist that only one surface appears in each étage (we only
use centered splittings). This does not affect the definition of towers,
though it increases the number of étages in a given tower. We do not use Bass-
Serre presentations, but we select conjugates of the $B_{i}$’s in Proposition
4.21.

Perin does not consider étages of free product type, but by Remark 4.31 _$G$
is an extended tower over $H$ in our sense if and only if $G$ is an extended
hyperbolic tower over $H*F$ in the sense of Perin [Per13] for some free
subgroup $F$. _

We now quickly review the definition of regular NTQ systems and NTQ groups
from [KM05, Section 7.6]. If
${\mathcal{S}}=\\{S_{1}(x_{1},\dots,x_{n}),\dots,S_{k}(x_{1},\dots,x_{n})\\}$
is a set of equations in the variables $x_{1},\dots,x_{n}$ (maybe with
constants in $G$), the coordinate group $G_{R({\mathcal{S}})}$ of
${\mathcal{S}}$ is the largest residually-$G$ quotient of $\langle
G,x_{1},\dots,x_{n}\mid{\mathcal{S}}\rangle$, i.e. the quotient of $G*\langle
x_{1},\dots,x_{n}\rangle$ by the set of words $r(x_{1},\dots,x_{n})$ (with
constants in $G$) such that any $g_{1},\dots,g_{n}\in G$ satisfying
$S_{i}(g_{1},\dots,g_{n})=1$ also satisfies $r(g_{1},\dots,g_{n})=1$. We will
restrict to the case where $G$ is a limit group (over ${\mathbb{F}}_{2}$).

The building blocks of regular NTQ systems are quadratic systems of equations
(corresponding to surface groups), and empty equations (corresponding to the
free product with a free group). More generally, a TQ system, as defined in
Definition 43 of [KM05], also allows commutation equations (corresponding to
extensions of centralizers, or abelian étages), but these are ruled out in the
definition of _regular_ NTQ systems. In regular NTQ systems, one requires
[KM05, Def. 6] that the quadratic equations correspond to non-exceptional
surfaces and that the equations have a non-commutative solution in $G$
(corresponding to retractions such that the surface groups have non-abelian
image). Thus, a group is the coordinate group $G_{R({\mathcal{S}})}$ of a
regular quadratic system of equations over $G$ if and only if it is a simple
étage of surface type over $G$. A group $L$ is the coordinate group of a
regular NTQ system over $G$ (including constants) if and only if it is a
simple tower over $G$. We note that, for $G={\mathbb{F}}_{r}$ with $r\geq 2$,
this class of groups depends on $r$ (all such groups have an epimorphism to
${\mathbb{F}}_{r}$). For $r=2$, this class includes the fundamental group of
all non-exceptional surfaces except $N_{4}$.

If a system of equations ${\mathcal{S}}$ does not involve constants in $G$,
one can additionally define a _constant-free_ coordinate group $G^{*}_{R(S)}$
as the largest residually-$G$ quotient of

$\langle x_{1},\dots,x_{n}\mid{\mathcal{S}}\rangle.$

A group $L$ is the constant-free coordinate group of a regular NTQ system
without constants over the free group ${\mathbb{F}}_{r}$ if and only if
$L*{\mathbb{F}}_{r}$ is a simple tower over ${\mathbb{F}}_{r}$. One can easily
check that this notion does not depend on $r\geq 2$. This yields a larger
class of groups, containing all non-exceptional surface groups (including the
fundamental group of $N_{4}$), elementarily free parachutes, ${\mathbb{Z}}$
and the trivial group.

##  5 Some properties of towers

In this section we assume that $G$ is CSA, torsion-free, finitely generated.

###  5.1 Making an étage simple

Simple towers are better than extended towers; in particular, Theorem 2.1
about elementary embeddings only applies to simple towers. It is therefore
desirable to make an étage simple. We show that this is possible if the
surface is complicated enough, or if we are prepared to enlarge $G$ by taking
a free product with a free group (Corollary 5.4).

######  Proposition 5.1.

Assume that $G$ has a _non-simple_ retractable centered splitting $\Gamma$
such that the associated surface $\Sigma$ has Euler characteristic satisfying
$\chi(\Sigma)\leq-3$.

Then $G$ has a retractable _simple_ splitting $\hat{\Gamma}$ whose base is
isomorphic to that of $\Gamma$ (so $G$ is a simple étage of surface type).

The proposition applies unless the non-exceptional surface $\Sigma$ is a
4-punctured sphere $S_{0,4}$, a thrice-punctured projective plane $N_{1,3}$,
or a twice-punctured Klein bottle $N_{2,2}$ (the twice-punctured torus cannot
appear in a non-simple retractable splitting by [GLS19], see Remark 4.25).
These surfaces cause pathologies that will be studied in Section 10.

Figure 10: Proof of Proposition 5.1.

###### Proof.

Let $B_{1},\dots,B_{n}$ be the bottom groups of $\Gamma$, with $n\geq 2$, and
$Q=\pi_{1}(\Sigma)$ the central group. Let $p:Q\to G$ be a non-degenerate
boundary-preserving map. Let $C_{1}$ and $C_{2}$ be two boundary components of
$\Sigma$ whose fundamental groups are contained in conjugates of $B_{1}$ and
$B_{2}$ respectively (see Figure 10). Using Lemma 4.9, we may find a (possibly
non-maximal) family ${\mathcal{C}}$ of pinched curves separating $\Sigma$ into
two connected surfaces $\Sigma^{1},\Sigma^{2}$ containing $C_{1}$ and $C_{2}$
respectively (on Figure 10, the family ${\mathcal{C}}$ consists of the single
curve $\delta$).

We choose an arc $\gamma\subset\Sigma$ joining $C_{1}$ to $C_{2}$ and meeting
${\mathcal{C}}$ exactly once, and we call $\delta$ the curve of
${\mathcal{C}}$ which meets $\gamma$. We view $Q$ as $\pi_{1}(\Sigma,x)$, with
a basepoint $x\in\gamma$, and using $\gamma$ to join $C_{1}$ and $C_{2}$ to
$x$ we can view $\pi_{1}(C_{1})$ and $\pi_{1}(C_{2})$ as subgroups of $Q$
which are well-defined, not just up to conjugacy (this is similar to the
choice of subgroups $\tilde{B}_{i}$ in the proof of Proposition 4.21).

Let $P\subset\Sigma$ be a pair of pants obtained as a regular neighborhood of
$C_{1}\cup\gamma\cup C_{2}$, with $\partial P$ consisting of $C_{1},C_{2}$,
and a third curve $C$. We shall define a boundary-preserving map $\hat{p}$
from $\pi_{1}(\hat{\Sigma})$ to $G$, where $\hat{\Sigma}$ denotes the surface
obtained from $\Sigma$ by removing $P$. Note that
$\chi(\hat{\Sigma})=\chi(\Sigma)+1\leq-2$, so $\hat{\Sigma}$ is non-
exceptional.

A regular neighbourhood of $C_{1}\cup\gamma\cup C_{2}\cup\delta$ is a
$4$-punctured sphere containing $P$; its boundary consists of $C_{1},C_{2}$
and curves $C^{\prime}_{1},C^{\prime}_{2}$ with
$C^{\prime}_{i}\subset\hat{\Sigma}\cap\Sigma^{i}$. Since $\delta$ is pinched,
the image of $\pi_{1}(C^{\prime}_{i})$ by $p$ is conjugate to that of
$\pi_{1}(C_{i})$, hence contained in a conjugate of $B_{i}$.

We cannot directly use the restriction of $p$ to $\pi_{1}(\hat{\Sigma})$
because it does not have to map $\pi_{1}(C)$ to a conjugate, so we first have
to modify $p$. Denote by $\Sigma_{\mathcal{C}}$ the pinched space (see
Definition 4.8), so that $p$ factors through
$p_{\mathcal{C}}:\pi_{1}(\Sigma_{\mathcal{C}})\rightarrow G$. Let
$\Sigma^{1}_{\mathcal{C}},\Sigma^{2}_{\mathcal{C}}$ be the images of
$\Sigma^{1},\Sigma^{2}$ in $\Sigma_{\mathcal{C}}$, and denote by
$p_{\mathcal{C}}^{i}$ the restriction of $p_{\mathcal{C}}$ to
$\pi_{1}(\Sigma^{i}_{\mathcal{C}})$. Since
$\pi_{1}(\Sigma^{1}_{\mathcal{C}})*\pi_{1}(\Sigma^{2}_{\mathcal{C}})$ is a
free factor of $\pi_{1}(\Sigma_{\mathcal{C}})$, one can modify
$p_{\mathcal{C}}$ by composing each $p_{\mathcal{C}}^{i}$ with a conjugation
so that the modified map $p^{\prime}=p_{\mathcal{C}}\circ\pi_{\mathcal{C}}$
from $\pi_{1}(\Sigma)$ to $G$ is a boundary-preserving map which is the
identity on $\pi_{1}(C_{1})$ and $\pi_{1}(C_{2})$, hence on $\pi_{1}(P)$.

The restriction $\hat{p}$ of $p^{\prime}$ to $\pi_{1}(\hat{\Sigma})$ is a
boundary-preserving map. It is associated to a splitting $\hat{\Gamma}$ having
one less vertex group than $\Gamma$: the vertex groups $B_{1}$ and $B_{2}$
have been replaced by a single vertex group isomorphic to $B_{1}*B_{2}$, in
particular the base of $\hat{\Gamma}$ is isomorphic to that of $\Gamma$. The
non-degeneracy of $\hat{p}$ follows from the fact that its image contains
those of the groups $\pi_{1}(C^{\prime}_{i})$, which are non-trivial and
contained in conjugates of $B_{i}$.

If $\hat{\Gamma}$ is not simple, we can iterate this process and obtain a non-
degenerate boundary-preserving map associated to a simple splitting. There
remains to check that the corresponding surface $\Sigma_{f}$ is non-
exceptional when $\chi(\Sigma)\leq-3$.

We denote by $g,b,n$ the genus of $\Sigma$, the number of boundary components
of $\Sigma$, and the number of bottom vertices of $\Gamma$ respectively.
Retractability of $\Gamma$ implies $g+b\geq 2n$ ([GLS19], see Remark 4.25). By
construction $\Sigma_{f}$ has genus $g$ and $b-n+1$ boundary components. We
assume that $\Sigma_{f}$ is exceptional (so $\chi(\Sigma_{f})=-1$ and $g\leq
2$), and we deduce that $\chi(\Sigma)=-2$.

First suppose that $\Sigma$ is orientable. This implies $g=0$ since non-planar
orientable surfaces are non-exceptional. Then $b\geq 2n$ and
$1=|\chi(\Sigma_{f})|=b-n-1\geq n-1$, so $n=2$, $b=4$, and $\Sigma$ is a
4-punctured sphere.

If $\Sigma$ is non-orientable with $g=1$, we have
$1=|\chi(\Sigma_{f})|=b-n\geq n-1$, so $n=2$, $b=3$, and $\Sigma$ is a thrice-
punctured projective plane.

If $\Sigma$ is non-orientable with $g=2$, we have
$1=|\chi(\Sigma_{f})|=b-n+1\geq n-1$, so $n=b=2$ and $\Sigma$ is a twice-
punctured Klein bottle. ∎

Figure 11: If $G$ is an extended étage with bottom groups $B_{1}$, $B_{2}$,
$B_{3}$, then $G*{\mathbb{F}}_{2}$ is a simple étage over a subgroup
isomorphic to $B_{1}*B_{2}*B_{3}$.

######  Proposition 5.2 (See Figure 11).

Let $\Gamma$ be a centered splitting of $G$. If it is retractable, there
exists a (possibly trivial) free group $F$ and a simple retractable splitting
$\Gamma_{s}$ of $G*F$, which makes $G*F$ a simple étage.

The splittings $\Gamma$ and $\Gamma_{s}$ are associated to the same surface.
The base of $\Gamma_{s}$ is isomorphic to the base of $\Gamma$, except that
$\Gamma_{s}$ has base $B_{1}*{\mathbb{Z}}$ if $\Gamma$ is simple with an
abelian base $B_{1}$.

Conversely, we will see (Corollary 5.18) that $G$ has a retractable splitting
whenever some $G*F$ has one.

###### Proof.

There is nothing to prove if $n=1$, except if $\Gamma$ has an abelian base
$B_{1}$. But then $G*{\mathbb{Z}}$ is a simple étage over
$B_{1}*{\mathbb{Z}}$.

Now assume $n\geq 2$ (see Figure 11). Let $\rho:G\to
H=\tilde{B}_{1}*\dots*\tilde{B}_{n}$ be a retraction provided by Proposition
4.21. Write ${\mathbb{F}}_{n-1}=\langle t_{2},\dots,t_{n}\rangle$ and define
$\varphi:H\to G*{\mathbb{F}}_{n-1}$ as being the identity on $\tilde{B}_{1}$
and conjugation by $t_{i}$ on $\tilde{B}_{i}$. It induces an isomorphism (also
denoted by $\varphi$) between $H$ and a subgroup
$H^{\prime}=\tilde{B}_{1}*\tilde{B}_{2}^{t_{2}}*\cdots*\tilde{B}_{n}^{t_{n}}\subset
G*{\mathbb{F}}_{n-1}$, with
$\tilde{B}_{i}^{t_{i}}=t_{i}\tilde{B}_{i}t_{i}^{-1}$.

The group $H^{\prime}$ is the base of a simple centered splitting of
$G*{\mathbb{F}}_{n-1}$, whose restriction to $G$ is $\Gamma$. The map
$\rho^{\prime}:G*{\mathbb{F}}_{n-1}\to H^{\prime}$ equal to $\varphi\circ\rho$
on $G$ and killing every $t_{i}$ is a retraction from $G*{\mathbb{F}}_{n-1}$
to $H^{\prime}$. ∎

######  Remark 5.3.

If $\Gamma$ is not simple, this construction makes $G*F$ a simple étage over a
subgroup $H^{\prime}$ which is isomorphic to $H$ but is not contained in $G$.

######  Corollary 5.4.

If $G$ is an extended tower over a group $H$, there exists a free group $F$
such that $G*F$ is a simple tower over a subgroup $H^{\prime}$ isomorphic to
$H*{\mathbb{F}}_{2}$. If $H$ is not abelian, one can take $H^{\prime}\simeq
H$.

###### Proof.

By Proposition 4.32, there exists a tower
$G=G_{0}>G_{1}\dots>G_{k}=H*F^{\prime}$ where $G_{i}$ is an extended étage of
surface type over $G_{i+1}$ and $F^{\prime}$ is free.

Assume first that $H$ is not abelian. Then no $G_{i}$ is abelian, so for all
$i<k$ there is a free group $F_{i}$ such that $G_{i}*F_{i}$ is a simple tower
over a subgroup $G^{\prime}_{i+1}$ isomorphic to $G_{i+1}$. Applying Remark
4.30 inductively, one finds a free group $F$ such that $G*F$ is a simple tower
over a subgroup isomorphic to $H*F^{\prime}$, hence over a subgroup isomorphic
to $H$.

If $H$ is abelian, then $G*{\mathbb{F}}_{2}$ is an extended tower over the
non-abelian group $H*{\mathbb{F}}_{2}$ by Remark 4.30, and the argument given
above applies. ∎

######  Remark 5.5.

In Corollary 5.4, assume that the tower is relative to a subgroup $A<H$, and
let $H=H_{1}*\dots*H_{k}*{\mathbb{F}}_{r}$ be a Grushko decomposition of $H$
relative to $A$. At each étage, every $H_{j}$ is contained in a conjugate of a
bottom group. It follows from the way that $\varphi$ was defined in the proof
of Proposition 5.2 that there is an isomorphism $\psi:H\to H^{\prime}$ whose
restriction to each $H_{j}$ agrees with the conjugation by some element of
$G*F$ (depending on $i$); in particular $H^{\prime}$ contains a conjugate of
$A$ (and we may choose $H^{\prime}$ so that it contains $A$). If the tower is
relative to $H$ itself, then $H^{\prime}$ is conjugate to $H$ in $G*F$.

By Remark 4.40, the tower is automatically relative to $A$ if $A<H$ is one-
ended; thus _any one-ended $A<H$ is contained in $H^{\prime}$ up to
conjugacy._ This also holds if $A$ is one-ended relative to a subgroup
$A^{\prime}<A$ and the tower is relative to $A^{\prime}$.

The following lemma will be used in Section 9.

######  Lemma 5.6.

Assume that $G$ is an extended tower over a subgroup $H<G$ relative to $A<H$,
and let $H=H_{1}*\dots*H_{k}*{\mathbb{F}}_{r}$ be the Grushko decomposition of
$H$ relative to $A$.

Then $(H_{1},\dots,H_{k})$ is a malnormal family in $G$: if $H_{i}^{g}\cap
H_{j}\neq\\{1\\}$, then $i=j$ and $g\in H_{i}$.

###### Proof.

First assume that the tower $G=G_{0}>G_{1}>\dots>G_{n}=H$ is simple. Then
$G_{i}$ is malnormal in $G_{i-1}$ for each $i\geq 1$ by Remark 4.4. It follows
that $H$ is malnormal in $G$, and the lemma follows from the malnormality of
the family of Grushko factors of $H$.

We now turn to the general case. By Corollary 5.4 and Remark 5.5, $G*F$ is a
simple tower over a group $H^{\prime}$, and there is an isomorphism from $H$
or $H*{\mathbb{F}}_{2}$ to $H^{\prime}$ whose restriction to each $H_{j}$ is
inner. The result follows from the case of a simple tower. ∎

###  5.2 Narrowing the range

In this subsection we consider a subgroup $H\subset G$ and a splitting of $H$.
We assume that the splitting is retractable in $G$ (Definition 4.23), and we
wish to show that it is retractable in $H$. Our results are based on the
following lemma.

######  Lemma 5.7.

Let $\pi_{1}(\Sigma)\subset H\subset G$, with $H$ malnormal. Let
$p:\pi_{1}(\Sigma)\to G$ be a non-degenerate boundary-preserving map, and let
${\mathcal{C}}$ be a maximal family of pinched curves, with
${\mathcal{C}}\neq\emptyset$.

Assume that, whenever $\Sigma_{\mathcal{C}}^{j}$ (as in Definition 4.8)
contains a component of $\partial\Sigma$, the image of
$\pi_{1}(\Sigma_{\mathcal{C}}^{j})$ is contained in a conjugate of $H$. Then
there is a pinching non-degenerate boundary-preserving map
$p^{\prime}:\pi_{1}(\Sigma)\to H$.

###### Proof.

We factor $p$ into $p=p_{\mathcal{C}}\circ\pi_{\mathcal{C}}$ with
$\pi_{\mathcal{C}}:\pi_{1}(\Sigma)\to\pi_{1}(\Sigma_{\mathcal{C}})$ and
$p_{\mathcal{C}}:\pi_{1}(\Sigma_{\mathcal{C}})\to G$ (see Definition 4.8). The
group $\pi_{1}(\Sigma_{\mathcal{C}})$ is isomorphic to a free product whose
factors are closed surface groups, cyclic groups, and the
$\pi_{1}(\Sigma_{\mathcal{C}}^{j})$’s for which $\Sigma_{\mathcal{C}}^{j}$
contains a component of $\partial\Sigma$; we fix such an isomorphism. We shall
modify $p_{\mathcal{C}}$ on each free factor, and $p^{\prime}$ will be
$p_{\mathcal{C}}\circ\pi_{\mathcal{C}}$.

We can modify $p_{\mathcal{C}}$ on each $\pi_{1}(\Sigma_{\mathcal{C}}^{j})$,
by composing with a conjugation of $G$, so that the image of
$\pi_{1}(\Sigma_{\mathcal{C}}^{j})$ is contained in $H$ (rather than a
conjugate); by malnormality of $H$, the modified
$p_{\mathcal{C}}\circ\pi_{\mathcal{C}}$ acts on each boundary subgroup of
$\Sigma$ as conjugation by some element of $H$. We can also modify
$p_{\mathcal{C}}$ on the other free factors so that the image of
$\pi_{1}(\Sigma_{\mathcal{C}})$ is contained in $H$. We now have to make sure
that this image is non-abelian.

If $\Sigma$ has two boundary components $C_{1},C_{2}$ belonging to different
surfaces $\Sigma_{\mathcal{C}}^{j_{1}}$, $\Sigma_{\mathcal{C}}^{j_{2}}$, the
images of $A_{1}=\pi_{1}(C_{1})$ and $A_{2}=\pi_{1}(C_{2})$ are conjugate to
subgroups of $\pi_{1}(\Sigma)$ (in $G$ hence in $H$ by malnormality); we can
modify $p_{\mathcal{C}}$ on $\pi_{1}(\Sigma_{\mathcal{C}}^{j_{1}})$ so that
$p_{\mathcal{C}}(A_{1})$ and $p_{\mathcal{C}}(A_{2})$ are two non-commuting
conjugates of $A_{1}$ and $A_{2}$ in $H$ (for instance by mapping $A_{1}$ and
$A_{2}$ to themselves in $\pi_{1}(\Sigma)\subset H$, since they do not commute
there).

If $\partial\Sigma$ is contained in a single $\Sigma_{\mathcal{C}}^{j}$, and
$\pi_{1}(\Sigma_{\mathcal{C}}^{j})$ has abelian image, we argue as in the
proof of Proposition 4.14. Recall that $G$ is torsion-free and
$\pi_{1}(\Sigma_{\mathcal{C}})$ has non-abelian image under the original
$p_{\mathcal{C}}$. Consider the factors in the decomposition of
$\pi_{1}(\Sigma_{\mathcal{C}})$ as a free product. Besides
$\pi_{1}(\Sigma_{\mathcal{C}}^{j})$, there must be at least a ${\mathbb{Z}}$
or the group of a closed surface different from $S^{2}$ and $P^{2}$. Such a
surface group maps onto ${\mathbb{Z}}$, and we can modify $p_{\mathcal{C}}$ to
make its image non-abelian. ∎

We can now state:

######  Proposition 5.8.

Assume that $H$ is a free factor of $G$. If a splitting of $H$ is retractable
in $G$, then it is retractable in $H$.

We view this as a special case of the following more precise result, which
will be needed later.

######  Lemma 5.9.

Let $\pi_{1}(\Sigma)\subset H\subset G$, with $H$ a free factor of $G$. If
there is a non-degenerate boundary-preserving map $p:\pi_{1}(\Sigma)\to G$,
there is a non-degenerate boundary-preserving map
$p^{\prime}:\pi_{1}(\Sigma)\to H$. Moreover, $p^{\prime}$ is pinching or
$p^{\prime}=\iota\circ p$ with $\iota$ an inner automorphism of $G$.

###### Proof.

If $p$ is non-pinching, its image is contained in a conjugate of $H$ by Lemma
3.7 and we take $p^{\prime}=\iota\circ p$ for some inner automorphism $\iota$
of $G$. Otherwise we apply Lemma 5.7, noting that the image of every
$\pi_{1}(\Sigma_{\mathcal{C}}^{j})$ is contained in a conjugate of $H$ by
Lemma 3.7. ∎

We now consider the following situation. Let $\Gamma$ be a retractable
centered splitting of $G$, and let $B$ be a bottom vertex group. Given a
surface subgroup $\pi_{1}(\Sigma)\subset B$, and a non-degenerate boundary-
preserving map $p:\pi_{1}(\Sigma)\rightarrow G$, we wish to construct a non-
degenerate boundary-preserving map $p^{\prime}$ with values in $B$.

Figure 12: Example 5.10: there is a non-degenerate boundary-preserving map
$p:\pi_{1}(\Sigma_{B})\rightarrow G$, but none from $\pi_{1}(\Sigma_{B})$ to
$B$.

The following simple example shows that additional assumptions are required.

######  Example 5.10 (see Figure 12).

As in Example 4.7, we consider a group
$B=B_{1}*_{g=[x_{B},y_{B}]}F(x_{B},y_{B})$ obtained by attaching a punctured
torus $\Sigma_{B}$ to a group $B_{1}$, but we now require that $g$ is _not_ a
commutator in $B_{1}$. Construct $G=B*_{g=[x,y]}F(x,y)$ by attaching a second
punctured torus $\Sigma$ to $B_{1}$ in the same way as $\Sigma_{B}$. There is
a non-degenerate boundary-preserving map $p:\pi_{1}(\Sigma_{B})\to G$ with
image $\pi_{1}(\Sigma)$, but no non-degenerate boundary-preserving map
$p:\pi_{1}(\Sigma_{B})\to B$ since $g$ is not a commutator in $B_{1}$.

######  Lemma 5.11.

Let $\Gamma$ be a retractable centered splitting of $G$, with central vertex
group $\pi_{1}(\Sigma)$, and let $B$ be a bottom vertex group. Let
$\pi_{1}(\Sigma_{B})<B$ be a surface subgroup, and let
$p:\pi_{1}(\Sigma_{B})\to G$ be a non-degenerate boundary-preserving map.

If $p$ satisfies one of the following conditions, there is a non-degenerate
boundary-preserving map $p^{\prime}:\pi_{1}(\Sigma_{B})\to B$:

  1. 1.

no conjugate of the image of $p$ contains a finite index subgroup of
$\pi_{1}(\Sigma)$;

  2. 2.

$p$ is pinching.

Moreover, $p^{\prime}$ is pinching or $p^{\prime}=\iota\circ p$ with $\iota$
an inner automorphism of $G$.

###### Proof.

Assume 1. If $p$ does not pinch, then it takes values in a conjugate of $B$ by
Lemma 4.11, so we can take $p^{\prime}=\iota\circ p$ for some inner
automorphism $\iota$. If $p$ is pinching, consider a maximal collection
${\mathcal{C}}$ of pinched curves on $\Sigma_{B}$. Lemma 4.11 ensures that the
fundamental group of every connected component of
$\Sigma_{B}\setminus{\mathcal{C}}$ is mapped into a conjugate of a bottom
group of $\Gamma$. One can then apply Lemma 5.7: if such a component contains
a component of $\partial\Sigma_{B}$, Remark 4.4 implies that this bottom group
has to be $B$ since $\pi_{1}(\Sigma_{B})\subset B$ and $p$ agrees with
conjugations on the boundary subgroups.

If 1 does not hold but $p$ is pinching, we define $\hat{p}=\rho\circ p$, with
$\rho:G\to\tilde{B}_{1}*\dots*\tilde{B}_{n}$ a retraction associated to
$\Gamma$ provided by Proposition 4.21. The image of $\hat{p}$ is non-abelian
because it is CSA and contains a finite index subgroup of the non-abelian
group $\rho(\pi_{1}(\Sigma))$, so $\hat{p}$ is a pinching non-degenerate
boundary-preserving map with values in $\tilde{B}_{1}*\dots*\tilde{B}_{n}$.
The group $B$ is conjugate to some $\tilde{B}_{i}$, and we apply Lemma 5.9 to
find $p^{\prime}$ with values in $B$. ∎

###  5.3 Retracting a Grushko factor

In this subsection we introduce a blowup of a centered splitting that will
enable us to understand how a freely decomposable group may be an étage of
surface type. In particular, we will show that, if $G$ is an extended étage of
surface type, so is one of its Grushko factors. As a corollary, we will see
that a free group has no retractable splitting (this is a main step in the
proof of homogeneity of free groups [PS12]).

Figure 13: The Grushko blowup $\Lambda$ of $\Gamma$, and the resulting
centered splitting $\Lambda_{\mathbb{Z}}$ with fundamental group $G_{1}$ and
base $B_{\Lambda_{\mathbb{Z}}}$.

######  Definition 5.12 (Grushko blowup of a centered splitting, see Figure
13).

Let $\Gamma$ be a centered splitting of $G$ with bottom vertex groups
$B_{1},\dots,B_{n}$ and central group $Q$. The _Grushko blowup_ of $\Gamma$ is
the splitting $\Lambda$ obtained by blowing up every bottom vertex of $\Gamma$
using a Grushko decomposition $\Lambda_{i}$ of $B_{i}$ relative to the
incident edge groups (see Subsection 3.4).

We denote by $\Lambda_{\mathbb{Z}}\subset\Lambda$ the union of all edges
carrying a non-trivial group (the star of the central vertex). The fundamental
group of $\Lambda_{\mathbb{Z}}$ is a free factor $G_{1}$ of $G$ containing
$Q$.

The definition is illustrated in Figure 13, which depicts a case where there
are two bottom vertex groups $B_{1},B_{2}$. If, for some $i$, the group
$B_{i}$ is freely decomposable relative to the incident edge groups, we choose
a relative Grushko decomposition $\Lambda_{i}$ with a central vertex $w_{i}$
carrying a trivial group joined by a single edge to each vertex with non-
trivial group. Some of these vertices (carrying $\hat{B}_{1}$, $\hat{B}_{2}$,
$\hat{B}_{3}$ on the figure) have at least one edge of $\Lambda_{\mathbb{Z}}$
attached to them, the base $B_{\Lambda_{\mathbb{Z}}}$ of
$\Lambda_{\mathbb{Z}}$ is the free product of their groups. The other vertex
groups of the $\Lambda_{i}$’s ($G_{2}$,…,$G_{4}$ on the figure) are Grushko
factors of $G$. There may also be loops attached at $w_{i}$, corresponding to
the free group in the relative Grushko decomposition of $B_{i}$. The edges
with trivial group induce a decomposition of the form
$G=G_{1}*G_{2}*\dots*G_{k}*F$ with $G_{1}=\pi_{1}(\Lambda_{\mathbb{Z}})$ and
$F$ free (of rank 4 on Figure 13).

Note that the splitting $\Lambda_{\mathbb{Z}}$ fails to be minimal if, for
some $i$, an edge $e$ of $\Gamma$ containing $v_{i}$ carries a group
$A_{e}\simeq{\mathbb{Z}}$ such that $B_{i}$ is a free product
$B_{i}=A_{e}*B^{\prime}_{i}$, and the groups carried by all other edges of
$\Gamma$ containing $v_{i}$ are contained (up to conjugacy) in the
complementary free factor $B^{\prime}_{i}$. If this does not happen,
$\Lambda_{\mathbb{Z}}$ is minimal and is a centered splitting of $G_{1}$.

The main result of this subsection is the following.

######  Proposition 5.13.

Let $\Gamma$ be a centered splitting of $G$, with central vertex $Q$ and base
$B$. If $\Gamma$ is retractable, then:

  1. 1.

the group $G_{1}=\pi_{1}(\Lambda_{\mathbb{Z}})$ is a Grushko factor of $G$
containing $Q$ (i.e. $G_{1}$ is one-ended);

  2. 2.

$\Lambda_{\mathbb{Z}}$ is minimal, and is a retractable centered splitting of
$G_{1}$;

  3. 3.

there is a Grushko decomposition $G=G_{1}*G_{2}*\dots*G_{k}*F$ such that

$B\simeq B_{\Lambda_{\mathbb{Z}}}*G_{2}*\dots*G_{k}*F^{\prime},$

with $B_{\Lambda_{\mathbb{Z}}}$ the base of $\Lambda_{\mathbb{Z}}$ (possibly
not one-ended) and $F^{\prime}$ a free factor of the free group $F$.

######  Remark 5.14.

By Assertion 1, collapsing all edges of $\Lambda$ with non-trivial stabilizer
yields a Grushko decomposition of $G$.

We derive several consequences of the proposition before proving it, starting
with two obvious corollaries.

######  Corollary 5.15.

If $G$ has a retractable splitting, so does one of its Grushko factors. ∎

######  Corollary 5.16 ([Per11, PS12]).

A free group is not an extended étage of surface type. ∎

If $G$ has a Grushko decomposition $G=G_{1}*G_{2}*\dots*G_{k}*F$, define
$r(G)$ as $G_{1}*G_{2}*\dots*G_{k}$.

######  Corollary 5.17.

If $G$ is an extended tower over $B$, then $r(G)$ is an extended tower over a
subgroup isomorphic to $r(B)$.

###### Proof.

It suffices to prove the result for an étage. It is clear if
$G=B*{\mathbb{Z}}$. If $G$ is an extended étage of surface type over $B$,
Proposition 5.13 says that $G_{1}$ is an extended étage over
$B_{\Lambda_{\mathbb{Z}}}$, so by Remark 4.30 $r(G)=G_{1}*G_{2}*\dots*G_{k}$
is an extended étage of surface type over
$B_{\Lambda_{\mathbb{Z}}}*G_{2}*\dots*G_{k}$, which is of the form
$r(B)*F^{\prime\prime}$ with $F^{\prime\prime}$ free.

∎

The following result is a partial converse to Proposition 5.2.

######  Corollary 5.18.

If $G*{\mathbb{F}}_{r}$ is an extended tower over a group $B$ for some $r\geq
1$, then $G$ is an extended tower over a group $B^{\prime}$ such that $B\simeq
B^{\prime}*{\mathbb{F}}_{s}$ for some $s\geq 0$. ∎

###### Proof.

By the previous corollary, $r(G)=r(G*{\mathbb{F}}_{r})$ is an extended tower
over $B^{\prime}:=r(B)$, and $G$ is an extended tower over $r(G)$. ∎

The rest of this subsection is devoted to the proof of Proposition 5.13. The
key step is the following fact.

######  Proposition 5.19.

Let $\Gamma$ be a centered splitting of $G$. If it is retractable, the group
$G_{1}$ of Definition 5.12 is one-ended.

###### Proof of Proposition 5.13 from Proposition 5.19.

Since $G_{1}$ is one-ended, it is a Grushko factor of $G$.

Minimality of $\Lambda_{\mathbb{Z}}$ will be proved in Lemma 5.20. Since
$\Gamma$ is retractable, there is a non-degenerate boundary-preserving map
from $Q$ to $G$. By Lemma 5.9 there is one with values in $G_{1}$, so
$\Lambda_{\mathbb{Z}}$ is a retractable splitting of $G_{1}$.

The structure of $B$ follows from the above description of the Grushko blowup;
the rank of $F^{\prime}$ is the sum of the first Betti numbers of the graphs
$\Lambda_{i}$ (the total number of loops at the vertices $w_{i}$); in the rank
of $F$ there is an extra term equal to the number of bottom groups of
$\Lambda_{\mathbb{Z}}$ minus the number of bottom groups of $\Gamma$. ∎

We now prove Proposition 5.19. This requires several lemmas.

######  Lemma 5.20.

If a centered splitting $\Gamma$ of $G$ is retractable, the splitting
$\Lambda_{\mathbb{Z}}$ of Definition 5.12 is minimal.

###### Proof.

We assume that $\Lambda_{\mathbb{Z}}$ is not minimal and we argue towards a
contradiction. As explained above, there is an edge $e=vv_{i}$ of $\Gamma$
carrying a group $A_{e}$ such that $B_{i}=A_{e}*B^{\prime}_{i}$, and the
groups carried by all other edges of $\Gamma$ containing $v_{i}$ are contained
(up to conjugacy) in $B^{\prime}_{i}$.

Let $C$ be the boundary component of $\Sigma$ associated to the edge $e$.
Fixing a non-degenerate boundary-preserving map $p$, we let ${\mathcal{C}}$ be
a maximal family of pinched curves on $\Sigma$, and we consider the component
$\Sigma^{\prime}$ of $\Sigma\setminus{\mathcal{C}}$ containing $C$. By Lemma
4.9 (and Remark 4.10) $p$ maps $\pi_{1}(\Sigma^{\prime})$ into $B_{i}$ (up to
conjugacy). Moreover, $\pi_{1}(C)$ is mapped surjectively onto a conjugate of
$A_{e}$, and if $C^{\prime}$ is another component of
$\Sigma^{\prime}\cap\partial\Sigma$ then $\pi_{1}(C^{\prime})$ is mapped into
a conjugate of $B^{\prime}_{i}$. Projecting $B_{i}$ to
$A_{e}\simeq{\mathbb{Z}}$ by killing $B^{\prime}_{i}$, we deduce that a
generator of ${\mathbb{Z}}$ is a product of commutators or squares (depending
on whether $\Sigma^{\prime}$ is orientable or not), a contradiction. ∎

The following remark will be used in Subsections 6.2.1 and 9.2.

######  Remark 5.21.

Assume that a cyclic splitting $\Gamma$ has a vertex $v$ which is as in
Definition 3.5 (surface-type vertices), except that the map from the set of
incident edges to the set of boundary components of $\Sigma$ is injective but
not surjective; in other words, $\Sigma$ has at least one “free” boundary
component (this may occur at a flexible vertex of a JSJ decomposition relative
to a cyclic subgroup $A$, with a free boundary component whose fundamental
group contains a conjugate of $A$). The argument used to prove the lemma then
says that there is no non-degenerate boundary-preserving map
$p:\pi_{1}(\Sigma)\rightarrow G$. In particular, if $\Gamma$ is a centered
splitting which is not minimal (thus contradicting our standing assumption),
it is not retractable.

######  Lemma 5.22.

Let $\Gamma$ be a centered splitting of $G$ with central vertex group $Q$.

If $J$ is a finitely generated one-ended subgroup of $G$, then some conjugate
of $J$ either is contained in one of the bottom vertex groups, or contains a
finite index subgroup of $Q$.

###### Proof.

If $J$ is elliptic in the Bass-Serre tree $T$ of $\Gamma$, it is contained (up
to conjugacy) in a bottom group $B_{i}$ (not in $Q$, which is a free group).
If not, the action of $J$ on its minimal subtree defines a non-trivial
splitting $\Gamma_{J}$. Using conjugations, we may assume that $\Gamma_{J}$
has a vertex group $J\cap Q$, with an incident edge group of the form $J\cap
C$ where $C$ is a maximal boundary subgroup of $Q=\pi_{1}(\Sigma)$. Since $J$
is one-ended, $J\cap C$ is non-trivial. It is maximal cyclic in $J\cap Q$
because $C$ is maximal cyclic in $Q$. By 1-acylindricity of $T$ and minimality
of $\Gamma_{J}$, we have $J\cap C\neq J\cap Q$. Lemma 3.6 then implies that
$J\cap Q$ has finite index in $Q$. ∎

######  Remark 5.23.

If $J$ is not elliptic in $T$, the proof shows that $J$ has a cyclic splitting
with a surface-type vertex whose associated surface is a finite cover of
$\Sigma$.

######  Lemma 5.24.

Let $\Gamma$ be a centered splitting of $G$. If each bottom group $B_{i}$ is
one-ended relative to the incident edge groups, then $G$ is one-ended.

######  Remark 5.25.

In this lemma, the implicit assumption that $\Gamma$ is a minimal splitting is
essential.

###### Proof.

By [Hor17, Lemma 6.11], if all Grushko factors of $G$ are elliptic in
$\Gamma$, then some vertex group of $\Gamma$ is freely decomposable relative
to its incident edge groups, contradicting our assumption. Thus, there is a
Grushko factor $J$ of $G$ which is not elliptic in $\Gamma$. Lemma 5.22