arXiv:1603.07187v2 [] 8 May 2017

THE STRUCTURE OF LIMIT GROUPS OVER
HYPERBOLIC GROUPS
DANIEL GROVES AND HENRY WILTON

Abstract. Let Γ be a torsion-free hyperbolic group. We study
Γ–limit groups which, unlike the fundamental case in which Γ is
free, may not be finitely presentable or geometrically tractable.
We define model Γ–limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic).
Given a strict resolution of an arbitrary Γ–limit group L, we canonically construct a strict resolution of a model Γ–limit group, which
encodes all homomorphisms L → Γ that factor through the given
resolution. We propose this as the correct framework in which to
study Γ–limit groups algorithmically. We enumerate all Γ–limit
groups in this framework.

Limit groups over a group Γ (otherwise known as Γ–limit groups)
arise naturally when studying algebraic geometry over Γ; that is to
say, sets of homomorphisms Hom(G, Γ), where G is a finitely generated group. We will be concerned with the case in which Γ is a
non-elementary, torsion-free hyperbolic group, a case which has been
studied extensively by Sela [46] and others [33, 38, 39, 20]. The results
of [46] extend Sela’s solution to Tarski’s problems in the case when Γ
is free in [45] et seq. (see also [32] et seq.).
Since every finitely generated subgroup of Γ is a Γ–limit group, the
hyperbolic case immediately presents new problems that do not arise
in the free case. One such problem is that hyperbolic groups typically
contain subgroups that are finitely generated but not finitely presented
[40]. Thus, it is not immediately clear how to give a finite description
of a Γ–limit group. Moreover, Γ–limit groups do not in general have
the nice geometric properties of limit groups over free groups (which
are all toral relatively hyperbolic [1, 8], in particular finitely presented).
The geometry of hyperbolic and relatively hyperbolic groups has been
integral to the solutions of many algorithmic problems [41, 43, 10, 12,
Date: April 6, 2018.
The work of the first author was supported by the National Science Foundation
and by a grant from the Simons Foundation (#342049 to Daniel Groves).
The second author was supported by the EPSRC.
1

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DANIEL GROVES AND HENRY WILTON

11]) and has been very useful in the algorithmic study of limit groups
(see, for example, [19, 20]).
Our goal in this paper is to associate to a Γ–limit group L a model
Γ–limit group M, which is toral relatively hyperbolic, contains L as a
subgroup, and most importantly, encodes a large set of homomorphisms
L → Γ in a natural way. (The reader should bear in mind that the
interest of Γ–limit groups arises precisely because they encode large
sets of homomorphisms to Γ; indeed, Sela proved that Γ–limit groups
are precisely the finitely generated fully residually Γ groups.) To give
the reader a flavour of our results, we state some properties of a model
M associated to a Γ–limit group L.
Theorem 0.1. Let L be a freely indecomposable Γ–limit group. There
exists a Γ–limit group M such that:
(1) there is an injection η : L ֒→ M;
(2) M is toral relatively hyperbolic, in particular finitely presented;
(3) there is a natural homomorphism of modular groups
Φ : Mod(L) → Mod(M);
(4) Φ intertwines η (that is,
η(α(g)) = Φ(α)(η(g))
for α ∈ Mod(L) and g ∈ L).
Recall that the modular automorphisms of a Γ–limit group are those
that arise naturally from the JSJ decomposition of Γ. They play a
special role in the theory of Γ–limit groups, and so items (3) and (4)
above demonstrate that M captures much of the information contained
in L. Items (1) and (2) are proved in Theorem 4.10 below. Items (3)
and (4) are proved in Theorem 5.1.
We now elaborate on the details of our construction. According to
[46, Section 2], all of the homomorphisms from L to Γ can be described using a collection of strict resolutions (see Definition 2.21 below). These take the form of a sequence of strict maps
L0 → L1 → · · · → Ln
where L0 is (a quotient of) L, the Li are all Γ–limit groups and Ln
admits a (fixed) strict map to Γ. In particular, Sela shows that every
Γ–limit group admits a strict resolution, and we use these to build
model Γ–limit groups canonically.
It is important to note that we need to relax two of Sela’s conditions.
Sela insists that the maps in resolutions are surjective, but we allow our
resolutions to include non-surjective maps. Moreover, Sela terminates

THE STRUCTURE OF LIMIT GROUPS

3

with a Γ–limit group which is a free product of a free group and freely
indecomposable subgroups of Γ. We do not know how to ensure this
and keep the features of our construction. These changes seems to be
necessary both in order to make model Γ–limit groups geometrically
tractable and also to have their structure closely mimic that of the Li .
Theorem 0.2. To every strict resolution L0 → L1 → . . . → Ln there
is canonically associated a strict resolution that fits into a commutative
diagram
// L1
// · · ·
// Ln
L0
η0



M0
//

η1



M1
//

ηn

···
//



Mn

where each (Mi , ηi ) satisfies the properties of Theorem 0.1.
This result follows from Theorems 4.10 and 5.1, along with our construction of models from a resolution of L0 .
λn−1
λ
λ
Suppose that R = L0 →0 L1 →1 . . . → Ln is a strict resolution of
L0 . The way in which R describes a large set of homomorphisms from
L0 to Γ is via homomorphisms which factor through R. This means
that a homomorphism can be described as a composition of modular
automorphisms of the Li and the given maps λi , finally composed with
a particular embedding of each of the freely indecomposable free factors
of Ln into Γ and an arbitrary map from the free free factor of Ln to
Γ. It is important to note that one needs modular automorphisms of
Ln in this description. From a homomorphism factoring through R,
the intertwining maps Φi : Mod(Li ) → Mod(Mi ) canonically induce a
homomorphism factoring through the model resolution.
We give a brief outline of this paper. In Section 1 we discuss some
differences between the work in this paper and a construction that appears in [33] and also in [31]. In Section 2 we recall some background
required for the results in this paper. In Section 3 we give the definition
of the (sequence of) model(s) built from a strict resolution of a Γ–limit
group L, in terms of the universal property that a model satisfies. In
Section 4 we provide the construction of models inductively in terms
of a graph of groups built from the Grushko and JSJ decompositions
of the limit groups appearing in the resolution of L, and prove this
construction satisfies the universal property from Section 3. In Section 5 we prove the remaining properties of model Γ–limit groups. In
Section 6 we introduce the notion of relatively immutable subgroups
of toral relatively hyperbolic groups, prove basic results about them
and provide an effective enumeration of them. In Section 7 we prove

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DANIEL GROVES AND HENRY WILTON

that quasi-convex enclosures (see Subsection 2.6) can be algorithmically computed. Finally, in Section 8 (See Theorem 8.3) we apply the
construction of models to algorithmically enumerate Γ–limit groups via
an enumeration of all of the pairs of resolutions as in Theorem 0.2. This
enumeration is analogous to the enumeration of limit groups (over free
groups) which we provided in [19].
Acknowledgements. We thank the referee for a very careful reading
and for many very helpful comments and corrections which improved
the manuscript in numerous ways.
1. A cautionary example
A somewhat similar construction to the one we perform in this paper
is described in [33, §§3.5] and [31, §§6.6]. (The construction in these
two papers appears to be identical; we refer to the numbering in [33].)
In order to highlight some differences between the construction in
[33, 31] and the one in this paper, in this section we describe certain
examples. It is worth remarking that these examples are very elementary, and the issues that they exemplify are entirely generic – they
will occur for many torsion-free hyperbolic groups Γ and many Γ–limit
groups.
We start with a basic fact about hyperbolic groups. For brevity, we
call a torsion-free hyperbolic group rigid if it admits no non-trivial free
or cyclic splitting.
Lemma 1.1. Any torsion-free word-hyperbolic group G embeds in a
rigid word-hyperbolic group Γ. Furthermore, if G is virtually special (in
the sense of Haglund–Wise [29]) then we can choose Γ to be virtually
special.
Proof. We may assume that G is non-elementary. Choose pairwise nonconjugate elements h1 , . . . , hn ∈ G (with n ≥ 2) such that, in every
nontrivial free or virtually cyclic splitting of G, some hi is hyperbolic.
(Finitely many hi suffice by standard accessibility results.) By the pingpong lemma, replacing the elements hi with proper powers, we may
assume that the subgroup H that they generate is free and quasiconvex.
Now fix any non-elementary hyperbolic group G0 without free or cyclic
splittings, and let H0 be a free, quasiconvex, malnormal subgroup of
G0 of the same rank as H. (Examples of such pairs abound.) The
amalgam
Γ := G ∗H ∼
=H0 G0
is now the fundamental group of a 2–acylindrical graph of hyperbolic
groups with quasiconvex edge groups, and hence is hyperbolic by [4].

THE STRUCTURE OF LIMIT GROUPS

5

In any free or cyclic splitting of Γ, G0 is elliptic by construction, hence
G is too. Since n ≥ 2, no edge is stabilized by H, and so the splitting
is trivial.
Finally, if G is virtually special, we may also choose G0 to be virtually
special, and then Γ is virtually special by the main theorem of [50]. 
We now describe our examples.
Example 1.2. Suppose that F is a finitely generated nonabelian free
group, and that w is an element in F which is not a proper power,
and so that F admits no splittings over {1} or Z in which w is elliptic.
Such elements w are entirely generic and easy to find—they are called
rigid by Cashen–Macura [7], and the procedure they describe can be
used to certify that a given element is rigid.
Let D be the double of F over w, and note that D = F ∗hwi F is the
JSJ of H. The group D is an ordinary limit group, since the natural
retraction D → F is strict. Applying Lemma 1.1, we obtain a torsionfree hyperbolic group Γ that admits no splittings over {1} or Z and in
which D embeds.
Now let u be a primitive element of D so that D admits no nontrivial
splitting over {1} or Z in which u is elliptic, and let D1 be the double
of D over hui. Let ρ1 : D1 → D be the map that identifies the two
copies of D in D1 . This is a strict map.
Let G1 be a finite-index subgroup of D1 which does not contain u,
and let C = huk i = G1 ∩ hui (so k > 1). Let G2 = ρ1 (G1 ) and note
that it is straightforward to check that G1 ։ G2 is a strict map, so
since G2 is a subgroup of Γ,
G1 → G2
is a strict resolution of G1 . In the description from [33], the group G2
is given a fixed (conjugacy class of) embedding into Γ, which can be
chosen to be the inclusion described above.
In this example there are many homomorphisms from G2 to Γ that
are not injections. However, by pre-composing the inclusion of G2 into
Γ with a sequence of Dehn twists in the edge group of the JSJ of G2 ,
we get an infinite sequence of non-conjugate embeddings of G2 into
Γ. Needless to say, since these maps are all injections, they do not
factor through a proper shortening quotient of G2 . Thus, in order to
describe all of the homomorphisms from G2 to Γ, and therefore all of
the homomorphisms from G1 to Γ that factor through G1 → G2 , one
cannot forget about the JSJ decomposition of G2 .

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DANIEL GROVES AND HENRY WILTON

Remark 1.3. It is also straightforward to come up with an example
of a pair G1 → G2 with a strict map, G2 a subgroup of a torsionfree hyperbolic group with nontrivial JSJ such that G2 has no proper
shortening quotients. One can do this with 3–manifolds, Property (T)
hyperbolic groups, the Rips construction, or other ways.
We now describe the construction from [33, §§3.5] for Example 1.2.
In their notation, n = 2 and G2 = H1 . Note that they explicitly insist
that all of the maps between the Gi are proper quotients, and that
the final Hi are given with fixed embeddings of the Hi into Γ (up to
conjugacy).
The primary JSJ of G1 contains the edge group huk i, whose image
in G2 is uk . (See §§2.2 for the definition of the canonical primary JSJ
decomposition.) The construction from [33] extends the centralizer of
uk in G2 . This extension of centralizers obviously has presentation
hG2 , t | [t, uk ] = 1i
This procedure is performed for each of the edges in the JSJ of G1
(which are G1 –conjugacy classes of intersections of hui with G1 ). Since
there are no abelian or QH subgroups in the JSJ decomposition of G1 ,
and since n = 2, we finish with the group G1 , which is this iterated
extension of centralizers. Clearly G1 is a Γ–limit group and contains
G1 as a subgroup.
It is then stated in [33]:
“We now extend each subgroup Hi by its quasi-convex closure Γi .
Denote the obtained group by N. Therefore N is NTQ and total (sic)
relatively hyperbolic. Each Γi is freely indecomposable.”
There is no proof that N is NTQ or toral relatively hyperbolic in [33]
or [31], only the above assertion. We now explain that when applying
this construction to Example 1.2 neither of these assertions is true. We
do not provide the definition of an ‘NTQ group’ here, but only note
that it is a particular kind of Γ–limit group. The group N constructed
in [33] for Example 1.2 is not a Γ–limit group, let alone an NTQ–group.
For Example 1.2, we have H1 = G2 , and only one Hi . It is not clear
precisely what “extend each subgroup Hi by its quasi-convex closure”
is supposed to mean, but it seems the only reasonable interpretation
in this case is to form the graph of groups
N = G1 ∗ G2 Γ 1 .
In this example, Γ admits no nontrivial splittings over {1} or Z so the
description of how to find Γi in [33, §3.4] yields Γ1 = Γ

THE STRUCTURE OF LIMIT GROUPS

7

However, the element uk has a nontrivial root in Γ, so if we glue Γ
onto G1 along G2 then the resulting group N is not CSA. Both Γ–limit
groups and toral relatively hyperbolic groups are CSA, so this means
that it is not a Γ–limit group (so certainly not NTQ) and also not toral
relatively hyperbolic. Thus the claim from [33] is false.
Moreover, as described above, the group N cannot be used to describe all of the homomorphisms from G2 to Γ, nor those from G1 to Γ
which factor through G1 → G2 , since not all injective homomorphisms
from G2 to Γ can be extended to homomorphisms from Γ to Γ.
As we stated above, this is a very simple example, and it exhibits
behavior that is entirely generic to the situation for Γ–limit groups for
torsion-free hyperbolic Γ.
The construction in [33, §§3.5] is very important to the construction
in that paper, and the above Example shows that Proposition 11 from
[33] (which is [31, Theorem 2]) is not proved in either of these papers, because the construction they make does not have the properties
that they claim it does. It appears to be a fundamental construction
upon which many of the other algorithms rely (see for example, [33,
§§4.4], where they use it to ‘construct algorithmically a finite number
of Γ–NTQ systems corresponding to branches b of the canonical Hom–
diagram. . . ’). In summary, this appears to be a very serious error in
[33].
The construction in this paper takes these issues (and others) into
account, and embeds a Γ–limit group (equipped with a strict resolution) into a toral relatively hyperbolic Γ–limit group, as described in
Theorems 0.1 and 0.2.
Remark 1.4. Since the first version of this paper was circulated, new
versions of [33] and [31] have been posted to the arXiv, which claim to
address the issues outlined here. In the interests of pointing out some
of the subtleties inherent in the construction made in this paper, we
decided to leave this section largely unchanged.

2. Preliminaries
Let Q be a group. A Q–limit group is a limit of finitely generated
subgroups of Q in the space of k–generated marked groups, for any fixed
k. It is convenient to work with the following notions, which help to
elucidate the connection between Q–limit groups and homomorphisms
from a finitely generated group G to Q.

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DANIEL GROVES AND HENRY WILTON

Definition 2.1. Suppose that Q and G are groups. A sequence of
homomorphisms {ρn : G → Q} is convergent1 if for any g ∈ G either
(i) g ∈ ker(ρn ) for all but finitely many n; or (ii) g 6∈ ker(ρn ) for all but
finitely many n.
The stable kernel of {ρn } is the set −
Ker
−→ (ρn ) of elements g ∈ G which
are in the kernel of ρn for all but finitely many n.
A Q–limit group is a group of the form L = G/Ker
−−→ (ρn ) where G is
a finitely generated group and {ρn : G → Q} is a convergent sequence
of homomorphisms.
Definition 2.2. A group G is equationally noetherian if for any finitely
b along with an
generated group H there is a finitely presented group H
b → H so that the map
epimorphism ρ : H
b G)
ρ∗ : Hom(H, G) → Hom(H,

induced by precomposition with ρ is a bijection.
b to G factors through
In other words, every homomorphism from H
ρ. Yet another way of expressing this is that once a tuple of elements
b it automatically also
of G satisfies the finitely many relations for H
satisfies the infinitely many relations for H. Thus, the ‘Hilbert Basis
Theorem’ holds for equations over G.
Theorem 2.3 (Sela; [46], Theorem 1.22). Torsion-free hyperbolic groups
are equationally noetherian.
Definition 2.4. Suppose that G and D are groups. We say that D is
fully residually–G if for every finite set A ⊂ D there is a homomorphism
φA : D → G which is injective on A.
It is well known (see, for example, [37, Theorem 2.1]) that if G is
equationally noetherian then the class of G–limit groups is exactly the
class of finitely generated fully residually–G groups. From this, the
following is straightforward.
Corollary 2.5. Let Γ be a torsion-free hyperbolic group and let L be a
Γ–limit group. Then there exists a convergent sequence {ρn : L → Γ}
with −
Ker
−→ (ρn ) = {1}.
2.1. Abelian subgroups. For the study of Γ–limit groups, the following definition and result are particularly useful.
Definition 2.6. A subgroup H of a group G is malnormal if for all
g ∈ G r H we have H ∩ H g = {1}.
1Other

authors call these ‘stable’ sequences. See, for example, [5, Definition 1.6].

THE STRUCTURE OF LIMIT GROUPS

9

A group G is called a CSA group if any maximal abelian subgroup
of G is malnormal.
A group G is called commutative transitive if whenever g1 , g2 , g3 ∈ G
with g2 6= 1 and [g1 , g2 ] = [g2 , g3 ] = 1 then [g1 , g3 ] = 1.
Our interest in torsion-free CSA groups comes from the following
result, which is due to Sela and implicit in [46, Section 1].
Theorem 2.7. Suppose that Γ is a torsion-free hyperbolic group and
that L is a Γ–limit group. Then L is torsion-free, CSA and all abelian
subgroups of L are finitely generated.
It is easy to see that a CSA group is commutative transitive.
Remark 2.8. Being CSA is a closed condition in the space of marked
groups, as is being torsion-free. So these properties are straightforward
to prove. However, the fact that abelian subgroups are finitely generated is much deeper, and is due to Sela. The reader is referred to [18,
Corollary 5.12] for a proof.
2.2. Primary JSJ decompositions. JSJ decompositions are one of
the fundamental tools needed to study Γ–limit groups. They were
developed in this context by Sela [46]; various other groups of authors
have developed related theories [42, 15, 16, 6], and the whole theory
was put into a unified context by Guirardel–Levitt [23, 24, 25, 28].
The specific decomposition that we will work with is called the
canonical primary JSJ decomposition. In this section, we shall adapt
the definitions and results of Guirardel–Levitt to define the canonical
primary JSJ decomposition and to explain some of its properties.
Definition 2.9. [12, Definition 3.31] Suppose that G is a torsion-free
CSA group. An abelian splitting of Γ is called essential if whenever E
is an edge group and γ ∈ G satisfies γ k ∈ E for some k 6= 0 we already
have γ ∈ E. An essential splitting is called primary if every noncyclic
abelian subgroup is elliptic.
In Guirardel–Levitt’s terminology, if we let E denote the set of rootclosed abelian subgroups and Anc the class of non-cyclic abelian subgroups then the Bass–Serre tree of a primary splitting is an E–tree
relative to Anc . Our goal is to construct a canonical, primary JSJ tree
for Γ-limit groups. A related JSJ tree for toral relatively hyperbolic
groups was used in [26, §5], in which the Bass–Serre trees of primary
splittings are called RC trees. Our JSJ decomposition will differ slightly
from the RC-JSJ decomposition of [26].
We give the definition and state and prove some of the properties of
the primary JSJ decomposition here.

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DANIEL GROVES AND HENRY WILTON

Definition 2.10. [28, Introduction] A primary tree is universally elliptic if every edge stabilizer is elliptic in every primary splitting. A
primary JSJ tree is a universally elliptic primary tree which is maximal
among all universally elliptic primary trees, meaning that it dominates
any other universally elliptic primary tree.
When we speak of a canonical primary JSJ splitting, we mean one
that is invariant under the natural action of the outer automorphism
group. The construction of the canonical primary JSJ tree goes via the
canonical abelian JSJ tree. An abelian JSJ tree is defined in the same
way as a primary JSJ tree, using the set A of all abelian subgroups
in the place of E; again, we only consider A-trees relative to Anc .
Guirardel–Levitt proved that a CSA group G (such as a Γ-limit group)
has a canonical abelian JSJ tree, and described its structure [25].
Theorem 2.11 (Guirardel–Levitt). Let G be a finitely generated, oneended, torsion-free, CSA group. There exists a canonical abelian JSJ
tree Tab , which is bipartite with the following structure:
(1) One class of vertices has maximal abelian stabilizers.
(2) The stabilizers of the other class of vertices are either rigid or
quadratically hanging (QH) (defined below).
(3) If e is an edge of Tab and x is an adjacent, non-abelian vertex,
then Ge is maximal abelian in Gx .
(4) Furthermore, the incident edge stabilizers at x form a malnormal family.
In particular, Tab is 2-acylindrical.
Proof. Existence of the canonical abelian JSJ tree is stated as [25,
Theorem 5]; see also Proposition 6.3 of the same paper, where it is
proved that Tab (denoted there by Tc ) is 2-acylindrical. The bipartite
structure follows from its description as a tree of cylinders.

The rigid vertices have the property that they admit no non-trivial
abelian splitting relative to Anc and their incident edge groups. Note
that below we will consider vertex groups which are rigid with respect
to primary splittings, rather than with respect to all abelian splittings,
and this will be the meaning of the word rigid after Definition 2.13.
The quadratically hanging vertices are fundamental groups of surfaces, with edge groups attached isomorphically to the infinite cyclic
subgroups corresponding to boundary components.
In order to describe the primary JSJ decomposition ∆, we must first
introduce the notion of a socket.

THE STRUCTURE OF LIMIT GROUPS

11

Definition 2.12. Let Σ be compact hyperbolic surface with k ≥ 0
boundary components and let d1 , . . . , dk be generators for the subgroups corresponding to the boundary components. Let n1 , . . . , nk be
positive integers. The group S obtained from π1 Σ by attaching an ni -th
root to di for each i is called a socket. When considered as a subgroup,
a socket will always refer to a vertex group in a graph-of-groups decomposition. If S is a vertex group in a graph of groups, we insist that
the incident edge groups are infinite cyclic with generators identified
with the roots attached to boundary components, and that every root
has an incident edge group attached to it. A socket is called maximal
if it is not properly contained in any larger socket which appears as a
vertex in a graph-of-groups decomposition.
It is a usual feature of JSJ decompositions that the flexible vertices
are QH subgroups. In the case of essential splittings, however, the
flexible vertices are sockets, because of the requirement that the edges
be root-closed.
There is an important difference between socket groups for abelian
splittings and socket groups for primary splittings, which is the difference between our primary JSJ splitting and the RC-JSJ from [26].
Edge stabilizers in the RC-JSJ tree are required to be elliptic in every
abelian splitting rel Anc , whether or not this splitting is RC (primary,
in our terminology). For the primary JSJ tree, we require edge stabilizers to be elliptic in every primary splitting. The difference can
be seen if a vertex group is a once-punctured Klein bottle, with the
adjacent edge group corresponding to the puncture. Cutting along the
non-separating, two-sided simple closed curve yields a primary splitting whose edge group is elliptic in all other primary splittings of the
once-punctured Klein bottle group (relative to the puncture group),
but not in all abelian splittings, as witnessed by cutting along a separating curve. In the RC-JSJ of [26], the once-punctured Klein bottle
group would be a flexible vertex, whereas in our primary JSJ decomposition, we cut along this curve and replace this once-punctured Klein
bottle with two vertex groups – one corresponding to an annular neighbourhood of the curve, and one corresponding to the rest of the surface,
and we have two edge groups corresponding to the two (isotopic) curves
that join these two vertex groups. The new vertex group corresponding
to the surface will be a rigid socket group.
As noted in [28, §9.5], this distinction happens on orbifolds exactly
when the surface is either a Klein bottle, a once-punctured Klein bottle,
or a Klein bottle with one orbifold point. Since we are in the torsion-free
CSA setting, the once-punctured Klein bottle is the only case that can

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DANIEL GROVES AND HENRY WILTON

arise. Moreover, if Γ is a one-ended torsion-free CSA group and there is
an abelian edge group E in some primary splitting of Γ which is elliptic
in all primary splittings of Γ but hyperbolic in some abelian splitting
Λ, then E must correspond to the non-separating, two-sided curve on
some once-punctured Klein bottle QH subgroup. Indeed, we must get
a hyperbolic-hyperbolic pair from E and some edge group in Λ, which
leads to a QH subgroup. If we take a maximal QH subgroup, then
there can be no non-separating, two-sided curve intersecting the curve
for E, which leads us to conclude that the QH subgroup corresponds
to a once-punctured Klein bottle.
We refer to these rigid vertex groups obtained from a Klein bottle
in the above way as an exceptional socket. Note that an exceptional
socket is not a maximal socket.
Definition 2.13. Suppose that K is a group and H is a collection
of subgroups. We say that (K, H) is rigid if K does not admit any
nontrivial primary splitting relative to H. If K is a nonabelian vertex
group in a graph-of-groups decomposition, we say that K is rigid if
(K, Ke ∪ Anc ) is rigid where Ke is the set of edge groups adjacent to K.
A vertex group of a primary decomposition that is neither abelian
nor rigid is called flexible.
Proposition 2.14. Suppose that G is freely indecomposable, finitely
generated, torsion-free and CSA. Any flexible vertex of a primary JSJ
decomposition of G is a maximal socket.
Proof. This can be proved by following the proof from [28, §6], which
builds a QH subgroup, then noting that in order to find a primary
splitting we need to adjoin the roots to the QH subgroup in order to
build a socket. This is also similar to [44, 13, 12], and can be proved
by similar techniques to any of those papers.

Note that the converse to the above proposition does not hold: there
can also be rigid vertices that are sockets, necessarily built by attaching
roots to the boundary components of a thrice–punctured sphere. When
discussing the JSJ below, it will still be important to insist that any
such vertices are either exceptional sockets or maximal sockets.
Remark 2.15. Note that our terminology differs from that of [26], in
which the non-abelian vertices are called rigid with sockets and QH
with sockets.
We apply these ideas in our setting. Let Γ be a torsion-free hyperbolic group and let L be a Γ–limit group. In particular, L is a
torsion-free CSA group and all abelian subgroups of L are finitely generated.

THE STRUCTURE OF LIMIT GROUPS

13

Proposition 2.16. If L is a freely indecomposable Γ–limit group then
L has a canonical primary JSJ decomposition ∆. Furthermore, ∆ has
the following properties.
(1) Vertices of ∆ fall into three classes (denoted by ui , vj and wk
respectively) with the following properties:
(a) ∆ui is rigid for each i (and, if it is a socket, is either an
exceptional socket or a maximal socket);
(b) ∆vj is a maximal socket, for each j;
(c) ∆wk is abelian.
(2) ∆ is bipartite, with only abelian vertices adjacent to rigid or
socket vertices.
(3) If v is a vertex associated to a nonabelian vertex group ∆v then
every edge group adjacent to ∆v is maximal abelian in ∆v .
(4) ∆ is 2–acylindrical.
Conversely, any primary splitting ∆ that satisfies the above properties
is the canonical primary JSJ.
Proof. We begin by showing that the primary JSJ decomposition exists. We start with the canonical abelian JSJ tree Tab , guaranteed by
Theorem 2.11, and explain how to construct T , the Bass–Serre tree of
the canonical primary JSJ decomposition ∆, following [26, §5].
For each edge e incident at some abelian vertex w, let Ge be the
subgroup consisting of the roots of elements of the edge stabilizer Ge .
The union of the translates Ge .e is a subtree with every edge adjacent
to w. For each edge e, fold the corresponding subtree to a single edge
ē incident at w, with stabilizer Ge . The tree T ′ is then the minimal
G-invariant subtree of the result of performing these folds. (More precisely: T ′ is trivial if G acts with a global fixed point on the folded
tree, and otherwise it is the minimal G-invariant subtree.)
The other modification to Tab which is required is as follows. Suppose that ∆vi is a QH-subgroup of Tab with underlying surface a oncepunctured Klein bottle. (See the above discussion for an explanation
of why the once-punctured Klein bottle is the only relevant case.)
Then ∆vi admits a single primary splitting relative to the adjacent
edge group, corresponding to cutting the Klein bottle along the twosided simple closed curve. Such a splitting is universally elliptic for
primary splittings, but not for abelian splittings. Thus, cut along this
curve and replace the once-punctured Klein bottle with an exceptional
socket group, along with an extra abelian vertex group corresponding
to the curve.
Having performed these modifications we have a tree T , which we
claim satisfies properties (1)–(4).

14

DANIEL GROVES AND HENRY WILTON

We first prove that the resulting tree satisfies properties (1)–(4) from
the statement of the result, and then that any splitting which satisfies
these properties is a canonical primary JSJ decomposition.
The vertex groups of T come in different kinds, depending on how
they arose from the tree Tab . It is clear from the construction that the
vertex groups of T can be partitioned into the required types from the
statement of (1), and that the resulting structure is bipartite according
to the scheme described in (2). Statement (3) follows immediately from
the construction and the corresponding property for Tab , and statement
(4) follows immediately from statement (3). It follows from conditions
(2)–(4) that T is equal to its own (collapsed) tree of cylinders, and then
it follows from [28, Corollary 7.4] that T is canonical.
We next prove the converse. Let ∆ be a primary splitting satisfying
items (1) to (4). We will first show that ∆ is a primary JSJ splitting; to
do this, we check first that it is universally elliptic among all primary
decompositions, and then that it is maximal among all universally elliptic primary decompositions.
To check that ∆ is universally elliptic, let Υ be a (without loss of
generality one-edge) primary splitting and let e be an edge of ∆. If
∆e is non-cyclic then it is elliptic in Υ by hypothesis, so we need only
consider the case in which ∆e is cyclic. Suppose that ∆e is not elliptic
in Υ. Let Λ be a primary JSJ decomposition of L. By [28, Lemma
2.8] some refinement Λ0 of Λ dominates ∆. Since edge stabilizers in ∆
and Λ0 are both abelian and Λ0 is primary, there is an edge group in
Λ0 which is conjugate to ∆e . However, ∆e is not universally elliptic,
so this edge group in Λ0 must correspond to a simple closed curve c in
a surface associated to a socket subgroup of Λ.
Let ∆v be the nonabelian vertex group adjacent to ∆e . The above
argument shows that every cyclic edge group of ∆ is elliptic in Λ0 , and
thus every edge group is elliptic, because Λ0 is primary. If ∆v is rigid,
this implies that ∆v is elliptic in Λ, and hence must be contained in
part of the socket containing the group associated to c. Thus, in this
case, ∆v is a socket which is neither exceptional nor maximal, which
contradicts property (1a). On the other hand, if ∆v is a socket, then it
is clearly not maximal, which contradicts property (1b). This proves
that ∆e is elliptic in Υ, and therefore ∆ is a universally elliptic splitting.
To prove that ∆ is maximal universally elliptic, we must prove that
all vertex groups of ∆ are elliptic in any universally elliptic primary
splitting Θ (see [28, Definition, p.4]). Since the edge groups are elliptic
in Θ, it is clear that the rigid vertex groups of ∆ must be elliptic in
Θ. That abelian vertex groups of ∆ are elliptic in Θ follows quickly

THE STRUCTURE OF LIMIT GROUPS

15

from the fact that G is one-ended and CSA, and the fact that Θ is a
primary splitting.
Finally, suppose that ∆vj is a socket vertex group of ∆. Then there
is a natural subgroup H ≤ ∆vj (the fundamental group of the surface
Σ from the definition, before the roots are added). It is clear that H is
a QH–subgroup, in the sense of [28, Definition 5.13]. Therefore, by [28,
Theorem 5.27], H is elliptic in the JSJ deformation space corresponding
to all abelian edge groups relative to noncyclic abelian edge groups.
However, since we are only considering primary splittings, this implies
that the socket of H is elliptic in Θ, as required. Thus ∆ is maximal
universally elliptic, and so ∆ is a primary JSJ decomposition.
Finally, we note that conditions (2) to (4) imply that the Bass–Serre
tree of ∆ is its own (collapsed) tree of cylinders, and hence is a canonical
primary JSJ tree by the argument applied to T above. This completes
the proof.

For brevity, we will call the decomposition ∆ produced by the above
theorem the JSJ decomposition of L. When a freely indecomposable
Γ–limit group L is not a subgroup of Γ (and often when it is) it admits
a nontrivial primary splitting. Therefore, unless L is a (socket of a)
surface group, the primary JSJ decomposition of L is nontrivial.
Let H be a family of subgroups of L, closed under conjugacy. We will
also sometimes need to work with the Grushko and JSJ decompositions
of L relative to H. These decompositions exist for the same reasons
as above – for example, instead of taking the JSJ decomposition for
(E, Anc)–trees, take the JSJ decomposition for (E, Anc ∪ H)–trees. The
work in [28] works in the relative setting also.
2.3. Modular automorphisms and envelopes. One important role
of the JSJ is to encode an associated collection of automorphisms of a
limit group L, namely the modular group Mod(L) (cf. [45, Definition
5.2]). We briefly recall the definition of a Dehn twist. Suppose a group
G = A ∗C B and z is in the centralizer Z(C). Then the assignments
δz (a) = a for all a ∈ A and δz (b) = zbz −1 for all b ∈ B define an
automorphism δz of G, called a Dehn twist in the splitting G = A∗C B.
Similarly, if G = A∗C with stable letter t, and z is in the centralizer
Z(C), then the assignments δz (a) = a for all a ∈ A and δz (t) = tz
define an automorphism δz of G, called a Dehn twist in the splitting
G = A∗C .
Definition 2.17. Let L be a Γ-limit group, and let ∆ be its canonical primary JSJ decomposition. An automorphism α of L is called a
generalized Dehn twist if it is of one of the following forms.

16

DANIEL GROVES AND HENRY WILTON

(1) A Dehn twist in a one-edge splitting of L obtained by contracting all but one edges of ∆.
(2) A Dehn twist in a one-edge splitting of L obtained by cutting
along a two-sided simple closed curve in a socket vertex of ∆.
(3) Let ∆w be an abelian vertex group of ∆. Then α restricted to
∆w fixes every edge group incident at ∆w (equivalently, α fixes
the peripheral subgroup P (∆w ) defined in Definition 2.18), and
α acts as the identity on every other vertex group and stable
letter of ∆.
The modular group of L, denoted by Mod(L), is the subgroup of Out(L)
generated by the equivalence classes of generalized Dehn twists. The
elements of the modular group are called modular automorphisms.
Note that modular automorphisms act as inner automorphisms on
rigid vertex groups. Furthermore, certain slightly larger subgroups
associated to a rigid vertex group are also preserved (up to conjugacy)
by modular automorphisms. These larger subgroups were named the
envelopes of rigid vertices by Bestvina–Feighn [5].
Working with the canonical JSJ, we are able to simplify slightly the
definition of an envelope.
Definition 2.18. Let ∆ be the JSJ of L, and A an abelian vertex
stabilizer. The subgroup P (A) ⊆ A is generated by the incident edge
stabilizers in A, and the peripheral subgroup P (A) is the minimal direct
factor of A containing P (A).
The envelope of V , E(V ), is then generated by V together with the
peripheral subgroups of the adjacent (abelian) vertex stabilizers.
2.4. Strict homomorphisms and resolutions. Together with the
JSJ decomposition, the second pillar of the structure theory of Γ–limit
groups is provided by the notion of a strict homomorphism. The following definition is rather lengthy, but the idea is that there exists a
homomorphism L → R, where R is a ‘simpler’ limit group, and the
homomorphism has prescribed behaviour on each part of the JSJ of L.
Definition 2.19 (Strict Homomorphism; Sela [45], Definition 5.9). Let
G be a finitely generated group, and let G = G1 ∗ G2 ∗ · · · ∗ Gn ∗ F be
the Grushko decomposition of G, where F is a free group and the Gi
are freely indecomposable. Suppose that Λi is a primary splitting of
Gi .
Let H be a finitely generated group. A homomorphism η : G →
H is strict with respect to the splittings Λi if there is a free product
decomposition H = H1 ∗ · · · ∗ Hn of H (no assumptions are made

THE STRUCTURE OF LIMIT GROUPS

17

about the decomposability of the Hi ) so that η(Gi ) = Hi and for each
i we have that:
(1) η is injective when restricted to envelopes of rigid vertex groups
of Λi ;
(2) The image under η of each socket vertex group of Λi is nonabelian in Hi ;
(3) If E is an edge group of Λi corresponding to a boundary component of a socket-type vertex then η|E is injective; and
(4) If A is an abelian vertex group of Λi , and P (A) is the peripheral
subgroup of A then η|P (A) is injective.
If G is a freely indecomposable Γ–limit group and Λ is the primary
JSJ decomposition of G then a homomorphism η : G → H is strict
if it is strict with respect to Λ. Note that in the case the Λi are the
canonical primary JSJ splittings, item 4 implies item 3.
Definition 2.20. Let G be a finitely generated group, L a Γ–limit
group and {fn : G → Γ} a convergent sequence of homomorphisms.
We say that {fn } converges to L if
L∼
= G/Ker
−−→ {fn }.
Definition 2.21. Suppose that G is a finitely generated group, equipped
with its Grushko decomposition and primary splittings Λi of the free
factors as in Definition 2.19. A strict resolution of G is a sequence of
epimorphisms
G ։ L0 ։ L1 ։ . . . ։ Ln ,
where Ln is a free product of subgroups of Γ and each map G ։ L0
and Li ։ Li+1 is a strict map (with respect to some free and primary
splittings of Li ).
The next theorem now characterizes Γ–limit groups in terms of strict
resolutions. Although this characterization is significantly more complicated than the original definition, it comes equipped with much more
information. In particular, it will allow us to describe a Γ–limit group
G inductively, in terms of ‘simpler’ Γ–limit groups.
Theorem 2.22. [46, Theorem 1.31], cf. [45, Theorem 5.12] A finitely
generated group G is a Γ–limit group if and only if it admits a strict
resolution.
Furthermore, each of the groups G and Li in the resolution are Γ–
limit groups, and we can take the abelian splittings of the free factors
of the Li to be the canonical primary JSJ decomposition of the free
factors.

18

DANIEL GROVES AND HENRY WILTON

Remark 2.23. By Theorem 2.22, we lose nothing by assuming up
front that the groups Li in a strict resolution of a finitely generated
group G are Γ–limit groups.
Once we assume that the groups involved are Γ–limit groups, there
is no reason to assume that each of the homomorphisms in a resolution
are surjective (since finitely generated subgroups of Γ–limit groups are
themselves Γ–limit groups).
Moreover, the terminal group Ln may be assumed merely to admit a
strict map to Γ, rather than be a free product of subgroups of Γ, since
Theorem 2.22 still implies that Ln is a Γ–limit group. In this case, we
may have two resolutions like λ : Ln → Γ and Ln → λn (Ln ) → Γ. In
case the freely indecomposable free factors of λ(Ln ) admit nontrivial
JSJ decompositions, this second resolution encodes more homomorphisms from Ln to Γ than the first one.
These two changes give us some extra flexibility which is central to
the constructions in this paper.
In this paper, we use a strict resolution R of a freely indecomposable
Γ–limit group L in order to embed L in a Γ–limit group MR so that
MR is finitely presented, and furthermore toral relatively hyperbolic.
This allows us to give an algorithmic enumeration of Γ–limit groups,
even though not all Γ–limit groups need be finitely presented.
We make the following definition in order to capture one of the features of strict homomorphisms in a more general setting.
Definition 2.24. Suppose that G is a group and P is a finite collection
of subgroups of G. We say that a homomorphism η : G → H (for some
group H) is non-degenerate with respect to P if P ∩ ker(η) = {1} for
all P ∈ P. Otherwise, we say that η is degenerate with respect to P.
Observation 2.25. Suppose that L is a freely indecomposable Γ–limit
group and that D is the set of edge groups of the primary JSJ decomposition of L. If η : L → R is a strict map then η is non-degenerate
with respect to D.
2.5. Relative hyperbolicity. One of the key properties of the model
Γ–limit groups which we construct in this paper is that they are toral
relatively hyperbolic, which means that there are many available algorithmic tools which can be applied to them (see, for example [9, 11, 12,
13]). In this section, we recall the basic definitions, and some results
which we need in order to prove that model Γ–limit groups are toral
relatively hyperbolic.

THE STRUCTURE OF LIMIT GROUPS

19

Definition 2.26. A group is called toral relatively hyperbolic if it is
torsion-free and hyperbolic relative to finitely generated abelian subgroups. There are many different equivalent definitions of relative hyperbolicity; we refer to [30] for a summary of many of them, and proofs
that they are equivalent.
A toral relatively hyperbolic group admits a canonical set of peripheral subgroups, namely a set of representatives for the conjugacy
classes of maximal noncyclic abelian subgroups. We always take this
to be the peripheral structure, and so we often leave the peripheral
structure unmentioned.
We will construct our relatively hyperbolic models by gluing. Combination theorems for relatively hyperbolic groups were proved by Alibegović [2] and Dahmani [8], and subsequently by Mj–Reeves [36], and
others. We state Dahmani’s combination theorem here.
Theorem 2.27. [8, Theorem 0.1]
(1) Let Γ be the fundamental group of a finite acylindrical graph of
relatively hyperbolic groups, whose edge groups are fully quasiconvex subgroups of the adjacent vertex groups. Let G be the
collection of images of the maximal parabolic subgroups of the
vertex groups in Γ, and their conjugates. Then (Γ, G) is a relatively hyperbolic group.
(2) Let G be a group which is hyperbolic relative to G, and let P ∈ G.
Let A be a finitely generated group which has P as a subgroup.
Then Γ = A ∗P G is hyperbolic relative to (H ∪ A), where H is
the set of Γ–conjugates of elements of G r P G and A is the set
of Γ–conjugates of A.
(3) Let G1 and G2 be relatively hyperbolic groups, let P be a maximal parabolic subgroup of G1 and suppose that P can be embedded in a parabolic subgroup of G2 . Then Γ = G1 ∗P G2 is
relatively hyperbolic.
(4) Let G be a relatively hyperbolic group and let P, P ′ be nonconjugate isomorphic parabolic subgroups of G. Then Γ = G∗P
is relatively hyperbolic.
The following converse to the combination theorem will also be useful.
Proposition 2.28. [27, Proposition 3.4] Suppose that G is a toral
relatively hyperbolic group and that Λ is a graph of groups decomposition
in which the edge groups are abelian and all noncyclic abelian subgroups
are elliptic.

20

DANIEL GROVES AND HENRY WILTON

Then the vertex groups of Λ are relatively quasiconvex, and hence
toral relatively hyperbolic.
2.6. Strong accessibility. Having decomposed a (relatively hyperbolic) group G into the freely indecomposable factors Gi of its Grushko
decomposition, and having computed the JSJ decompositions ∆i of
those factors, one has in a sense reduced the study of G to the study of
the rigid vertex groups Gi,j of the ∆i . It is now natural to apply this
procedure again to each Gi,j , and to continue recursively. The assertion
that this process terminates is called strong accessibility.
We now make this precise. To a toral relatively hyperbolic group G
and a family of subgroups P, we associate a tree TG,P of subgroups as
follows.
(1) We start by labelling a node by (G, P).
(2) For a node v labelled by a relatively freely decomposable pair
(H, Q), we compute the Grushko decomposition H1 ∗. . .∗Hk ∗F
of H (with the Hi well defined subgroups up to conjugacy)
relative to Q; if k = 1 and F = {1} we proceed to the next step;
otherwise, we attach k edges to v, and label the other ends of
the edges by the (Hi , Qi ), where Qi consists of all the elements
of Q contained in Hi .
(3) For a node v labelled by a relatively freely indecomposable subgroup (H, Q), we compute the primary JSJ decomposition ∆ of
H relative to Q.
(a) If ∆ is trivial, then add no edges to v.
(b) If ∆ is non-trivial, and H1 , . . . , Hk are the rigid vertices
of ∆ (again, well defined up to conjugacy), then attach
k edges to v, and label the other ends of the edges by
the (Hi , Qi ), where Qi consists of all the elements of Q
contained in Hi .
(4) Apply steps 2 and 3 to any remaining nodes.
The Strong Accessibility theorem [35, Theorem 2.5] now asserts that
this construction terminates after finitely many steps.
Theorem 2.29 (Strong Accessibility, Louder–Touikan). If G is a toral
relatively hyperbolic group and P is any family of subgroups then the
tree TG,P is finite.
We apply this to deduce that any subgroup of a toral relatively hyperbolic group has a quasi-convex enclosure. Let H be a non-abelian
subgroup of a relatively hyperbolic G and D a finite collection of subgroups of H. We suppose that H does not admit a nontrivial primary
splitting relative to D. We may now use the hierarchy TG,D to define a

THE STRUCTURE OF LIMIT GROUPS

21

sequence of subgroups
H ⊆ Gn ⊆ Gn−1 ⊆ . . . ⊆ G0 = G
and, for each i, an Gi –tree Ti . Indeed, we set vn to be the unique leaf
of TG,D that contains (a conjugate of) H and let the embedded path
from vn to the root of TG,D be vn , . . . , v0 . For each i, let Gi be such
that the label of vi is (Gi , D) (where if necessary we replace Gi by a
suitable conjugate so that H ⊆ Gi ) and let Ti be the Bass–Serre tree
of the splitting of Gi given in the definition of TG,D (taking Tn to be a
point). Note that each Ti has a unique vertex ui stabilized by H, and
Gi+1 is the stabilizer of ui .
Definition 2.30. With the above notation, the quasiconvex enclosure
of H (relative to D) is (with a slight abuse of notation) denoted by H
and defined to be the subgroup Gn .
While this paper was in preparation, the authors discovered that a
similar definition is made in [33, §§3.4], where a “quasi-convex closure”
is defined similarly, although that definition is not relative to a set of
subgroups D. The application of the definition in [33] is similar in spirit
to the one in this paper, although, as we explain in §1 above, considerable care is needed to ensure that the groups constructed really do
have the good geometric features one would like, and also that simultaneously one does not lose control over the sets of homomorphisms one
considers.
Corollary 2.31. Suppose that G is a toral relatively hyperbolic group,
that H is a finitely generated non-abelian subgroup, and D a finite
collection of subgroups of H. Suppose that H admits no abelian splitting
relative to D. Then:
(1) the quasiconvex enclosure H is the maximal subgroup of G that
contains H and does not admit any essential abelian splitting
relative to D; and
(2) H is relatively quasiconvex (and hence itself toral relatively hyperbolic).
Proof. The first assertion is immediate from the definition. The second
follows from Proposition 2.28 and induction.

2.7. Basics on algorithms with toral relatively hyperbolic groups.
Proposition 2.32. [9, Theorem 0.2] There is a one-sided algorithm
which takes as input finite presentations P and terminates if and only
if the group defined by P is toral relatively hyperbolic. In case it is toral

22

DANIEL GROVES AND HENRY WILTON

relatively hyperbolic, the algorithm provides a basis for each peripheral
subgroup.
Suppose that we are given a finite presentation of a toral relatively
hyperbolic group Υ, which is witnessed to be such by the above algorithm. Then we have an explicit solution to the word problem for
Υ. One way to see this is [10, Theorem 0.1], which proves that the
existential theory of Υ is decidable.
The next theorem is not immediately implied by the statement of
[19, Theorem 3.5]. However the proof there applies verbatim to prove
the following stronger result.
Theorem 2.33 (cf. Theorem 3.5, [19]). There exists an algorithm that,
given as input a presentation for a toral relatively hyperbolic group Γ,
and an element γ ∈ Γ, outputs a minimal set of generators for ZΓ (γ).
We remark that in case Γ is toral relatively hyperbolic, ZΓ (γ) is a
finitely generated free abelian group. Therefore the cardinality of a
minimal generating set determines the group up to isomorphism. In
particular, it tells us the rank and a presentation can be easily (and
algorithmically) written down for ZΓ (γ).
3. Model Γ–limit groups
In this section we make the required definitions of model Γ–limit
groups, which are the central object of concern in this paper.
We start by introducing the main category we work with. Recall the
definition of a non-degenerate map from Definition 2.24.
Definition 3.1. The category Ccsa is defined as follows:
• The objects of Ccsa are pairs (G, P), where G is a torsion-free CSA
group and P is a finite collection of non-trivial abelian subgroups of G.
Denote the set of objects by Obj(Ccsa ).
• The morphisms from (G, P) to (H, Q) are the homomorphisms
φ : G → H that are non-degenerate with respect to P and such that,
for each P ∈ P, φ(P ) is conjugate into an element of Q. If X1 , X2 ∈
Obj(Ccsa ) then MorCcsa (X1 , X2 ) denotes the set of morphisms between
X1 and X2 .
Lemma 3.2. Ccsa is a category.
Proof. The only thing that needs to be checked is that morphisms
compose. They do, since the distinguished subgroups of one group
map into the distinguished subgroups of another, so the composite
homomorphism remains injective on distinguished subgroups.


THE STRUCTURE OF LIMIT GROUPS

23

The construction of model Γ–limit groups we give in the next section
relies on colimits in Ccsa .
In this section we give the definition of models in terms of a universal property, leaving open the question of existence. In Section 4 we
provide the construction of models.
Let L0 be a Γ–limit group and let
λ

λ

λn−1

L0 →0 L1 →1 · · · → Ln
be a strict resolution of L0 , where Ln admits a (given) strict map
λn : Ln → Γ. Call this resolution (along with the map λn ) R0 . For
i ≥ 0, let Ri denote the resolution obtained from R0 by starting at Li ,
and finally appending the map λn : Ln → Γ.
Below we define the model Γ–limit groups of Li with respect to Ri ,
denoted MRi . The definition is made by (reverse) induction on i.
Before we describe the construction, we note a few important properties of models, which will be proved throughout the subsequent sections.
Theorem 3.3. Let R0 be as above, and let Ci be the edge groups of the
JSJ decomposition of the freely indecomposable factors of the Grushko
decomposition of Li . For i ≥ 0, there exists a canonical group MRi ,
and these groups satisfy the following properties:
(1) MRi is a toral relatively hyperbolic Γ–limit group;
(2) There is a strict map µi : MRi → MRi+1 ;
(3) There is a distinguished family DRi of abelian subgroups of MRi
so that (MRi , DRi ) ∈ Obj(Ccsa );
(4) There are injective maps ηi : Li → MRi so that µi ◦ηi = ηi+1 ◦λi ;
(5) The map ηi takes Ci to DRi and is non-degenerate with respect
to Ci . In other words, ηi ∈ Mor(Ccsa ).
The claims of Theorem 3.3 are contained in Theorems 4.10 and 5.1
below.
We now give the definition of models. First suppose that L0 is freely
indecomposable.
Suppose inductively that we have defined MR1 , along with a homomorphism η1 : L1 → MR1 . Consider the primary JSJ decomposition
∆ of L0 , and let {∆v } be the collection of rigid vertex groups. Let ∆v
be such a rigid vertex group. Let Cv = {∆e1 , . . . , ∆es } be the set of
adjacent edge groups. We remark that these edge groups are maximal
abelian subgroups of ∆v .
We define the relative model of ∆v , denoted M (∆v ), to be the maximal subgroup of MR1 which does not admit any essential abelian splitting relative to (η1 ◦ λ0 )(Cv ). If we suppose by induction that MR1

24

DANIEL GROVES AND HENRY WILTON

is a toral relatively hyperbolic group, such a maximal subgroup exists
by Corollary 2.31, and indeed is the quasiconvex enclosure η1 ◦ λ0 (∆v )
(relative to (η1 ◦ λ0 )(Cv )). In particular, M (∆v ) is toral relatively hyperbolic, and hence finitely presented, torsion-free and CSA by that
same result.

Consider the elements {(∆v , Cv )}, (M (∆v ), C v ) of Obj(Ccsa ), where
C v is the set {(η1 ◦ λ0 )(C) | C ∈ Cv }. Also consider the set C of all
peripheral subgroups of edge groups of the JSJ of L0 , and the element
(L0 , C) ∈ Obj(Ccsa ).
The model MR0 is then the group that appears in the colimit of
the following diagram (labelling only groups and not the attendant
subgroups), in the category Ccsa (supposing for the moment that such
a colimit exists):
∆v1

η1 ◦λ0

ι

//
:: L0
✈
✈
ι ✈✈✈
✈✈
✈
✈✈

∆vn



M (∆v1 )

η1 ◦λ0



M (∆vn )
The map η0 : L0 → MR0 is the one furnished by the definition of
colimit. Further, the groups M (∆vi ) are subgroups of MR1 , and we
have a map η1 ◦ λ0 : L0 → MR1 . Temporarily define a set of maximal
abelian subgroups D of MR1 by taking images in MR1 of C and of the
C v . Note that this will typically not be the set of maximal abelian
subgroups of MR1 that it was furnished with when it was defined as a
pushout in Ccsa .
Since the maps η1 ◦ λ0 and inclusion of M (∆vi ) into MR1 agree on
each ∆vi , and these maps are all in Mor(Ccsa ), the universal property
of the colimit yields a map µ0 : MR0 → MR1 so that µ0 ◦ η0 = η1 ◦ λ0 .
Note also that (so long as it exists) the universal property of MR0 ,
together with the distinguished subgroups of the groups ∆v , M (∆v )
and L0 , furnish MR0 with a collection of distinguished subgroups.
Now suppose that L0 is freely decomposable
L0 = H1 ∗ . . . ∗ Hr ∗ Fk

THE STRUCTURE OF LIMIT GROUPS

25

where each Hi is freely indecomposable, and Fk is a free group. Let
R0,i denote the resolution obtained by restricting R0 to the subgroup
Hi of L0 .
The definition above gives the models MR0,i along with maps to the
model of L1 . The model of L0 is
MR0 = MR0,1 ∗ . . . ∗ MR0,r ∗ Fk
and the map η0 is defined in the obvious way.
Remark 3.4. At the moment, it is not obvious that the colimit above
exists. If it does, it is not clear that the model should be a Γ–limit
group, or what its properties should be (except that if it exists it is
clearly torsion-free and CSA). Much of the rest of the paper is spent
proving that models exist and understanding the properties of models,
in order to see what they are and why they are useful.
If we apply the definition of model to the strict resolution Γ → Γ
given by the identity map, we can see that with respect to any collection
P of abelian subgroups, the model of (Γ, P) is (Γ, P).
Proposition 3.5. For any collection of nontrivial abelian subgroups
P of Γ, the model of (Γ, P) (with respect to the trivial resolution) is
(Γ, P).
Proof. In case Γ is freely indecomposable, this is immediate from the
observation that M (∆v ) = ∆v . The freely decomposable case is also
immediate.

4. Abelian graphs of groups in Ccsa and the construction
of models
Let L0 be a freely indecomposable Γ–limit group, equipped with a
strict resolution R0 . In this section, we give a construction of MR0
which is amenable to understanding the structure of the model of L0 .
Let Ccsa be the category defined in Definition 3.1 above. In order
to prove our structural results about the model of L0 , we investigate
certain kinds of colimits in Ccsa . Along the way, we exhibit various
subtleties about building torsion-free CSA groups by taking graphs of
groups. We will then show (in case L0 is freely indecomposable) that
we can construct MR0 as a graph of groups with abelian edge groups.
4.1. Some colimits. The following simple example illustrates that the
class of torsion-free CSA groups is not closed under taking pushouts in
the category of groups.

26

DANIEL GROVES AND HENRY WILTON

Example 4.1. Let A1 = A2 = C = Z and let ιi : C → Ai be index 2
embeddings for i = 1, 2. Then the pushout in the category of groups is
A1 ∗C A2 = Z ∗2Z Z which is not commutative transitive.
In the category of abelian groups, the pushout is Z × Z/2Z, which
is not torsion-free.
In the category of free abelian groups, the pushout is Z, with the
maps from the Ai being the same isomorphism and the map from C
being the index 2 embedding.
The next example shows that we do not want to merely take pushouts
in the category of torsion-free CSA groups, where all homomorphisms
are allowed, but required our maps to be non-degenerate.
Example 4.2. Let A = ha1 , a2 | [a1 , a2 ]i ∼
= Z2 , B = hb1 , b2 | [b1 , b2 ]i ∼
=
Z2 and C = hc | ∅i ∼
= Z. Let ι1 : C → A be defined by ι1 (c) = a1 and
ι2 : C → B by ι2 (c) = b1 . Then the pushout in the category of groups
is G = hx, y, z | [x, y], [y, z]i ∼
= F2 × Z, which is not CSA.
Certainly, Gab = Z3 is a torsion-free CSA quotient of G. If we added
peripheral structures PC = {C} and the images of C in A and B to A
and B, then this would be the pushout in Ccsa .
On the other hand, killing y gives F2 = hx, z | ∅i as a torsion-free
CSA quotient of G. These are incomparable quotients, in the sense
that there is no torsion-free CSA quotient of G which ‘lies above’ them
both.2
To remedy these difficulties, we consider the pushout in the category
Ccsa , and in particular we restrict to non-degenerate maps.
First, we explicitly describe the pushout in the category of finitely
generated abelian groups. It is easy to see that pushouts exist in the
category of abelian groups (the pushout is the abelianization of the
pushout in the category of groups). However, we are interested in free
abelian groups rather than merely abelian groups.
Lemma 4.3. Let A1 , A2 , C be finitely generated nontrivial free abelian
groups, and let ιi : C → Ai be embeddings, for i = 1, 2. There exists a
pushout of the diagram
2Of

course, this phenomenon is well known from the construction of Makanin–
Razborov diagrams since Gab and F2 , equipped with the canonical quotient maps
from G form the first level of the MR diagram for G (over a nonabelian free group
F). It is also one of the key reasons for the distinguished abelian subgroups and
the requirement that homomorphisms be non-degenerate.

THE STRUCTURE OF LIMIT GROUPS

C
//

27

A1



A2
in the category of free abelian groups.
Proof. This is straightforward. One way to proceed is to pass to direct
factors of the Ai which contain the image of C as finite-index subgroups,
embed as lattices in Rk , and take the span of these two lattices.

The following lemma is straightforward.
Lemma 4.4. If M is the pushout of a collection of finitely generated
free abelian groups {Ai } (with appropriate maps), and H is a finitely
generated free abelian group with injective maps Ai ֒→ H for each i,
making the appropriate diagram commute, then the induced map from
M to H is injective.
This is useful because of the following.
Corollary 4.5. If all groups are free abelian and all maps are injective
then the pushout of the diagram
// A
C


B
in the category of finitely generated free abelian groups is also the
pushout of that diagram in Ccsa so long as the set of distinguished subgroups of C is nonempty, and each map is a morphism in Ccsa .
In Subsection 4.2 below, we will need a slightly more general construction than a colimit, involving a graph of groups.
4.2. Construction of models.
Definition 4.6. Suppose that G is a finitely generated, torsion-free
CSA group. A graph of groups decomposition ∆ of G is JSJ–like if it
satisfies the following properties:
(1) ∆ is universally elliptic, i.e. the edge stabilizers of ∆ are elliptic
in every primary splitting of G.
(2) Every non-cyclic abelian subgroup of L0 is elliptic in ∆.
(3) ∆ is bipartite, with one class of vertex stabilizers abelian and
the other either (nonabelian) rigid (in the sense of Definition
2.13) or a maximal socket (with every boundary component
used). We denote the set of abelian vertices by {w}, the rigid
vertices by {u} and the socket vertices by {v}.

28

DANIEL GROVES AND HENRY WILTON

(4) Every edge group is maximal abelian in the adjacent vertex
group which is not abelian. Furthermore, ∆ is 2–acylindrical.
Note that in case G is freely indecomposable, it follows immediately
from Proposition 2.16 that the canonical primary JSJ decomposition
of G is JSJ–like, and that every JSJ–like decomposition is in fact a
canonical primary JSJ. We introduce this definition for the slight extra
flexibility of applying it in case that we do not know that G is freely
indecomposable.
Suppose that L is a freely indecomposable Γ–limit group, that ∆ is
a JSJ–like decomposition of L and that ρ : L → M is a strict map with
respect to ∆, where M is a toral relatively hyperbolic group. In our
applications, M will be the model of L1 , where L → L1 is the beginning
of a strict resolution of L.
b as follows:
Given ∆ and ρ, we build a new graph of groups ∆
For a rigid vertex u of ∆, we consider the map ρ|∆u : ∆u → M.
b u to be the quasiconvex
Since λ0 is strict, ρ|∆u is injective. We define ∆
enclosure of ρ(∆u ) in M (relative to the images under ρ of the adjacent
b u.
edges groups). Let ηu = ρ|∆u be the natural inclusion from ∆u to ∆
Define an equivalence relation on the edges of ∆ as follows: Two
edges e1 and e2 adjacent to a rigid vertex u are equivalent if Z∆b u (ηu (∆e1 ))
b u . Edges not adjacent to rigid verand Z∆b u (ηu (∆e2 )) are conjugate in ∆
tices are equivalent only to themselves.
Suppose that e1 and e2 are adjacent to a nonabelian vertex y and are
equivalent. Let Z1 = Z∆b y (ηy (∆e1 )) and Z2 = Z∆b y (ηy (∆e2 )). By CSA,
b y which conjugates Z1 to Z2 induces the same isomorphism,
any γ ∈ ∆
so there is a canonical isomorphism between Z1 and Z2 .
b is equal to the quotient graph of the
The underlying graph of ∆
underlying graph of ∆ obtained by identifying edges which are equivalent according to the above relation. Note that there is a natural way
of labeling each vertex of this graph as exactly one of abelian, rigid
or socket. If y is a vertex of ∆ we let [y] denote the corresponding
b under the quotient map; likewise, the edges of ∆
b are the
vertex in ∆
equivalence classes [e] where e is an edge of ∆. Note, however, that
the equivalence classes [u] of rigid vertices and [v] of socket vertices are
singletons, and so we will usually write u for [u] and v for [v].
b as follows:
We now define the edge and vertex groups of ∆
b u defined above.
(1) A rigid vertex u is labelled by the group ∆
b v , obtained
(2) A socket vertex v is labelled by the socket group ∆
from ∆v by keeping the same surface, but adding any extra
roots that exist in M to the boundary components.

THE STRUCTURE OF LIMIT GROUPS

29

(3) Let [e] be an edge joining an abelian vertex [w] to a vertex y.
b [e] = Z b (ηy (∆e )).
Note that y is not an abelian vertex. Then ∆
∆y
(4) Finally, let [w] be an abelian vertex, and let [e1 ], . . . , [el ] be the
adjacent edges. If [w] = {wk } and [ei ] = {ei,j } then we have
two sets of inclusions of free abelian groups:
b [e ] for each i, j;
(a) ιi,j : ∆ei,j → ∆
i
(b) κi,j : ∆ei,j → ∆wk whenever ei,j is incident at wk .
We let these data define a graph of groups Λ[w] in the natural
way: the underlying graph is bipartite, with vertex set the disjoint union of the {wk } and {[ei ]}, and edge set {ei,j } equipped
with the natural incidence relations; the vertex group of wk is
b [e ] , the edge group of ei,j is
∆wk , the vertex group of [ei ] is ∆
i
∆ei,j , and the edge maps are given by ιi,j and κi,j . The vertex
b [w] is now defined to be the abelianization of π1 Λ[w] .
group ∆
b [e] used the representative e of [e];
We remark that the definition of ∆
however we have already noted that the appropriate edge groups are
canonically isomorphic, so there is no ambiguity in this definition.
b are defined in the natural way using the
Finally, the edge maps of ∆
canonical isomorphisms of conjugacy representatives of edge groups and
either inclusion (in the case of rigid and socket vertices) or the natural
inclusion of a vertex group in the fundamental group of a graph of
groups (in the abelian case).
b the expansion of ∆ with respect to
We call the graph of groups ∆
ρ : L → M.
Lemma 4.7. Suppose that L is a freely indecomposable Γ–limit group,
and that ∆ is a JSJ–like decomposition of L. Suppose further that
ρ : L → M is a strict map, that M is toral relatively hyperbolic, and
b is the expansion of ∆ with respect to ρ. Then π1 (∆)
b is toral
that ∆
relatively hyperbolic, and in particular torsion-free and CSA.
Proof. By construction, the edge groups incident at each rigid vertex
form a malnormal family. In particular, we may apply Dahmani’s Combv
bination Theorem 2.27 together with the fact that the vertex groups ∆
b
are toral relatively hyperbolic (by Corollary 2.31) to deduce that π1 ∆
is toral relatively hyperbolic, in particular torsion-free and CSA.

b defined as follows.
There is a natural homomorphism η : L → π1 (∆)
b u . For socket
For rigid vertices ∆u , the map η is given by ηu : ∆u → ∆
vertex groups ∆v , η is defined to be the inclusion of the socket ∆v
b v . For abelian vertex group ∆w , note that ∆w is a
into the socket ∆
vertex group of Λ[w], and so η is defined to be the composition of the

30

DANIEL GROVES AND HENRY WILTON

inclusion map ∆w → π1 Λ[w] with abelianization. Finally, after picking
a maximal tree in the underlying graph of ∆, each stable letter either
b or to an element of π1 Λ[w] for some abelian
maps to a stable letter of ∆
vertex group w.
b is an injection.
Proposition 4.8. In the above situation η : L → π1 ∆
Proof. In order to prove injectivity of the map η it is convenient to
modify ∆ to produce a new graph of groups ∆′ . The underlying graph
of ∆′ is the same as the underlying graph of ∆. If v is a (flexible)
socket vertex of ∆ then we take ∆′v = ∆v , and likewise if w is an
abelian vertex we take ∆′w = ∆w . For u a rigid vertex of ∆, we set
b u ) (recalling from Definition 2.18 that E(∆u )
∆′u = E(∆u ) ∩ ρ−1 (∆
denotes the envelope of ∆u ). Finally, for e an edge adjoining an abelian
vertex w and a socket or rigid vertex y, we take ∆′e = ∆′w ∩ ∆′y . We
make three remarks about the construction of ∆′ .
First, ∆′ was obtained from ∆ by pulling certain elements across
edges, and so π1 ∆′ = L.
Second, if y is a rigid or socket vertex in ∆, then it remains the case
that the incident edge groups in ∆′ form a malnormal family in ∆′y ; in
particular, ∆′ is also 2–acylindrical.
b [y] ,
Third, for each vertex or edge y of ∆, we have that η(∆′y ) ⊆ ∆
′
by the definition of ∆y in the different cases. It follows that there is
b
a morphism of trees α : T ′ → Tb (the Bass–Serre trees of ∆′ and ∆
respectively) that intertwines η – that is, α(g.x) = η(g)α(x) for any
g ∈ L and x ∈ T ′ .
To show that η is injective, it now suffices to prove that η is injective
on vertex groups of ∆′ (which follows from strictness of ρ) and that α
is a local injection (i.e. does not factor through a fold). In order to
prove that α does not factor through a fold, we must prove that for
b [e] ) ∩ ∆′ = ∆′ . Of course,
each edge e and each vertex y we have η −1 (∆
y
e
b [e] ) ∩ ∆′ , and it is the reverse inclusion that
it is clear that ∆′e ⊆ η −1 (∆
y
must be proved.
b [e] ) ∩ ∆′ . As usual, there are three
So, suppose that g ∈ η −1 (∆
y
cases, depending on the type of y. In case y is a socket vertex, the
b y is that there may be extra roots of
only difference between ∆′y and ∆
boundary components of the surface which exist in M but not in L.
Thus in this case we clearly have g ∈ ∆′ e , as required. Suppose then
b y ) and
that y is a rigid vertex group. By definition ∆′y = E(∆y )∩ρ−1 (∆
′
′
′
∆e = ∆w ∩ ∆y , where w is the vertex at the other end of e. Also, by
b [e] = Z b (ηy (∆e )) (for a suitable choice of representative
definition, ∆
∆y

THE STRUCTURE OF LIMIT GROUPS

31

edge e). But g ∈ ∆′y , so g ∈ Z∆′y (∆e ) = ∆′e , by construction and
malnormality.
Finally, suppose that y is an abelian vertex (so ∆′y = ∆y ), and let x
be the vertex at the other end of e. If x is a socket vertex then there
is nothing to prove, so we assume that x is rigid.
b [y] is the abelianizaThe defining graph of groups π1 Λ[y] , of which ∆
tion, has a quotient obtained from killing each peripheral subgroup of
b [e ] . By abelianizing,
each ∆yi , and also each vertex group of the form ∆
i
b [y] . In this quotient, the image of each ∆
b [e ]
we obtain a quotient of ∆
i
b [e] ), its
is trivial, and the image of ∆y is ∆y /P (∆y ). Since g ∈ η −1 (∆
image in this quotient is trivial. Therefore, g ∈ P (∆y ) ⊆ E(∆x ) and
b [e] ⊆ ∆
b x , we have
so, since η(g) is contained in ∆
b x ) = E(∆x ) ∩ ρ−1 (∆x ) = ∆′
g ∈ E(∆x ) ∩ η −1 (∆
x

whence, since g is also contained in ∆′y , we have that g ∈ ∆′x ∩∆′y = ∆′e
as required.

Proposition 4.9. Suppose that L0 is a freely indecomposable Γ–limit
group and that R0 is a strict resolution of L0 , starting with the strict
map λ0 : L0 → L1 and continuing with the strict resolution R1 of L1 .
Suppose further that MR1 exists, and satisfies:
(1) The natural map η1 : L1 → MR1 is an injection; and
(2) MR1 is toral relatively hyperbolic.
Let ρ : L0 → MR1 be the map ρ = η1 ◦ λ0 , let ∆ be a JSJ–like decomb be the expansion of ∆ with respect to ρ.
position of L0 and let ∆
b with the distinguished set D b of peripheral subgroups of
Equip π1 ∆
∆
b
b
abelian vertex groups of ∆. Then (π1 ∆, D∆b ) is the model of L0 with
respect to R0 .
b D b ) to be the required
Proof. We briefly recall what it means for (π1 ∆,
∆
model and what we have to prove. Recall that we have, for each rigid
vertex v of ∆, two pairs:
(1) (∆v , Cv ), where ∆v is the vertex group, and Cv is the family of
adjacent edge groups; and
(2) (MR1 (∆v ), C¯v ), where MR1 (∆v ) is the relative model (quasiconvex enclosure) of ∆v and C¯v is the set of centralizers in MR1 (∆v )
of the images of Cv .
We also have the pair (L0 , D∆ ), where D∆ is the set of peripheral
subgroups of abelian vertex groups of ∆.
b with the collection D b of peripheral subgroups of abelian
Equip π1 ∆
∆
b We claim that (π1 ∆,
b D b ) is the required colimit
vertex groups of ∆.
∆

32

DANIEL GROVES AND HENRY WILTON

in Ccsa . Indeed, suppose that (H, Q0 ) is an object in Ccsa and α ∈
MorCcsa ((L0 , D∆ ), (H, Q0 )), βv ∈ MorCcsa ((MR1 (∆v ), C¯v ), (H, Q0 )) are
morphisms that make the diagram below commute.
ι
//
∆u1
99 L0
t
tt
tt
t
tt
tt
ι

∆un

η1 ◦λ0

α



MR1 (∆u1 ) ❱

η1 ◦λ0

❱❱❱❱
❱❱❱❱βv1
❱❱❱❱
❱❱❱❱
❱❱❱❱

❱❱** 
// H
MR (∆u )
1

n

βvn

b → H is defined as follows.
The map π : π1 ∆
b u = MR (∆u ) and π| b = βu .
(1) For each rigid vertex u, ∆
0
∆u
b
(2) For each socket vertex v, ∆v is obtained from attaching extra
roots to boundary elements of the socket ∆v . This means that
b v is the colimit of a diagram obtained by restricting the maps ι
∆
and η1 ◦ λ0 to subgroups. The universal property of this colimit
defines the map π|∆b u
i
(3) Consider an abelian vertex [w]. The commutative diagram
above shows that the maps α and βu naturally induce a unique,
non-degenerate map π1 Λ[w] → H. Non-degeneracy, together
with the hypothesis that H is CSA, implies that this map factors through the abelianization.
(4) We may choose a maximal tree for ∆ that maps to a maximal
b Therefore, each stable letter b
b corresponds to
tree for ∆.
t of ∆
a stable letter t in ∆ and we define π(b
t) = α(t).
It is easy to check that π is a homomorphism. It remains to check
that π respects the associated families of distinguished subgroups. But
this is clearly the case, because the homomorphisms α and βv do, and
b are images of distinguished subthe distinguished subgroups of π1 ∆
groups of L0 and MR1 (∆v ).

We can now prove the following theorem by induction on the length
of a strict resolution.
Theorem 4.10. As above, let L0 be a Γ–limit group with strict resolution R0 .
(1) MR0 exists.

THE STRUCTURE OF LIMIT GROUPS

33

(2) The natural map η0 : L0 → MR0 is an injection.
(3) MR0 is a toral relatively hyperbolic group, in particular is finitely
presentable.
Proof. We have already observed in Proposition 3.5 that, when R0 has
length 0 and L0 = Γ, the model for Γ, with respect to any collection
of distinguished subgroups, is itself Γ. Therefore, we may assume that
R0 is of length at least one. We further assume by induction that MR1
exists and satisfies the three properties in the statement of the theorem.
Also, it is clear that the theorem is stable under passing to free
products, so we may assume that L0 is freely indecomposable.
Let ∆ be the canonical primary JSJ of L0 , and recall that ∆ is JSJ–
like. Let λ0 : L0 → L1 be the first map from R0 and η1 : L1 → MR1
be the natural map from L1 to its model. By induction, we know that
b be the expansion
η1 is injective, so ρ = η1 ◦ λ0 is a strict map. Let ∆
of ∆ with respect to ρ.
b is the model of L0 with
It follows from Proposition 4.9 that π1 ∆
respect to R0 . The other two properties in the statement of the theorem
follow immediately from Lemma 4.7 and Proposition 4.8.

5. Further properties of model Γ–limit groups
The purpose of this section is to prove further properties of models,
which are summarized in the next result. In order to state the result, we
adopt the following notation. Let L be a Γ–limit group and R a strict
resolution of L. Let ∆ be the canonical primary JSJ decomposition
of L. Let λ : L ։ L1 be the first map in R. Let MR be the model
of L with respect to R, let µ : MR → MR1 be the associated map
between models and let η : L ֒→ MR be the canonical inclusion. Let
b be the expansion
η1 : L1 ֒→ MR1 be the canonical inclusion and let ∆
b
of ∆ with respect to η1 ◦ λ, so that MR = π1 ∆.

Theorem 5.1. Let L be a freely indecomposable limit group with a
strict resolution R and let MR be the associated model for M.
(1) MR is freely indecomposable.
b is the primary JSJ decomposition for MR .
(2) ∆
(3) The map µ : MR → MR1 is strict.
(4) The model MR is a Γ–limit group.
(5) There is a canonical homomorphism Φ : Mod(L) → Mod(MR )
that intertwines the injection η; that is, for α ∈ Mod(L) and
g ∈ L, we have that η(α(g)) = Φ(α)(η(g)).

The proof of this theorem is spread out over a variety of results in
this section. The reader should note the key implication: the universal

34

DANIEL GROVES AND HENRY WILTON

property of MR allows one to take a factorization of a homomorphisms
f : L → Γ through R and induce a factorization of a homomorphism
fb: MR → Γ through the induced resolution Rb of MR . This is the key
point of the construction of models.
We start with item (1), which is a consequence of a standard result
that characterizes when graphs of groups admit free splittings.
Lemma 5.2. With the above notation, MR is freely indecomposable.
Proof. For a contradiction, we suppose that MR acts on a tree T with
b 1, . . . , ∆
bk
trivial edge stabilizers and without a global fixed point. Let ∆
b
be the components of ∆ created when we delete all non-cyclic abelian
b i acts on T without a global fixed point.
edge groups. For some i, π1 ∆
b i (and hence of ∆)
b whose
By [49, Theorem 18], there is a vertex v of ∆
vertex group admits a non-trivial free splitting relative to the incident
cyclic edge groups. However, non-cyclic abelian groups do not split
freely, and so this splitting is also relative to the incident non-cyclic
b
abelian edges. But this contradicts the definition of ∆.

The next result implies item (2).
Proposition 5.3. If ∆u is a rigid vertex group of ∆ (in the sense of
b u of ∆
b is rigid.
Definition 5.3) then the corresponding vertex group ∆
Proof. Consider an MR –tree T which has abelian edge stabilizers and
is so that every noncyclic abelian subgroup is elliptic. Consider the
induced L–action coming from the canonical embedding of L into MR .
This is an action of the same type, and ∆u is a rigid vertex group ∆,
b u was chosen so that
so ∆u fixes a point in T . However, the group ∆
it did not admit any essential abelian splittings relative to (the images
b u fixes a point of T ,
of) the edge groups of ∆ adjacent to ∆u . Thus ∆
as required.

b is the canonical primary JSJ
Corollary 5.4. The graph of groups ∆
decomposition of MR .
Proof. We have proved that MR is freely indecomposable. Therefore,
b R is a canonical
Proposition 2.16 implies that in order to show that ∆
primary JSJ decomposition we have to verify Conditions (1)–(4) from
that result. Proposition 5.3 shows that the vertices labelled ‘rigid’
are indeed rigid. Suppose that ∆v is a (maximal) socket group of ∆,
b v of ∆.
b Clearly ∆
b v is a socket, and
with corresponding vertex group ∆
we claim that it is a maximal socket. Suppose not. Then there is an
b v of valence 2 which is isomorphic to
abelian vertex group adjacent to ∆

THE STRUCTURE OF LIMIT GROUPS

35

each of its edge groups, and the other nonabelian vertex group adjacent
to this abelian vertex group is also a socket. By construction, it cannot
be a flexible socket, so it must be a rigid socket. However, in this case,
it arose from a vertex group of ∆ which contained the boundary curve
b v , and also powers of the other two boundary curves. This
from ∆
would means that the socket group ∆v was not maximal, contrary to
assumption.
b is bipartite with the required
It is clear from the construction that ∆
structure. Moreover, by construction, any edge group adjacent to a
nonabelian vertex group ∆v is maximal abelian in ∆v . It follows that
b is 2–acylindrical and so, by Proposition 2.16, ∆
b is the canonical
∆
primary JSJ decomposition of MR , as required.

We turn our attention to item (3).
Proposition 5.5. The map µ is strict.
Proof. There are three conditions to check, for the different flavours of
b
vertex groups of ∆.
For socket vertices, we must show that the image under µ is nonabelian, and this is immediate from the construction, because the
b and λ : L → L1
socket vertices of L embed in the socket groups of ∆
is strict.
b [w] is an abelian vertex group of ∆,
b and let
Suppose now that ∆
b [w] ) be the peripheral subgroup. Then P (∆
b [w] ) is a colimit of the
P (∆
b [e ] with maps
peripheral groups P (∆wk ) and the incident edge groups ∆
i
from the edge groups of ∆ to each of these. That µ is injective on
b [w] ) follows from the injectivity of the maps from P (∆w ) and ∆
b [e ] .
P (∆
i
k
So it remains to check that µ is injective on the envelopes of rigid
vertices. We therefore consider a rigid vertex v and its envelope E =
b v ). Note that the Bass–Serre tree Tb of ∆
b induces a splitting of E:
E(∆
the subgroup E acts cocompactly on its minimal invariant subtree S,
b v and
and the quotient graph is a star, with a central vertex labelled ∆
the leaves labelled by the peripheral subgroups of the adjacent abelian
vertices.
For convenience we write M = MR1 and consider the subgroup N =
µ(E). The hierarchy TM,µ(C v ) induces a hierarchy TN,µ(C v ) which, as in
the paragraph after Theorem 2.29, defines a chain of subgroups
Nn ⊆ Nn−1 ⊆ . . . ⊆ N0 = N
and abelian Ni –trees Ti so that Ni+1 is the stabilizer of a unique vertex
b v is an isomorphism
ui of Ti . By definition, the restriction of µ to ∆

36

DANIEL GROVES AND HENRY WILTON

b v → Nn . Let Ei = µ−1 (Ni ) and let Si be the minimal Ei –invariant
∆
subtree of S.
It is not hard to see that the Ni are all freely indecomposable, and
hence that the Ni –trees Ti induced from Grushko trees are trivial. We
therefore remove these redundancies from this list in the obvious way,
and henceforth assume that each Ti is induced from a primary JSJ tree.
We define canonical equivariant maps qi : Si → Ti by sending the
b v to ui , and by sending a vertex w of Si
vertex of Si stabilized by ∆
with abelian stabilizer to the unique vertex of Ti whose stabilizer is the
maximal abelian subgroup of Ni containing µ(StabEi (w)).
We now prove that µ : E → N is injective by induction. It suffices
to show that if µ|Ei : Ei → Ni is injective then qi−1 : Si−1 → Ti−1 does
not factor through a fold. Indeed, the malnormality of the family of
edge groups incident at the central vertex of Si−1 shows that a folded
edge has non-abelian stabilizer, which contradicts the fact that edge
stabilizers in Ti−1 are abelian. This completes the proof.

b The proof of Proposition
Remark 5.6. Consider a rigid vertex v of ∆.
5.5 uses the hierarchy induced on the image N of the envelope E =
b v ) under the map µ, which the proof shows is injective on E. The
E(∆
hierarchy TN,µ(C v ) for N therefore yields a hierarchy TE,Cv for E, which
we will make use of later. As in the proof of the proposition, we may
assume that every level of TE,Cv is induced by a JSJ tree.
Item (4) is an immediate consequence of Theorem 2.22 and Proposition 5.5.
Corollary 5.7. The group MR is a Γ–limit group.
Before proving item (5), we first construct a map Φ : Mod(L) →
Out(MR ). Let α ∈ Mod(L). After suitably conjugating the natural
maps MR (∆v ) → MR , we obtain a commutative diagram, and the
universal property of MR guarantees a unique map α
b : MR → MR .
We set Φ(α) = α
b.

Proposition 5.8. The map Φ : Mod(L) → Out(MR ) is a homomorphism and, for α ∈ Mod(L) and g ∈ L, we have that η(α(g)) =
Φ(α)(η(g)). Furthermore, its image lies in Mod(MR ).
Proof. Uniqueness in the universal property of MR implies in the usual
way that the assignment Φ(α) = α
b is a homomorphism. From the
commutativity of the diagram,
for all g ∈ L, as required.

η(α(g)) = α
b(η(g))

THE STRUCTURE OF LIMIT GROUPS

37

It therefore remains to prove that α
b ∈ Mod(MR ). By Definition
2.17, it suffices to show that if α ∈ Mod(L) is a generalized Dehn twist
then so is α
b ∈ Out(MR ). We need to check the three cases of Definition
2.17. The second is immediate: if α is a Dehn twist in an essential twosided simple closed curve in a socket vertex of ∆, the same curve arises
b and we can also Dehn twist in that.
on a socket vertex of ∆,
We deal with Cases (2) and (3) of Definition 2.17 simultaneously,
but first we need some notation. For a group G with a collection of
subgroups P = {P1 , . . . , Pn }, we write Aut(G, P) for the subgroup of
Aut(G) consisting of those automorphisms α so that, for each i, there
is an inner automorphism ιi ∈ Aut(G) so that α|Pi = ιi . Let Out(G, P)
be the image of Aut(G, P) in Out(G).
b and let [e] be an incident edge.
Let [w] be an abelian vertex of ∆
Then we may assume that α is a generalized Dehn twist either coming
from Case (2) of Definition 2.17 and associated to an edge e of ∆, or
coming from Case (3) of Definition 2.17 and associated to an abelian
vertex w of ∆. Let Λ[w] be the graph of abelian groups that arises in
b [w] , as in §§4.2.
the construction of ∆
We next delete the equivalence class [w], thought of as a set of abelian
vertices of ∆, and let ∆1 , . . . , ∆n be the graphs of groups corresponding
to the resulting connected components. Let P1 = π1 ∆i for each i, and
let P = {P1 , . . . , Pn }, and note that α is contained in Out(L, P).
b with a similar family of subgroups. Let
We next equip MR = π1 ∆
b
b
b after delet∆1 , . . . , ∆m be the connected components obtained from ∆
b j for each j, and let Q = {Q1 , . . . , Qm }.
ing the vertex [w], let Qj = π1 ∆
b [w] is defined to be the abelianization of π1 Λ[w] , there is a
Since ∆
natural map Out(L, P) → Out(MR , Q). But the latter is generated
by Dehn twists associated to edges incident at [w] and by generalized
Dehn twists associated to [w]. In particular, α is indeed contained in
Mod(MR ), as claimed.

6. Relatively immutable subgroups
6.1. Limiting actions on R–trees and the Rips machine. We
recall from [17, 18] that if Υ is a toral relatively hyperbolic group, G is
finitely generated group and {ρi : G → Υ} a sequence of non-conjugate
homomorphisms with stable kernel K = −
Ker
−→ (ρi ) then there is a limit
action of L = G/K on an R–tree T∞ with no global fixed point, abelian
edge stabilizers and trivial tripod stabilizers (see [17, Theorem 6.5]).
Since the L–action on T∞ is superstable, the Rips machine (see [22,

38

DANIEL GROVES AND HENRY WILTON

Main Theorem]) applies and L admits a nontrivial splitting over an
abelian subgroup.
Lemma 6.1. The group L = G/Ker
−−→ (ρi ) described above admits a nontrivial primary splitting.
Proof. This follows from the way that the L–action on T∞ is turned
into a simplicial action of L on a tree. There are essentially three cases
to consider: axial, surface and simplicial. The axial and surface cases
are straightforward and the simplicial case follows quickly from the fact
that tripod stabilizers are trivial. One needs to be slightly careful using
the tree of actions decomposition that [22, Main Theorem] produces,
since the splitting may come from a point which is the intersection
of two vertex trees, so the edge group needn’t fix an arc in the tree.
This splitting may not be primary, but the classification of the vertex
trees in this decomposition makes it straightforward to find a different
nontrivial primary splitting of L.

Below, when we consider relatively immutable subgroups, it will be
useful to consider homomorphisms that act as inner automorphisms on
particular subgroups. To this end, we make the following definition.
Definition 6.2. Suppose that G is a group, and H ≤ G a subgroup.
Suppose further that P is a collection of subgroups of H. Denote by
HomP (H, G) the set of maps λ : H → G so that for each P ∈ P the
restriction of λ to P is the restriction of an inner automorphism of G
to P .
Note that the inner automorphism associated to each P might be
different for different P .
Lemma 6.3. Let Υ be a toral relatively hyperbolic group, H a finitely
generated subgroup and P a finite collection of finitely generated subgroups of H. Suppose that {ρi } is a convergent sequence of nonconjugate homomorphisms from HomP (H, Υ) with trivial stable kernel.
Each element of P fixes a point in the limiting R–tree T∞ .
Moreover, H admits a nontrivial primary splitting relative to P.
Proof. Consider the homomorphisms {ρi : H → Υ} as in the statement
of the lemma, let P ∈ P and suppose p ∈ P . Then ρi (p) is always
conjugate to p in Υ, which implies that there is some xi ∈ Υ (the
conjugating element) so that dΥ (ρi (p).xi , xi ) ≤ K, where K is the word
length of p, and in particular doesn’t depend on i. By [12, Lemma 5.4]
this means that p fixes a point in T∞ . If we take another q ∈ P then
it will fix the same point of T∞ . Applying this argument to a finite
generating set of each P ∈ P shows that there are all elliptic in T∞ .

THE STRUCTURE OF LIMIT GROUPS

39

It now follows from the relative version of the Rips machine in [21]
that H admits a nontrival abelian splitting relative to P, and a nontrivial primary relative splitting can be found in the same way as for
Lemma 6.1 above.

6.2. Relatively immutable subgroups.
Definition 6.4. Let G be a group, H ≤ G a subgroup and P a collection of subgroups of H. The pair (H, P) is relatively immutable in G
if the set HomP (H, G) contains only finitely many conjugacy classes of
injective homomorphisms.
Lemma 6.5. Suppose that Υ is toral relatively hyperbolic, that H ≤ Υ
is finitely generated and that P is a finite nonempty collection of nontrivial finitely generated subgroups of H. Then (H, P) is relatively immutable if and only if H admits no nontrivial primary splitting relative
to P.
Proof. If HomP (H, Υ) contains infinitely many conjugacy classes of injective homomorphisms then Lemma 6.3 implies that H admits a nontrivial primary splitting relative to P.
On the other hand, if H does admit a nontrivial primary splitting
relative to P then there are infinitely many conjugacy classes of automorphisms of H which are contained in HomP (H, H). This can be
seen, for example, by a straightforward adaptation to the relative case
of [12, Lemma 3.34].

The following result is similar to results in [12, § 4,5] and also to
[20, Lemma 7.5]. In [12] it is assumed that the domain is finitely
presented, and in [20] it is assumed that the target is hyperbolic and it
is considering immutable rather than relatively immutable subgroups.
However, the technical details are similar.
Proposition 6.6. Suppose that Υ is a toral relatively hyperbolic group,
that H is a finitely generated subgroup of Υ and that P is a finite,
nonempty collection of nontrivial finitely generated subgroups of H.
Then (H, P) is relatively immutable if and only if there exist finitely
many maps ρ1 , . . . , ρk ∈ HomP (H, Υ) and a positive integer D so that
for any map η ∈ HomP (H, Υ) either
(1) η is conjugate to ρi for some i; or
(2) η is not injective on the ball of radius D about 1 in the Cayley
graph of H (with respect to the chosen finite generating set).
Proof. If (H, P) is not relatively immutable then there are infinitely
many conjugacy classes of injective maps in HomP (H, Υ), so in this

40

DANIEL GROVES AND HENRY WILTON

case there does not exist a D and a finite list as in the statement of
the result.
Conversely, suppose that the statement of the result does not hold.
Then there is a sequence {φi } of non-conjugate maps in HomP (H, Υ) so
that φi is injective on the ball of radius i about 1 in H. We can pass to
a convergent subsequence of {φi } and obtain a faithful action of H on
a limiting R–tree T∞ . By Lemma 6.3 H admits a nontrivial primary
splitting relative to P and so (H, P) is not relatively immutable, by
Lemma 6.5.

6.3. Equations with rational constraints. In this subsection we
recall some results from [10] and [12], following the point of view of
[20]. What we do in this subsection is similar to [20, §6], working
with toral relatively hyperbolic groups instead of torsion-free hyperbolic groups. In particular, see [20, §2] for an introduction to systems
of equations and inequations over groups, and their relationship with
homomorphisms. In this section, we will always consider a solution to
a system of equations as a homomorphism.
Throughout this section, we fix a toral relatively hyperbolic group Υ.
The main theorem in this section is Theorem 6.13 which asserts that
we can algorithmically recognize when a finite system of equations and
inequations over Υ has finitely many conjugacy classes of solutions
(and list conjugacy–representatives of solutions in this case). In fact,
the algorithm is uniform in the sense that it takes a finite presentation
for Υ as input, along with the equations and inequations.
Proving Theorem 6.13 involves understanding certain conjugacy classes
of homomorphisms from a group HΣ (where Σ = 1 are the equations) to
Υ. It is straightforward to check (using [10, Theorem 0.1]) whether or
not every homomorphism from HΣ to Υ has abelian image, and in this
case Theorem 6.13 is straightforward. Therefore, we henceforth deal
with situations where HΣ has nonabelian images in Υ, and thereby can
(effectively) choose a pair of elements a, b ∈ HΣ which do not commute
(this will be witnessed by non-commuting images in Υ).
Definition 6.7. Let H be a group with a fixed generating set X. A
homomorphism ψ : H → Υ is compatible if it is injective on the ball of
radius 8 in H (with respect to X).
Definition 6.8. Let k be a natural number and let BF (X) (k) be the
ball of radius k in the free group F (X). A system of equations and
inequations Σ = 1, Λ 6= 1 forces the ball of radius k if there is a subset
S ⊆ BF (X) (k) such that S ⊆ Σ and BF (X) (k) \ S ⊆ Λ.

THE STRUCTURE OF LIMIT GROUPS

41

Suppose that Σ = 1, Λ 6= 1 is a system of equations and inequations.
If X is finite, there are finitely many subsets of BF (X) (8). For any
S ⊆ BF (X) (8), let ΣS = Σ ∪ S and ΛS = Λ ∩ BF (X) (8) \ S . For any
solution ψ to Σ = 1, Λ 6= 1 there is a unique S so that ψ is a solution
to ΣS = 1, ΛS 6= 1. Conversely, for any S, a solution to ΣS = 1,
ΛS 6= 1 is obviously a solution to Σ = 1, Λ 6= 1. Moreover, for any
S, any solution φ to ΣS = 1, ΛS 6= 1 corresponds to a compatible
homomorphism HΣS → Υ.
Therefore, by replacing Σ = 1, Λ 6= 1 by finitely many systems of
equations and inequations, we may consider only solutions corresponding to compatible homomorphisms.
Fix once and for all a pair a, b ∈ H of noncommuting elements,
which exist if H is nonelementary. Given such a and b, [12, Remark
4.8] defines a condition Ω, and [12, Definition 4.5] (first defined in
[10]) defines the notion of an acceptable lift, which is a choice of words
representing the images of the ball of radius 2 under a homomorphism.
See [12], [10] for more details.
Definition 6.9. A homomorphism ψ : HΣ → Υ is fairly short if it is
compatible and has an acceptable lift ψ̃ : BHΣ (2) → F that satisfies
Ω. A homomorphism ψ : HΣ → Υ is very short if it is compatible and
every acceptable lift ψ̃ : BHΣ (2) → F satisfies Ω.
The following result is implied by [12, Lemma 4.6, Proposition 4.7].
Lemma 6.10. Any conjugacy class of compatible homomorphism HΣ →
Υ contains at least one very short homomorphism and at most finitely
many fairly short homomorphisms.
As explained in [20, §6], the following is now a consequence of [10,
Proposition 1.5].
These notions are useful because of the following proposition, which
is an immediate consequence of a theorem of Dahmani [10]. We refer
the reader to [12, Theorem 3.22] for the statement.
Proposition 6.11. There exists an algorithm that takes as input:
(1) a finite presentation for a toral relatively hyperbolic group Υ
and a pair of non-commuting elements a, b ∈ Υ,
(2) two finite sets of variables x = {xi }, y = {yj },
(3) a finite system (*) consisting of equations Σ(x) = 1 and inequations Λ(x) 6= 1, and equations Θ(x, y) = 1 with coefficients in
Υ
and always terminates with answer ‘Yes’ or ‘No’. In case the answer
is ‘Yes’, there exists a solution to (*) in Υ with an acceptable lift of x

42

DANIEL GROVES AND HENRY WILTON

that satisfies Ω. In case the answer is ‘No’, there is no solution to (*)
in Υ for which every acceptable lift of x satisfies Ω.
A special case of this result (where the variables y and the system Θ
are not required) is the following.
Proposition 6.12. There exists an algorithm that, given a finite presentation for a toral relatively hyperbolic group, a finite system of equations Σ = 1 and a finite system of inequations Λ 6= 1, each with coefficients in Υ, always terminates with answer ‘Yes’ or ‘No’. In case the
answer is ‘Yes’, there exists a fairly short solution to Σ = 1, Λ 6= 1
in Υ. In case the answer is ‘No’, there does not exist any very short
solution.
The following now has exactly the same proof as [20, Theorem 6.9].
We remark that the analogous result there is stated as if the algorithm
depends on which group Υ is being considered. However, as noted in
[20, Remark 1.5] all algorithms in that paper are uniform, in the sense
that the finite presentation of Υ can be taken as input (with a single
algorithm). We state the uniform version below.
Theorem 6.13. There is an algorithm that takes as input a finite
presentation for a toral relatively hyperbolic group Υ, a finite system
of equations and inequations Σ = 1, Λ 6= 1 and terminates if and only
if there are finitely many conjugacy classes of solutions to the system.
In case the algorithm terminates, it outputs a list consisting of exactly
one representative of each conjugacy class of solutions.
Note that the proof from [20] uses the solution to the simultaneous conjugacy problem in Υ. One way to see that the simultaneous
conjugacy problem is solvable in toral relatively hyperbolic groups is
that this is a special case of an equation, which can be solved by [10,
Theorem 0.1].
Question 6.14. In [20], we proved that there exists a ‘complementary’
algorithm to the one above, which terminates if and only if the system
has infinitely many conjugacy classes of solutions. Is there such a complementary result in the setting of toral relatively hyperbolic groups?
6.4. Enumerating relatively immutable subgroups. In this section we adapt the proof of [20, Proposition 7.8] to enumerate relatively
immutable subgroups (in the toral relatively hyperbolic setting). Immutable subgroups H of a hyperbolic group Γ were characterized as
those subgroups for which there is a positive integer D such that the
set Hom(H, Γ) contained only finitely many Γ–conjugacy classes of homomorphisms injective on a ball of radius D. We then recognized this

THE STRUCTURE OF LIMIT GROUPS

43

condition by translating it into a set of equations and inequations over
Γ, and using the rational constraint Ω to restrict conjugacy classes to
finitely many representatives.
Our strategy here is similar. Proposition 6.6 characterizes relatively
immutable subgroups (H, P) of a toral relatively hyperbolic group Υ
as those for which the set HomP (H, Υ) contains only finitely many Υ–
conjugacy classes injective on a ball of radius D (for some D). We
will translate this into a system of equations and inequations over Υ
and again use the rational constraint Ω to restrict conjugacy classes to
finitely many representatives.
Proposition 6.15 (cf. [20], Proposition 7.8). There exists a Turing
machine that takes as input a finite presentation hX | Ri for a toral
relatively hyperbolic group Υ, a finite subset A of Υ, and a tuple of
finite subsets {Q1 , . . . , Qk } so that Qi ⊆ hAi for each i, and terminates
if and only if (hAi, Q) is relatively immutable in Υ, where Q = {hQi i}i .
Proof. Let H = hAi ≤ Υ. We run the following arguments in parallel
for increasing values of a positive integer D. For such a positive integer
b D be the group generated by A with relations equal to all the
D, let H
loops in the ball of radius D in H. These relations can be enumerated
D
using a solution to the word problem in Υ. Let {QD
1 , . . . , Qk } be the
b D corresponding to {Q1 , . . . , Qk }.
tuples of elements of H
b D → Υ which is injective on the ball of radius
A homomorphism H
b D can be characterized by a finite system of equations and
D in H
bD ,
inequations over Υ. The equations correspond to the relations in H
bD .
and the inequations separate the elements of the ball of radius D in H
To be precise, we take variables x corresponding to the elements of A, a
set of elements ΣD (x) in the free group F (x) on x corresponding to the
b D , and a set of elements ΛD (x) ⊆ F (x)
loops in the ball of radius D in H
corresponding to the differences between distinct elements of the ball of
b D . Homomorphisms in Hom(H
b D , Υ) which are injective
radius D in H
on the ball of radius D are then in natural bijection with solutions to
the system
(1)

ΣD (x) = 1 , ΛD (x) 6= 1

over Υ.
b D , Υ),
We now need to restrict attention to elements of HomQD (H
D
D
where QD = {hQ1 i, . . . , hQk i}. To do this, we add one variable yi for
each i, together with extra equations which say that, for each i, the
conjugate of each element of QD
i by yi is equal to the corresponding
element in Qi . To be precise, for each i, let Qi = {qij } ⊆ Υ and for

44

DANIEL GROVES AND HENRY WILTON

b
each j let ηij (x) be the corresponding element of QD
i ⊆ HD (written
as a word in the generators x). For each i and j we add the equation
with coefficients
(2)

ΘD (x, y) := yi ηij (x)yi−1qij−1 = 1

over Υ to the system (1), to obtain a system that is denoted by (3).
We consider the projection map from the set of solutions to the
system (3) to the set of solutions to the system (1) that we obtain by
forgetting the values of the variables yi . When next argue that we can
recognize when the image of this projection consists of finitely many
Υ–conjugacy classes.
As long as D ≥ 8 the system of equations (1) forces the ball of radius
8, and therefore any solution necessarily corresponds to a compatible
homomorphism. We now apply the algorithm from Proposition 6.11 to
the system (3). If it terminates with ‘No’, there is no very short solution
x to the system (1) in the image of the projection map from the solutions to (3). In particular, there are no conjugacy classes in the image
b D , Υ)
of the projection, by Lemma 6.10, and it follows that HomQD (H
contains no conjugacy classes that are injective on the ball of radius
D.
If the algorithm terminates with ‘Yes’, then there is a fairly short
solution to (1) in the image of the projection, and a simpleminded
search will find a solution (x0 , y 0 ) to (3) which projects to a fairly short
solution to (1). We now add further inequations (with coefficients) to
(1), stipulating that x 6= x0 , and repeat the procedure.
b D → H induces an Υ–equivariant
Since the natural quotient map H
b D , Υ), if this algorithm terminates
injection HomQ (H, Υ) ֒→ HomQD (H
then there is a D such that HomQ (H, Υ) contains only finitely many
conjugacy classes that are injective on the ball of radius D, and hence
(H, Q) is relatively immutable.
Conversely, suppose that (H, Q) is relatively immutable. Then for
large enough D the conclusion of Proposition 6.6 is satisfied. Since Υ is
equationally Noetherian, for sufficiently large D every homomorphism
b D to Υ factors through the natural quotient map from H
b D to
from H
H. Therefore, for sufficiently large D we know that the projection of
the solutions of system (3) has only finitely many conjugacy classes. It
follows that the algorithm terminates exactly when (H, Q) is relatively
immutable, as required.

In our enumeration of Γ–limit groups using models in Section 8, we
will require the following elementary observations.

THE STRUCTURE OF LIMIT GROUPS

45

Lemma 6.16. Suppose that Υ is a toral relatively hyperbolic group,
that H ≤ Υ and that P is a collection of nontrivial abelian subgroups
of H. If (H, P) is relatively immutable then so is (H, P ′ ), where P ′ =
{ZH (P ) | P ∈ P}.
Also, if (H, P) is relatively immutable, then so is (H, P0 ), where P0
consists of a collection of H-conjugacy representatives of the elements
of P.
7. Calculation of quasi-convex enclosures
Suppose that Υ is a toral relatively hyperbolic group. In this section,
we explain how to take a finite subset A of Υ, along with finitely many
finite subsets Q1 , . . . , Qk (all given as words in the generators of Υ) and
algorithmically calculate the quasi-convex enclosure of hAi relative to
{hQ1 i, . . . , hQk i}. Let Q = {hQ1 i, . . . , hQk i}.
The main difficulty is resolved by Theorem 7.2, which asserts that
we can compute relative Grushko and JSJ decompositions for toral relatively hyperbolic groups. The absolute version of this theorem follows
immediately from [11, Theorem 1.4] and [12, Theorem D]. However,
the strategy employed in [11] to compute Grushko decompositions (using connectivity of the Bowditch boundary) does not extend easily to
the setting of relatively one-ended groups. Instead, we use the following theorem, which enables us to handle free and abelian splittings
simultaneously.
Theorem 7.1. [49, Theorem 18] Let Υ be finitely generated, and the
fundamental group of a graph of groups with infinite cyclic edge groups.
Then Υ is one-ended if and only if every vertex group is freely indecomposable relative to the incident edge groups.
Theorem 7.1 generalizes a result of Shenitzer, and various similar
statements and proofs have appeared in the literature [47, 14, 16, 34,
48]. The first author learned of Theorem 7.1 from Sela, who intended
to use it to prove Theorem 7.2 in the setting of torsion-free hyperbolic
groups, as part of his unpublished proof of the isomorphism problem
for those groups, and Fujiwara explained a proof to the second author.
Theorem 7.2. There is an algorithm which takes as input a finite
presentation hA | Ri of a toral relatively hyperbolic group Υ and a finite
collection of finite tuples {Q1 , . . . , Qn } of elements of Υ and outputs
a Grushko decomposition for Υ relative to the subgroups Qi = hQi i
and, for each free factor Υ0 which is freely indecomposable relative to
those Qi conjugate into Υ0 , it also outputs the relative primary JSJ
decomposition for Υ0 .

46

DANIEL GROVES AND HENRY WILTON

The proof of Theorem 7.2 follows the strategy of the proof of [12,
Theorem D], with two important modifications. We outline the strategy and explain the modifications.
The algorithm in parallel attempts to find non-trivial primary relative splittings of Υ (using the obvious adaptation of [12, Theorem 5.15]
to the relative setting) and to prove that Υ admits no such splittings
using the algorithm from Proposition 6.15. If a non-trivial splitting is
found then we pass to the vertex groups, add the incident edge groups
to the relative structure, and repeat. By [3], this procedure eventually
terminates in a primary decomposition Λ for Υ in which every vertex
group is immutable relative to those Qi that are conjugate into it and
also to the incident edge groups.
Collapsing the non-trivial edge groups of Λ, one obtains a free splitting ΛG of Υ in which every vertex group is the fundamental group of
a graph of groups with non-trivial abelian edge groups and relatively
immutable vertex groups. Note that ΛG is the relative Grushko decomposition of Υ. Indeed, if some vertex group Υv were to admit a
relative free splitting then, since non-cyclic abelian subgroups would
necessarily be elliptic in such a splitting, it would follow from Theorem 7.1 that every edge group incident at Υv would be elliptic. Hence,
Υv would admit a non-trivial relative free splitting, contradicting the
relative immutability of Υv .
Since we have now found the Grushko decomposition, we restrict
attention to a freely indecomposable vertex group, which we shall for
notational brevity also call Υ. The splitting Λ of Υ that we have found
has the property that every vertex group is relatively immutable; no
vertex group admits a non-trivial relative primary splitting. Such a
decomposition is obtained from a relative primary JSJ of Υ by cutting
each socket vertex along a maximal non-peripheral essential multicurve
in the associated surface. The procedure described after [12, Proposition 6.3] explains how to reassemble the socket pieces into a primary
JSJ decomposition as required.
Finally, we need to compute the canonical primary JSJ decomposition ∆ associated with Λ by computing the associated tree of cylinders.
We explain how to do this in the following result, which completes the
proof of Theorem 7.2.
Proposition 7.3 (cf. [13], Lemma 2.34). There is an algorithm that
takes as input a presentation for a toral relatively hyperbolic group
Υ and a primary graph-of-groups decomposition Λ for Υ and outputs
the decomposition ∆ corresponding to the associated (collapsed) tree of
cylinders.

THE STRUCTURE OF LIMIT GROUPS

47

Proof. Using the word problem in Υ, we can determine which vertex
groups of Λ are abelian. We list the non-abelian vertices x1 , . . . , xk
together with presentations of their associated stabilizers ∆xi = Λxi .
For each i, we also record the images of incident edge maps in Λxi .
We next compute conjugacy representatives of the cylinders of Λ as
follows. For each pair of edges e1 and e2 , we determine whether or not
there exists γ ∈ Υ such that γΛe1 γ −1 commutes with Λe2 , by encoding
this as a system of equations over Υ. If so, we write e1 ∼ e2 . We now
introduce vertices w1 , . . . , wl , one for each equivalence class of edges
[ej ]. For each wj , we choose a representative edge ej and label wj with
the centralizer ∆wj = C(Λej ).
Finally, we need to describe the edges of ∆. For each non-abelian
vertex xi , we partition the centralizers of incident edge groups into ∆xi –
conjugacy classes. Each such class is conjugate into a unique cylinder
corresponding to some wj . These define the edge groups and the attaching maps.

It is now straightforward to compute quasiconvex enclosures: one
iteratively computes relative Grushko and JSJ decompositions, and
Theorem 2.29 ensures that the procedure terminates. Therefore we
have the following result.
Proposition 7.4. There is an algorithm which takes as input a finite
presentation hA | Ri of a toral relatively hyperbolic group Υ and a
finite subset S of Υ (given as a set of words in A± ) and outputs a
finite presentation for the quasi-convex enclosure of hSi in Υ.
8. Enumerating Γ–limit groups
We now turn to the enumeration of Γ–limit groups. First, we need to
describe the data that we will use to specify a Γ–limit group L = L0 .
Our enumeration will be non-unique, and indeed our limit group L0
will come equipped with a strict resolution R0 . The philosophy is to
specify L0 by specifying its model M0 := MR0 , together with certain
nicely specified subsets of M0 that generate L0 . The data will also
include the models of all the limit groups in the resolution R0 , forming
a strict resolution of M0 .
Definition 8.1. A model pair of (strict) resolutions is a diagram of
maps of strict resolutions of the form
L0

λ0

//

L1

η0



M0

λ1

···
//

λn−1

//

Ln

η1
µ0

//



M1

λn

Γ
//

ηn
µ1

//

···

µn−1

//



Mn

id
µn

//



Γ

48

DANIEL GROVES AND HENRY WILTON

where the first line is a strict resolution R0 of L0 followed by a strict
map λn : Ln → Γ, and the second line consists of the canonically associated resolution of models (along with a strict map µn : Mn → Γ).
The length of this model pair is n.
In such a model pair of strict resolutions, the pair of Γ–limit groups
η0 : L0 → M0 is said to be a model pair of Γ–limit groups built over
R1 , where R1 is the resolution of L1 induced by truncating R0 .
By abuse of indices, we refer to the identity map Γ → Γ as a model
pair of (strict) resolutions of length −1. Note that by Proposition 3.5,
when Γ is considered with respect to the trivial resolution, the model
of Γ is Γ.
Definition 8.2. An effective pair of resolutions of length n (specifying a resolution R0 ) consists of a commutative diagram of the form
L0

λ0

//

L1

η0



λ1

//

···

λn−1

//

Ln

η1
µ0



λn

//

Γ

ηn
µ1

µn−1



id
µn



// Mn
// Γ
// M1
// · · ·
M0
where each Mi is specified by a finite presentation hAi | Ri i, each Li
is specified by a finite set Si , each map ηi is specified by a map Si →
F (Ai ), each map µi : Mi → Mi+1 is specified by a map Ai → F (Ai+1),
and each map λi is specified by a map Si → F (Si+1 ).
An effective model pair of resolutions is an effective pair of resolutions
that defines a model pair.

We remark that, on the face of it, there is no effective procedure for
checking whether or not the maps λi and µi that we specify are strict,
or that Mi is the model of Li (with respect to Ri ). However, we will
describe a Turing machine that enumerates effective model pairs for
which these properties do hold.
Theorem 8.3. There is a Turing machine that takes as input a presentation for a torsion-free hyperbolic group Γ and outputs a list of
effective pairs of resolutions over Γ such that:
• every effective resolution on the list defines a model pair of strict
resolutions;
• for every strict resolution R over Γ, every effective pair of resolutions that specifies the model pair canonically associated to
R appears on the list.
Proof. We begin our enumeration with the unique effective model pair
of resolutions of length −1, which is specified by the given presentation

THE STRUCTURE OF LIMIT GROUPS

49

for Γ (considered as the model), the given generating set for Γ (considered as the Γ–limit group) and the identity map on this generating set
(specifying the map from the Γ–limit group to its model).
We now provide a parallel enumeration of all effective model pairs
of resolutions. Suppose that we have an effective pair of resolutions of
length n − 1 (for some n ≥ 0) given by
L1

λ1

//

···

λn−1

//

Ln

η1



λn

//

Γ

ηn
µ1

µn−1



id
µn



// Mn
// Γ
// · · ·
M1
along with data Ai , Ri and maps as in Definition 8.2. (We start the
indices with 1 so that we can append a model pair η0 : L0 → M0 built
over the model pair of resolutions.)
We now begin the enumeration of the model pairs η0 : L0 → M0 built
over the above effective pair. We proceed to enumerate in parallel all
of the possible collections of data as follows:

(1) A finite bipartite graph ∆0 with red and blue vertices, and a
choice of maximal tree in ∆0 .
(2) A decomposition of the blue vertices of ∆0 into two disjoint
families: the ‘socket’ vertices, and the ‘rigid’ vertices. The red
vertices of ∆0 are all ‘abelian’ vertices.
(3) For each edge e of ∆0 , a finite set of words Qe in S1± which
define nontrivial commuting elements of M1 . If an edge e is
adjacent to a socket vertex of ∆0 , the set Qe consists of a single
element qe .
(4) For each rigid vertex u, with adjacent edges e1 , . . . , ek , a finite
set of words Wu in S1± so that
(a) Qe1 , . . . , Qek are all in the subgroup hWu i; and
(b) (hWu i, {hQei i}) is relatively immutable in M1 .
(5) For a socket vertex v, a presentation of the socket of the fundamental group Σv of a surface with boundary, with boundary
components in bijection with certain powers of generators qe of
edge groups for edges adjacent to v. There is also a map from
Σv to M1 with image lying in hS1 i so that the generator of a
boundary component maps to a power of the corresponding qe
(under the bijection between boundary components and edges
adjacent to v) and the image of Σv is nonabelian.
(6) For each edge e of ∆0 we choose elements τe ∈ M1 , and insist
that τe = 1 if e is in the maximal tree of ∆0 .

50

DANIEL GROVES AND HENRY WILTON

(7) For each abelian vertex group w, a finite set of words Ww in
S1± which define nontrivial commuting elements of M1 so that
hWw i contains each of the τe Qe τe−1 for edges e adjacent to w.
Note that each such piece of data can be enumerated in a simple-minded
manner, with the verification that it satisfies the appropriate property
also being verified by the appropriate simple-minded enumeration, or
an application of Proposition 6.15 in the case of relatively immutable
subgroups.
We will define a model Γ–limit group M as the fundamental group
b which will be constructed from the above data
of a graph of groups ∆
as in Section 4. The subgroup L of which M is the model will also
be defined as the fundamental group of a graph of groups ∆. We will
b algorithmically. However, ∆ need not be equal to ∆0 , and
construct ∆
we do not know how to effectively construct ∆. On the other hand, we
will obtain an explicit finite list of generators for L, as required.
b as follows. Suppose that u is a
We now define the graph of groups ∆
b u be
rigid vertex of ∆0 , and consider the subgroup hWu i of M1 . Let ∆
bu
the quasiconvex enclosure of hWu i in M1 . A finite presentation for ∆
can be calculated by Proposition 7.4. If e is an edge in ∆ adjacent to u,
b e to be the centralizer in ∆
b u of hQe i, which can be calculated
we define ∆
b u is toral relatively hyperbolic, it
by Theorem 2.33 (note that since ∆
is CSA, so the centralizer of hQe i is the centralizer of any nontrivial
element of Qe ).
We define an equivalence relation on the edges of ∆0 by saying that
b e1
two edges e1 and e2 , adjacent to a rigid vertex u are equivalent if ∆
b e2 are conjugate in ∆
b u . Since these subgroups are nontrivial
and ∆
b u is toral relatively hyperbolic, we can deterabelian subgroups, and ∆
mine this equivalence using CSA and the algorithm to solve equations
b u (note that both ∆
b e1 and ∆
b e2 are maximal abelian subgroups
over ∆
b u , by construction).
of ∆
b is the quotient graph of ∆0 obtained
As in Section 4, the graph of ∆
b is naturally exactly
by identifying equivalent edges. Each vertex in ∆
one of rigid, socket or abelian. The definition of the graph of groups
b is now exactly as in Section 4. We have already defined the rigid
∆
b are the sockets obtained
vertex groups. The socket vertex groups in ∆
from the socket vertices of ∆0 by attaching maximal roots in M to
generators of boundary components. The edge groups are centralizers
in the same way as in Section 4, and the abelian vertex groups are
abelianizations of the fundamental group of a graph of groups exactly
b
as in Section 4. Then we let M = π1 ∆.

THE STRUCTURE OF LIMIT GROUPS

51

We next specify the finite set S and the map S → M that defines
L. The data listed above naturally defines a graph of groups structure
on ∆0 . Note that we do not have access to presentations for the vertex
and edge groups of ∆0 (indeed, the vertex groups may not be finitely
presentable). However, we may take S to be a natural generating set
for π1 ∆0 corresponding to the disjoint union of:
• the subsets Qe (as e ranges over the edges of ∆0 );
• the Wu (as u ranges over all rigid vertices);
• the generating sets specified by the chosen presentation for Σv
(as v ranges over all socket vertices);
• the Ww (as w ranges over all abelian vertices);
• the choices of stable letters τe for edges e of ∆0 not in the chosen
maximal tree.
The map S → M is now given by the construction of M.
It follows immediately from Proposition 4.9 that M is the model of
L (with respect to the resolution L → L1 → . . . Ln → Γ).
Enumerating these data enumerates all freely indecomposable effective model pairs built over the above effective pair of resolutions. To
enumerate all effective model pairs L0 → M0 , we enumerate finite sets
of such data, together with finitely generated free groups, and take
their free products.
Proceeding in this way, recursively and in parallel, over all effective
pairs of resolutions that we build, we obtain a list of effective pairs of
resolutions over Γ. It is clear that every model pair arises in this list,
and we have shown that every pair of resolutions that we build is a
model pair.

We end by remarking that we do not know how to algorithmically decide if a given (strict) map λn : Ln → Γ is an embedding of the freely
indecomposable free factors of Ln , nor do we know how to algorithmically compute the JSJ decomposition of a given finitely generated
subgroup of Γ. Therefore, terminating with a strict map λn : Ln → Γ
is the best that we can do. This introduces some extra redundancy
in the list of resolutions, as in Remark 2.23, but we do not believe it
affects the utility of the construction.
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Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Offices
(M/C 249), 851 S. Morgan St., Chicago, IL 60607-7045, USA
E-mail address: groves@math.uic.edu
DPMMS, Centre for Mathematical Sciences, Wilberforce Road,
Cambridge, CB3 0WB, UK
E-mail address: h.wilton@maths.cam.ac.uk

