Extending group actions on metric spaces

arXiv:1703.03010v2 [] 26 Nov 2017

C. Abbott, D. Hume, D. Osin

Abstract
We address the following natural extension problem for group actions: Given a group
G, a subgroup H ≤ G, and an action of H on a metric space, when is it possible to extend
it to an action of the whole group G on a (possibly different) metric space? When does
such an extension preserve interesting properties of the original action of H? We begin by
formalizing this problem and present a construction of an induced action which behaves
well when H is hyperbolically embedded in G. Moreover, we show that induced actions can
be used to characterize hyperbolically embedded subgroups. We also obtain some results
for elementary amenable groups.

Contents
1 Introduction

2

2 Preliminaries

6

2.1

Metric spaces and group actions: notation and terminology . . . . . . . . . . .

6

2.2

Hyperbolic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.3

Hyperbolically embedded subgroups . . . . . . . . . . . . . . . . . . . . . . . .

8

2.4

Weights, length functions, and metrics on groups . . . . . . . . . . . . . . . . .

11

3 The extension problem and induced actions

13

3.1

The extension problem for group actions on metric spaces . . . . . . . . . . . .

13

3.2

Equivalence of group actions

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.3

Induced action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

3.4

Incompressible subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4 Proofs of the main results

27

4.1

The extension problem for elementary amenable groups . . . . . . . . . . . . .

27

4.2

Lipschitz retractions to hyperbolically embedded subgroups . . . . . . . . . . .

30

4.3

Extending actions of hyperbolically embedded subgroups . . . . . . . . . . . . .

33

1

1

Introduction

1.1. The extension problem for group actions on metric spaces. All actions of groups
on metric spaces considered in this paper are assumed to be isometric by default. Thus, by a
group action on a metric space we mean a triple (G, S, ϕ), where G is a group, S is a metric
space, and ϕ is a homomorphism from G to Isom(S), the group of isometries of S. If no
confusion is possible, we omit ϕ from the notation and denote the action (G, S, ϕ) by G y S.
The main goal of this paper is to address the following natural question.
Question 1.1 (Extension Problem). Given a group G, a subgroup H ≤ G, and an action
H y R of H on a metric space R, does there exist an action of G on a (possibly different)
metric space that extends H y R?
There are several ways to formalize the notion of an extension. Since our interest in this
question arose in the context of geometric group theory, we accept a “coarse” definition, which
focuses on large scale invariants of groups and spaces.
Let H be a group acting on metric spaces, (R, dR ) and (S, dS ). Recall that a map f : R → S
is said to be
– coarsely H–equivariant if for all x ∈ R, we have
sup dS (f (hx), hf (x)) < ∞;

(1)

h∈H

– a quasi-isometric embedding if there is a constant C such that for all x, y ∈ R we have
1
dR (x, y) − C ≤ dS (f (x), f (y)) ≤ CdR (x, y) + C;
C

(2)

if, in addition, S is contained in the ε–neighborhood of f (R) for some constant ε, f is
called a quasi-isometry.
Definition 1.2. Let H be a subgroup of a group G and let H y R be an action of H on a
metric space R. An action G y S of G on a metric space S is an extension of H y R if there
exists a coarsely H–equivariant quasi-isometric embedding R → S. We say that the extension
problem is solvable for the pair H ≤ G if the answer to Question 1.1 is affirmative for every
action of H on a metric space.
The extension problem is natural enough to be of independent interest and is also motivated
by applications to the study of hyperbolic structures on groups, see [1]. In many cases the
answer is negative, the most striking example being the following.
Example 1.3. Every countable group H embeds in Sym(N), the group of all permutations of
natural numbers. However it is known that every action of Sym(N) on a metric space has
bounded orbits [8]. Thus no action of H with unbounded orbits can be extended to an action
of Sym(N)
On the other hand, there are many examples of pairs H ≤ G for which the answer is
obviously affirmative.
2

Example 1.4. (a) The extension problem for a pair H ≤ G is solvable whenever H is finite.
Indeed, for any action of a finite group H on a metric space R, the trivial action of G on
R is an extension of H y R.
(b) Let H be a retract of G and let ρ : G → H be a homomorphism such that ρ|H ≡ idH . It
is easy to see that for every action H y R, the action of G on the same metric space R
defined by gx = ρ(g)x for all g ∈ G and x ∈ R is an extension of H y R. Indeed the
identity map R → R is an H-equivariant isometry.
If the group G is finitely generated, solvability of the extension problem imposes strong
restrictions on the geometry of the embedding H ≤ G.
Proposition 1.5 (Prop. 3.7). Let G be a finitely generated group. Suppose that the extension
problem is solvable for some H ≤ G. Then H is finitely generated and undistorted in G.
It is worth noting that the extension problem may not be solvable even for finite index
subgroups of finitely generated groups, which are always undistorted.
Example 1.6. Let
G = ha, b, t | t2 = 1, t−1 at = bi ∼
= F (a, b) n Z2
and H = ha, bi ∼
= F (a, b), where F (a, b) is the free group with basis {a, b}. Then the action of
H that factors through the translation action of hai ∼
= Z on R does not extend to an action
of G. Indeed, if such an extension G y S existed, the subgroup hbi (respectively, hai) would
have bounded (respectively, unbounded) orbits in S. However this is impossible since a and b
are conjugate in G.
Proposition 1.5 implies that solvability of the extension problem for all subgroups of a
given group is a rather rare phenomenon. For example, we prove the following in Section 4.1.
Theorem 1.7. Let G be a finitely generated elementary amenable group. Then the extension
problem is solvable for all subgroups of G if and only if there exists a finite index free abelian
subgroup A ≤ G and the action of G on A by conjugation factors through the action of Z/2Z
by inversion.
We then turn to the opposite side of the group theoretic universe and study Question 1.1
for groups with hyperbolic-like properties. Our main result in this direction is the following.
Theorem 1.8 (Cor. 4.10). Let G be a group, H a hyperbolically embedded subgroup of G.
Then the extension problem for H ≤ G is solvable.
The notion of a hyperbolically embedded subgroup was introduced by Dahmani, Guirardel,
and Osin in [9]. For the definition we refer to the next section. Examples include peripheral
subgroups of relatively hyperbolic groups and maximal virtually cyclic subgroups containing
a pseudo-Anosov element (respectively, a fully irreducible automorphism) in mapping class
groups of closed surfaces (respectively, Out(Fn )), etc. For details and more examples see [9].
We mention one application of Theorem 1.8 to hyperbolic groups. Recall that a subgroup
H of a group G is almost malnormal if |H ∩ g −1 Hg| < ∞ for all g ∈ G \ H.
3

Corollary 1.9 (Cor. 4.18). Let G be a hyperbolic group.
(a) Suppose that H is quasiconvex in G and either virtually cyclic or almost malnormal.
Then the extension problem is solvable for H ≤ G.
(b) Conversely, if the extension problem is solvable for a subgroup H ≤ G, then H is quasiconvex.
Notice that the extension problem for the pair F2 ≤ F2 × Z/2Z, where F2 is the free group
of rank 2, is solvable (see Example 1.4 (b)) while F2 is neither virtually cyclic nor almost
malnormal in F2 × Z/2Z. Thus the sufficient condition from part (a) is not necessary for the
extension problem to be solvable. On the other hand, Example 1.6 shows that the necessary
condition from part (b) is not sufficient.
1.2. Induced actions. Our proof of Theorem 1.8 and the “if” part of Theorem 1.7 is based
on the construction of an induced action. We only present it for geodesic metric spaces in this
paper, although a similar theory can be developed in general settings. Restricting to geodesic
spaces makes our exposition less technical and still allows us to answer Question 1.1 in the full
generality since any action of any group on a metric space extends to an action of the same
group on a geodesic metric space (see Proposition 3.10).
Induced actions can be defined in a functorial way if we consider group actions on metric
spaces up to a natural equivalence relation. In order to avoid dealing with proper classes, we
fix some cardinal number c ≥ c, where c is the cardinality of the continuum and, henceforth,
we assume all metric spaces to have cardinality at most c. All results proved in this paper
remain true for every such c. For most applications, it suffices to take c = c.
Definition 1.10. We say that two actions G y R and G y S of a group G on metric spaces
R and S are equivalent (and write G y R ∼ G y S), if there exists a coarsely G–equivariant
quasi-isometry R → S.
It is easy to see that ∼ is indeed an equivalence relation (see Proposition 3.13). The
equivalence class of an action G y R is denoted by [G y R]; we also denote by A(G) the set
of all equivalence classes of actions of G on geodesic metric spaces (of cardinality at most c).
Suppose that a group G is generated by a subset X relative to a subgroup H; that is,
G = hX ∪ Hi. Associated to such a triple (G, H, X) is a natural map
IndX : A(H) → A(G),
called the induced action. Our construction is especially easy to describe in the particular case
when H is generated by a set Y and A ∈ A(H) is the equivalence class of the natural action
of H on its Cayley graph Γ(H, Y ). Then
IndX (A) = [G y Γ(G, X ∪ Y )].
In the general case, the definition is a bit more technical: IndX ([H y R]) is defined by mixing
the left action of G on the coset graph of H (with respect to X) and the given action H y R
4

in a natural way. This involves several additional parameters, but the resulting action is
independent of them up to the equivalence introduced above. If G is finitely generated modulo
H, IndX turns out to be independent of the choice of a finite relative generating set X and is
denoted simply by Ind.
Given H ≤ G, we say that B ∈ A(G) is an extension of A ∈ A(H) if some (equivalently,
any) action G y S ∈ B is an extension of some (equivalently, any) H y R ∈ A. Obviously
IndX (A) cannot always be an extension of A ∈ A(H), as such an extension may not exist at
all. However, our next theorem shows that the induced action is, in a certain sense, the best
thing we can hope for. Here we state our result for relatively finitely generated groups and
refer to Theorem 3.30 for the general case.
Theorem 1.11. Let G be group and let H ≤ G. Assume that G is finitely generated relative
to H. Then the following conditions are equivalent.
(a) The extension problem for the pair H ≤ G is solvable.
(b) For every action A ∈ A(H), Ind(A) is an extension of A.
(c) The subgroup H is incompressible in G.
The notion of an incompressible subgroup is introduced in Section 3.4. One can think
of it as a stronger version of the notion of an undistorted subgroup of a finitely generated
group. Unlike solvability of the extension problem, the property of being incompressible can
be defined for a subgroup H of a group G in intrinsic terms, without mentioning any actions
at all. Incompressibility and a generalization of Theorem 1.11 are instrumental in all proofs of
extendability results in our paper.
Since hyperbolicity of a geodesic space is a quasi-isometry invariant, we can define hyperbolic
elements of A(G) to be equivalence classes of G-actions on hyperbolic spaces. The following
theorem shows that the construction of the induced action behaves well for hyperbolically
embedded subgroups and, moreover, it can be used to characterize hyperbolic embeddings.
Theorem 1.12. Assume that a group G is generated by a set X relative to a subgroup H.
(a) If H is hyperbolically embedded in G with respect to X, then for every A ∈ A(H), IndX (A)
is an extension of A; if, in addition, A is hyperbolic then so is IndX (A).
(b) Suppose that H is countable and for every hyperbolic A ∈ A(H), IndX (A) is a hyperbolic
extension of A. Then H is hyperbolically embedded in G with respect to X.
Remark 1.13. The countability assumption in part (b) cannot be dropped, see Example 4.12.
Also the condition that IndX (A) is a hyperbolic extension of A for every hyperbolic A ∈ A(H)
cannot be replaced with the assumption that every action of H on a hyperbolic space extends
to an action of G on a hyperbolic space, see Example 4.13.
Acknowledgements. The first author was supported by NSF RTG award DMS-1502553.
The second author was supported by the grant ANR-14-CE25-0004 “GAMME”, and the NSF
5

grant DMS-1440140 while the author was in residence at the Mathematical Sciences Research
Institute in Berkeley, California, during the Fall 2016 semester. The third author was supported
by the NSF grant DMS-1308961.

2

Preliminaries

2.1

Metric spaces and group actions: notation and terminology

In this paper we allow distance functions on metric spaces to take infinite values. More precisely, we extend addition and ordering from [0, ∞) to [0, ∞] in the natural way: c + ∞ =
∞ + c = ∞ and c ≤ ∞ for all c ∈ [0, +∞]. Following the standard terminology, by an extended
metric space we mean a set S endowed with a function dS : S × S → [0, +∞] that satisfies the
standard axioms of a metric, where addition and ordering are extended as described above.
The function dS is called an extended metric.
Given an (extended) metric space S, we always denote the (extended) metric on S by dS
unless another notation is introduced explicitly. All graphs (not necessarily connected) are
considered as extended metric spaces with respect to the standard combinatorial metric.
Given a path p in a metric space in a metric space S, we denote by p− and p+ the origin
and the terminus of p, respectively. The length of p is denoted by `(p). A path p in a metric
space S is called (λ, c)–quasi–geodesic for some λ ≥ 1, c ≥ 0 if
`(q) ≤ λdS (q− , q+ ) + c
for any subpath q of p.
For a subset X of a group G, we denote by Γ(G, X) the Cayley graph of G with respect to
X. We do not assume that X generates G here and therefore Cayley graphs are not necessarily
connected. However we always assume that all subsets X ⊆ G used to form Cayley graphs,
as well as all generating sets of G, are symmetric. That is, given x ∈ X, we always assume
that x−1 ∈ X. By dX (respectively, | · |X ) we denote the extended word metric (respectively
length) on G associated to a subset X ⊆ G. That is, |g|X is the usual word length if g ∈ hXi
and ∞ otherwise; the metric is defined by dX (a, b) = |a−1 b|X .
In this paper we understand properness of actions in the metric sense. That is, an action
of a group G on a metric space S is called
– proper if for every bounded subset B ⊆ S the set {g ∈ G | gB ∩ B 6= ∅} is finite;
S
– cobounded if there exists a bounded subset B ⊆ S such that S = g∈G gB;
– geometric if it is proper and cobounded.
We will need the following well-known fact about actions of finitely generated groups. We
provide the proof for convenience of the reader.

6

Lemma 2.1. Let G be a group generated by a finite set X. For every action of G on a metric
space S and every s ∈ S, there exists a constant M such that, for all g ∈ G, we have
dS (s, gs) ≤ M |g|X .

(3)

Proof. Let
M = max dS (s, xs).
x∈X

Suppose that an element g ∈ G decomposes as g = x1 x2 . . . xn for some x1 , x2 , . . . , xn ∈ X and
n = |g|X . Then we have
dS (s, gs) ≤ dS (s, x1 s) + dS (x1 s, x1 x2 s) + · · · + dS (x1 · · · xn−1 s, x1 · · · xn s)
(4)
≤ dS (s, x1 s) + dS (s, x2 s) + · · · + dS (s, xn s) ≤ M n = M |g|X .

2.2

Hyperbolic spaces

In this paper, we say that a geodesic metric space S is δ-hyperbolic for some δ ≥ 0 if for every
geodesic triangle ∆ in S, every side of ∆ belongs to the union of the δ-neighborhood of the
other two sides.
We will use two properties of hyperbolic spaces. The first one is well-known and can be
found, for example, in [5, Theorem 1.7, p.401].
Lemma 2.2. For any δ ≥ 0, λ ≥ 1, c ≥ 0, there exists a constant κ = κ(δ, λ, c) ≥ 0 such that
every two (λ, c)–quasi–geodesics in a δ-hyperbolic space with the same endpoints belong to the
closed κ-neighborhoods of each other.
The next lemma is a combination of a simplified version of Lemma 10 from [14] and the
fact that in a δ-hyperbolic space every side of a geodesic quadrilateral belongs to the closed
2δ-neighborhood of the other 3 sides.
Lemma 2.3. Let S be a subset of the set of sides of a geodesic n–gon P = p1 p2 . . . pn in
a δ–hyperbolic space. Assume that the total lengths of all sides from S is at least 103 cn for
some c ≥ 30δ. Then there exist two distinct sides pi , pj , and subsegments u, v of pi and pj ,
respectively, such that pi ∈ S, min{`(u), `(v)} > c, and u, v belong to 15δ-neighborhoods of
each other.
We will also make use of group actions on combinatorial horoballs introduced by Groves
and Manning [13].
Definition 2.4. Let Γ be any graph. The combinatorial horoball based on Γ, denoted H(Γ),
is the graph formed as follows:
1) The vertex set H(0) (Γ) is Γ(0) × ({0} ∪ N).
7

2) The edge set H(1) (Γ) consists of three types of edges. The first two types are called
horizontal, and the last type is called vertical.
(a) If e is an edge of Γ joining v to w then there is a corresponding edge ē connecting
(v, 0) to (w, 0).
(b) If k ∈ N and 0 < dΓ (v, w) ≤ 2k , then there is a single edge connecting (v, k) to
(w, k).
(c) If k ∈ N and v ∈ Γ(0) , there is an edge joining (v, k − 1) to (v, k).
By dΓ and dH(Γ) we denote the combinatorial metrics on Γ and H(Γ), respectively. The
following results were proved in [13, Theorem 3.8 and Lemma 3.10].
Lemma 2.5.

(a) For every connected graph Γ, H(Γ) is hyperbolic.

(b) For every two vertices a, b ∈ Γ, there exists a geodesic p = p1 qp2 between a and b such
that p1 and p2 entirely consist of vertical edges and q has length at most 3.
Finally will need a well-known homological variant of the isoperimetric characterization of
hyperbolic graphs. Let Σ be a graph. Given a loop c in Σ, we denote its homology class in
H2 (Σ, Z) by [c]. Let `(c) and diam(c) denote the length and the diameter of c, respectively.
Proposition 2.6 (Bowditch, [4].). For any graph Σ the following conditions are equivalent.
(a) Σ is hyperbolic.
(b) There are some positive constants M , L such that if c is a simple loop in Σ, then there
exist loops c1 , . . . , ck in Σ with diam(ci ) ≤ M for all i = 1, . . . , k such that
[c] = [c1 ] + . . . + [ck ]

(5)

and k ≤ L`(c).

2.3

Hyperbolically embedded subgroups

Hyperbolically embedded subgroups were formally introduced by Dahmani, Guirardel, and the
third author in [9] although the idea goes back to the paper [17], where this notion was studied
in the context of relatively hyperbolic groups.
Let G be a group with a fixed collection of subgroups {Hλ }λ∈Λ . Given a (symmetric)
subset X ⊆ G such that G is generated by X together with the union of all Hλ ’s, we denote
by Γ(G, X t H) the Cayley graph of G whose edges are labeled by letters from the alphabet
X t H, where
G
H=
Hλ .
(6)
λ∈Λ

That is, two vertices g, h ∈ G are connected by an edge going from g to h which is labeled by
a ∈ X t H iff a represents the element g −1 h in G.

8

Notice that the unions in the definition above are disjoint. This means, for example, that
for every h ∈ Hλ ∩ Hµ , the alphabet H will have two letters representing the element h in G:
one in Hλ and the other in Hµ . It can also happen that a letter from H and a letter from X
represent the same element of G.
In what follows, we think of the Cayley graphs Γ(Hλ , Hλ ) as naturally embedded complete
subgraphs of Γ(G, X t H).
Definition 2.7. For every λ ∈ Λ, we introduce an extended metric dλ : Hλ × Hλ → [0, +∞]
as follows. Let
∆λ = Γ(G, X t H) \ E(Γ(Hλ , Hλ ))
be the graph obtained from Γ(G, X tH) by excluding all edges (but not vertices) of Γ(Hλ , Hλ ).
Then for h, k ∈ Hλ , dλ (h, k) is the length of a shortest path in ∆λ that connects h to k (we
think of h and k as vertices of Γ(G, X t H) here). If no such a path exists, we set dλ (h, k) = ∞.
Clearly dλ satisfies the triangle inequality.
Definition 2.8. Let G be a group, X a (not necessary finite) symmetric subset of G. We say
that a collection of subgroups {Hλ }λ∈Λ of G is hyperbolically embedded in G with respect to X
(we write {Hλ }λ∈Λ ,→h (G, X)) if the following conditions hold.
(a) The group G is generated by X together with the union of all Hλ and the Cayley graph
Γ(G, X t H) is hyperbolic.
(b) For every λ ∈ Λ, the extended metric space (Hλ , dλ ) is proper, i.e., any ball of finite
radius in Hλ contains finitely many elements.
Further we say that {Hλ }λ∈Λ is hyperbolically embedded in G and write {Hλ }λ∈Λ ,→h G if
{Hλ }λ∈Λ ,→h (G, X) for some X ⊆ G.
For details and examples we refer to [9]. The following proposition relates the notions of a
hyperbolically embedded subgroup and a relatively hyperbolic group. Readers unfamiliar with
relatively hyperbolic groups may think of it as a definition.
Proposition 2.9 ([9, Proposition 4.28]). Let G be a group, {Hλ }λ∈Λ a finite collection of
subgroups of G. Then G is hyperbolic relative to {Hλ }λ∈Λ if and only if {Hλ }λ∈Λ ,→h (G, X)
for some finite subset X ⊆ G.
We will need the following.
Lemma 2.10 ([9, Corollary 4.27]). Let G be a group, {Hλ }λ∈Λ a collection of subgroups of G,
X1 , X2 two relative generating sets of G modulo {Hλ }λ∈Λ such that X1 4 X2 is finite. Then
{Hλ }λ∈Λ ,→h (G, X1 ) if and only if {Hλ }λ∈Λ ,→h (G, X2 ).
In Sections 4.2 and 4.3 we will use terminology and results that first appeared in [16] and
[18] in the context of relatively hyperbolic groups and then were generalized to hyperbolically
embedded subgroups in [9].

9

Definition 2.11. Let q be a path in the Cayley graph Γ(G, X t H). A (non-trivial) subpath
p of q is called an Hλ –subpath, if the label of p is a word in the alphabet Hλ . An Hλ –subpath
p of q is an Hλ –component if p is not contained in a longer Hλ –subpath of q; if q is a loop, we
require in addition that p is not contained in any longer Hλ –subpath of a cyclic shift of q.
Two Hλ –subpaths p1 , p2 of a path q in Γ(G, X t H) are called connected if there exists an
edge c in Γ(G, X t H) that connects some vertex of p1 to some vertex of p2 is labeled by an
element of Hλ . In algebraic terms this means that all vertices of p1 and p2 belong to the same
left coset of Hλ . A component of a path p is called isolated in p if it is not connected to any
other component of p.
It is convenient to extend the extended metric dλ : Hλ × Hλ → [0, ∞] defined above to the
whole group G by assuming

dλ (f −1 g, 1),
if f −1 g ∈ Hλ
dλ (f, g) : =
dλ (f, g) = ∞, otherwise.
Given a path p in the Cayley graph Γ(G, X t H), we let p− , p+ denote the origin and
the terminal point of p, respectively. The following result, which is a simplified version of [9,
Proposition 4.13], will play a crucial role in Section 4.2.
Lemma 2.12. Assume that {Hλ }λ∈Λ ,→h (G, X). Then there exists a constant D such that
for any geodesic n–gon p in Γ(G, X t H), any λ ∈ Λ, and any isolated Hλ –component a of p,
we have dλ (a− , a+ ) ≤ Dn.
We will also use an isoperimetric characterization of hyperbolically embedded subgroups,
which generalizes the corresponding definition of relatively hyperbolic groups. We briefly recall
all necessary definitions and refer to [9] for more details.
Let G, {Hλ }λ∈Λ , H, and X be as above. The group G is a quotient group of the free
product
F = (∗λ∈Λ Hλ ) ∗ F (X),
(7)
where F (X) is the free group with the basis X. Suppose that kernel of the natural homomorphism F → G is a normal closure of a subset R in the group F . For every λ ∈ Λ, we denote
by Sλ the set of all words over the alphabet Hλ that represent the identity in Hλ . Then the
group G has the presentation
hX, H | S ∪ Ri,
(8)
S
where S =
Sλ . In what follows, presentations of this type are called relative presentations
λ∈Λ

of G with respect to X and {Hλ }λ∈Λ .
Definition 2.13. A relative presentation (8) is called bounded if the lengths of words from
the set R are uniformly bounded; if, in addition, for every λ ∈ Λ, the set of letters from Hλ
that appear in words R ∈ R is finite, the presentation is called strongly bounded.
Let ∆ be a van Kampen diagram over (8). As usual, a 2-cell of ∆ is called an R-cell
(respectively, a S-cell) if its boundary is labeled by a word from R (respectively S).
10

Given a word W in the alphabet X t H such that W represents 1 in G, there exists an
expression
k
Y
W =F
fi−1 Ri±1 fi
(9)
i=1

where the equality holds in the group F , Ri ∈ R, and fi ∈ F for i = 1, . . . , k. The smallest
possible number k in a representation of the form (9) is called the relative area of W and is
denoted by Arearel (W ).
Obviously Arearel (W ) can also be defined in terms of van Kampen diagrams. Given a
diagram ∆ over (8), we define its relative area, Arearel (∆), to be the number of R-cells in
∆. Then Arearel (W ) is the minimal relative area of a van Kampen diagram over (8) with
boundary label W .
Definition 2.14. A function f : N → N is a relative isoperimetric function of (8) if for every
n ∈ N and every word W of length at most n in the alphabet X t H representing 1 in G, we
have Arearel (W ) ≤ f (n). Thus, unlike the standard isoperimetric function, the relative one
only counts R-cells.
The following can be found in [9, Theorem 4.24].
Theorem 2.15. Let G be a group, {Hλ }λ∈Λ a collection of subgroups of G, X a relative
generating set of G with respect to {Hλ }λ∈Λ . Then {Hλ }λ∈Λ ,→h (G, X) if and only if there
exists a strongly bounded relative presentation of G with respect to X and {Hλ }λ∈Λ with linear
relative isoperimetric function.

2.4

Weights, length functions, and metrics on groups

Throughout this section, let G denote a group. When dealing with metrics on G, it is often
convenient to restrict to those metrics which take values in non-negative integers. For this
reason, we formally introduce the following.
Definition 2.16 (The set of integral valued left invariant metrics on a group). Let M(G)
denote the set of all left invariant metrics on G taking values in N ∪ {0}. That is, a function
d : G × G → N ∪ {0} belongs to M(G) if and only if it satisfies the following conditions.
(M1 ) d is a metric on G.
(M2 ) For all f, g, h ∈ G, d(f g, f h) = d(g, h).
One way to define a metric on a group is through weight functions.
Definition 2.17 (Weights on groups). A weight function on G is a map w : G → N ∪ {0, ∞}
satisfying the following conditions for all g ∈ G.
(W1 ) w(g) = 0 if and only if g = 1.
(W2 ) w(g) = w(g −1 ).
11

To every weight function w on G one can associate a function dw : G → N ∪ {0, ∞} by
letting
( k
)
X
−1
dw (f, g) = min
w(xi ) x1 , . . . , xk ∈ G, x1 · · · xk = f g
(10)
i=1

for every f, g ∈ G. Here the minimum is taken over all possible decompositions f −1 g = x1 · · · xk
of f −1 g. If the minimum in (10) is attained at a decomposition f −1 g = x1 · · · xk , we call it a
geodesic decomposition of f −1 g.
It is easy to see that dw ∈ M(G) if and only if the set
supp(w) = {g ∈ G | w(g) < ∞}
generates G. In this case we call dw the metric on G associated to the weight function w.
We now discuss the notion of an induced metric used later in this paper. Let {Hλ }λ∈Λ be a
collection of subgroups of G, X a generating set of G relative to {Hλ }λ∈Λ . That is, we assume
that
!+
*
[
Hλ
.
(11)
G= X∪
λ∈Λ

Suppose that we are also given a collection C = {dHλ }λ∈Λ of metrics dHλ ∈ M(Hλ ).
First, for every g ∈ G, we define
Λ(g) = {λ ∈ Λ | g ∈ Hλ }.
Further, let

0,



1,
wC,X (g) =
min{dHλ (1, g) | λ ∈ Λ(g)},



∞,

if g = 1,
if g ∈ X \ {1},
if Λ(g) 6= ∅ and g ∈
/ X,
in all other cases.

Obviously wC,X is a weight function on G. Let dC,X = dwC,X be the metric on G associated
to wC,X .
Definition 2.18. In the notation introduced above, we call dC,X ∈ M(G) the metric on G
induced by the collection C (and corresponding to a relative generating set X).
Remark 2.19. Alternatively, the induced metric can be defined in the following non-constructive
way. Given two metrics d1 , d2 ∈ M(G), we write d1  d2 if d1 (f, g) ≤ d2 (f, g) for all f, g ∈ G.
Obviously  is a partial order on M(G). Then dC,X is the greatest element of the subset
{d ∈ M(G) | d(1, x) = 1 ∀ x ∈ X \ {1} and d|Hλ  dHλ ∀ λ ∈ Λ}.

(12)

Indeed it is clear that dC,X defined above belongs to the subset described by (12). The fact
that it is the greatest element follows immediately from (10) and the triangle inequality.
We conclude with an elementary example.
Example 2.20. Let G be a group and let H be a subgroup of G. Suppose that Y is a generating
set of H and let dY denote the corresponding word length. Let C = {dY }. Let X be a relative
generating set of G with respect to H. Then the corresponding induced metric dC,X on G is
dX∪Y , the word metric on G corresponding to the generating set X ∪ Y .
12

3

The extension problem and induced actions

Throughout the rest of the paper, we deal with collections of subgroups of a given group.
Sometimes we require the collections to be finite while in other cases our arguments go through
without this assumption. To make it easier for the reader to distinguish between these situations we adopt the following convention: we use {H1 , . . . , Hn } to denote a finite collection of
subgroups and {Hλ }λ∈Λ to denote collections which are not necessarily finite.

3.1

The extension problem for group actions on metric spaces

We begin by formalizing the extension problem for group actions on metric spaces in the most
general situation. Let G be a group and let {Hλ }λ∈Λ be a (finite or infinite) collection of
subgroups of G.
Definition 3.1 (Extension of subgroups’ actions). We say that an action G y S of a group
G on a metric space S is an extension of a collection of actions {Hλ y Rλ }λ∈Λ of subgroups
Hλ ≤ G on metric spaces Rλ , λ ∈ Λ, if for every λ ∈ Λ, there exists a coarsely Hλ -equivariant
quasi-isometric embedding Rλ → S.
Problem 3.2 (Extension problem). Given a group G, a collection of subgroups {Hλ }λ∈Λ of
G, and a collection of actions {Hλ y Rλ }λ∈Λ as above, does there exist an extension of
{Hλ y Rλ }λ∈Λ to a G-action on a metric space?
Definition 3.3. Given a group G and a collection of subgroups {Hλ }λ∈Λ of G, we say that the
extension problem is solvable for {Hλ }λ∈Λ and G if the answer to Problem 3.2 is affirmative
for every collections of actions {Hλ y Rλ }λ∈Λ .
Remark 3.4. If the extension problem is solvable for a collection of subgroups {Hλ }λ∈Λ of a
group G, then it is obviously solvable for every Hλ ≤ G. The converse is false. Indeed there is
an obvious obstacle: for any two subgroups H1 , H2 from the given collection, the given actions
of H1 ∩ H2 on the corresponding spaces R1 and R2 must be equivalent for the extension to
exist.
As we already mentioned in the introduction, the extension problem may not be solvable
even for a single subgroup. Let us discuss some further examples.
Recall that a group G is called strongly bounded if every isometric action of G on a metric
space has bounded orbits. It is easy to show that a countable group is strongly bounded if
and only if it is finite. However, there are plenty of examples of uncountable strongly bounded
groups: Sym(N), ω1 –existentially closed groups, and (unrestricted) infinite powers of finite
perfect groups. For details on these we refer to [8].
Example 3.5. It is clear that whenever G is strongly bounded and H ≤ G, no action of H with
unbounded orbits extends to a G–action.
We now turn to finitely generated groups. Recall that a finitely generated subgroup H of
a finitely generated group G is undistorted if the inclusion H → G induces a quasi-isometric
embedding (H, dY ) → (G, dX ), where dX and dY are word metrics associated to some (equivalently, any) finite generating sets X and Y of G and H, respectively. This is obviously
13

equivalent to the requirement that there exists a constant C such that |h|Y ≤ C|h|X for all
h ∈ H.
Lemma 3.6. Let G be a group generated by a finite set X, H ≤ G, and let d be a left invariant
metric on H. Suppose that the left action of H on the metric space (H, d) extends to an action
of G. Then there exists a constant K such that
d(1, h) ≤ K|h|X
for all h ∈ H.
Proof. Let G y S be an extension of H y (H, d) and let f : H → S be the corresponding
coarsely H-equivariant quasi-isometric embedding. Since f is a quasi-isometric embedding,
there is a constant C such that for every h ∈ H, we have
d(1, h) ≤ CdS (f (1), f (h)) + C
≤ C(dS (f (1), hf (1)) + dS (hf (1), f (h)) + C

(13)

≤ CdS (f (1), hf (1)) + CD + C,
where D = suph∈H dS (hf (1), f (h)) < ∞ by coarse H-equivariance of f . Further by Lemma
2.1 applied to s = f (1) we have
dS (f (1), hf (1)) ≤ M |h|X
for some constant M independent of h. Combining this with (13) we obtain the required
inequality.
Proposition 3.7. For any finitely generated group G and any H ≤ G, the following hold.
(a) If the extension problem is solvable for H ≤ G, then H is finitely generated.
(b) Suppose that H is finitely generated. Then some (equivalently, any) geometric H–action
on a geodesic metric space extends to a G–action if and only if H is undistorted in G.
Proof. (a) Let X denote a finite generating set of G and let dX be the corresponding metric
on G. Arguing by contradiction, assume that H is not finitely generated. Then we can find
an infinite generating set Y = {yi | i ∈ N} of H with the property that
yi+1 ∈
/ hy1 , . . . , yi i

(14)

for all i ∈ N. We choose any increasing sequence (Mi ) ⊆ N such that
lim Mi /|yi |X = ∞

i→∞

and define a weight function on G by the rule

Mi , if g = yi for some i ∈ N,
w(g) =
∞, otherwise.
14

(15)

Let dw be the associated metric on H. By our assumption, the left action of H on itself
endowed with the metric dw extends to an action of G. Therefore, by Lemma 3.6 we have
dw (1, h) ≤ K|h|X for some constant K independent of h. In particular, we have dw (1, yi ) ≤
K|yi |X = o(Mi ) as i → ∞ by (15). On the other hand, we have dw (1, yi ) ≥ Mi for all i; indeed
every decomposition of yi into a product of elements of Y must contain an element of weight
at least Mi by (14). A contradiction.
(b) Let X (respectively, Z) be a finite generating set of G (respectively, H). If H is undistorted in G, it is straightforward to verify that G y (G, dX ) is an extension of H y (H, dZ ).
By the Svarc-Milnor Lemma (see Proposition 8.19 in Chapter I.8 of [5]), every geometric Haction H y R on a geodesic space R is equivalent to H y (H, dZ ); therefore, G y (G, dX ) is
an extension of H y R as well. Conversely, suppose that H y (H, dZ ) extends to a G–action.
By Lemma 3.6, there exists a constant K such that |h|Z ≤ K|h|X for all h ∈ H. Thus H is
undistorted in G.
Example 3.8. Let
G = ha, b | b−1 ab = a2 i.
Then the extension problem for H = hai and G is unsolvable. Moreover, the natural action of
H∼
= Z on R does not extend to an action of G. Indeed it is well-known and easy to see that
H is (exponentially) distorted in G.
Recall that solvability of the extension problem does not pass to finite index subgroups
(see Example 1.6). However we have the following result, which will be used in the proof of
Corollary 1.9.
Lemma 3.9. Let G be a group, H a subgroup of G, K a finite index subgroup of H. Suppose
that the extension problem is solvable for K ≤ G. Then it is solvable for H ≤ G.
Proof. Let H y R = (H, R, α) be an action of H on a metric space R and let K y R =
(K, R, α|K ). Let G y S denote an extension of K y R and let f : R → S be a coarsely
K–equivariant quasi-isometric embedding. Fix some r ∈ R. Let T be a finite subset of H such
that H = KT . Further let
M = max dS (f (tr), tf (r))
t∈T

and
C = max sup dS (f (ktr), kf (tr)).
t∈T k∈K

Since f is coarsely K–equivariant, C is finite. For every h ∈ H, we have h = kt for some t ∈ T
and k ∈ K. Therefore,
dS (f (hr), hf (r)) ≤ dS (f (ktr), kf (tr)) + dS (kf (tr), ktf (r)) ≤ C + dS (f (tr), tf (r)) ≤ C + M.
Thus f is also coarsely H–equivariant and hence G y S is an extension of H y R.
We conclude this section with an elementary observation which allows us to reduce the
extension problem to the particular case of group actions on geodesic metric space.

15

Proposition 3.10. Let G be a group acting on a metric space R. Then there exists an
extension G y S of G y R such that S is geodesic.
Proof. Let S be the complete graph with the vertex set V (S) = R. For every x, y ∈ R such
that x 6= y, we identify the edge connecting x and y with a segment of length dR (x, y). This
induces a metric on S in the obvious way. The action of G on V (S) = R extends (in the usual
sense) to an isometric action on S and it is straightforward to check that the identity map
R → R = V (S) induces a G-equivariant isometric embedding R → S.

3.2

Equivalence of group actions

We begin by showing that the relation ∼ introduced in Definition 1.10 is indeed an equivalence
relation. This is fairly elementary and straightforward to prove. Nevertheless, we decided to
provide a complete proof since this concept is central to our paper.
Definition 3.11. Let X and Y be two metric spaces and let α : X → Y be a map. We say
that a map β : Y → X is a (right) coarse inverse of α if
dY (α ◦ β(y), y) ≤ ε

(16)

for all y ∈ Y .
Lemma 3.12. Let G y X and G y Y be two actions of a group G on metric spaces. Let
α : X → Y be a coarsely G–equivariant quasi-isometry. Then there exists a coarse inverse
β : Y → X of α and every coarse inverse of α is a coarsely G–equivariant quasi-isometry.
Proof. Since α is a quasi-isometry, Y coincides with the ε–neighborhood of α(X) for some
constant ε. Therefore, to every y ∈ Y we can associate a point β(y) ∈ X such that (16) holds.
Thus we get a map β : Y → X, which is a coarse inverse of α.
Let now C denote the quasi-isometry constant of α (as defined in the Introduction) and let
β : Y → X be any map satisfying (16). Given y ∈ Y , let
D = sup dY (α(gβ(y)), gα ◦ β(y)).
g∈G

Then for every g ∈ G, we have
dX (β(gy), gβ(y)) ≤ C(dY (α ◦ β(gy), α(gβ(y))) + C) ≤
C(dY (α ◦ β(gy), gα ◦ β(y)) + C + D) ≤
C(dY (α ◦ β(gy), gy) + dY (gy, g(α ◦ β(y))) + C + D) ≤ C(2ε + C + D).
Thus β is coarsely G–equivariant.
Further, for every y1 , y2 ∈ Y , we have
dX (β(y1 ), β(y2 )) ≤ C (dY (α ◦ β(y1 ), α ◦ β(y2 )) + C) ≤ C (dY (y1 , y2 ) + 2ε + C) .
16

Similarly, we obtain
1
1
(dY (α ◦ β(y1 ), α ◦ β(y2 )) − C) ≥ (dY (y1 , y2 ) − 2ε − C) .
C
C
Finally, we note that X belongs to the closed C(ε + C)–neighborhood of β(Y ). Indeed for
every x ∈ X, we have
dX (β(y1 ), β(y2 )) ≥

dX (x, β ◦ α(x)) ≤ C (dY (α(x), α ◦ β ◦ α(x)) + C) ≤ C(ε + C)
by (16). Thus β is a quasi-isometry.
Proposition 3.13. The relation ∼ introduced in Definition 1.10 is an equivalence relation.
Proof. Reflexivity and transitivity are obvious. That ∼ is symmetric follows from Lemma
3.12.
Recall that given a connected graph Γ, we can think of it as a metric space with respect to
the standard combinatorial metric (that is, the metric obtained by identifying every edge of Γ
with [0, 1]). The fact stated below will be used in the next section.
Proposition 3.14. For any group G, any action of G on a geodesic metric space is equivalent
to an action of G on a graph (endowed with the combinatorial metric) with trivial vertex
stabilizers.
Proof. Suppose that G acts on a geodesic metric space S. Let Γ be the graph with vertex set
V (Γ) = G × S and the set of edges consisting of all pairs {(g1 , s1 ), (g2 , s2 )} ⊆ G × S such that
dS (s1 , s2 ) ≤ 1. The given action of G on S and the left action of G on itself extend (in the
usual sense) to an action on V (Γ), which in turn can be extended to a G–action on Γ since we
define edges in a G-equivariant way. It is straightforward to verify that the action of G on V (Γ)
is free and the map s → (1, s) induces a coarsely G-equivariant quasi-isometry S → Γ.
Remark 3.15. Note however that the action of G on Γ constructed in Proposition 3.14 may
not be free as edges may have non-trivial (setwise) stabilizers generated by involutions. We
could make the action of G free by doubling these edges, but having free action on the vertex
set is sufficient for our goals.
Definition 3.16. Given a group G, we denote by A(G) and a collection of all equivalence
classes of actions of G on geodesic spaces (of cardinality at most c).
Definition 3.17. Given a group G, a collection of subgroups {Hλ }λ∈Λ , and A ∈ A(G),
(Bλ )λ∈Λ ∈ ×λ∈Λ A(Hλ ) we say that A is an extension of B if every action A ∈ A is an
extension of every action Bλ ∈ Bλ for all λ ∈ Λ.
The following proposition allows us to replace ‘every’ with ‘some’ in the definition above.
Proposition 3.18. Let G be a group and {Hλ }λ∈Λ a collection of subgroups of G. Let A ∈
A(G) and (Bλ )λ∈Λ ∈ ×λ∈Λ A(Hλ ). Then A is an extension of B if and only if for all λ ∈ Λ,
some action A ∈ A is an extension of some action Bλ ∈ Bλ .
Proof. This follows immediately from the fact that a composition of a quasi-isometry and a
quasi-isometric embedding (in any order) is again a quasi-isometric embedding.
17

3.3

Induced action

We are now ready to introduce the concept of an induced action. Throughout this section, let
G be a group, {H1 , . . . , Hn } a finite collection of subgroups of G, and let X be a generating
set of G relative to {H1 , . . . , Hn }. Proposition 3.10 and Proposition 3.14 imply that in order
to solve the extension problem, it suffices to deal with actions on connected graphs whose
restrictions to vertex sets are free. Henceforth, we fix a collection of actions
A = {H1 y R1 , . . . , Hn y Rn }
on graphs R1 , . . . , Rn such that the restrictions of these actions to the vertex sets V (R1 ), . . . ,
V (Rn ) are free.
We denote by Γ(G, X) the Cayley graph of G with respect to X. Notice that Γ(G, X) is
not necessarily connected since X may not generate G by itself.
Roughly speaking, the induced action of G associated to these data is the natural action
of G on the space obtained from Γ(G, X) by gluing copies of Ri to all cosets gHi along a
fixed Hi –orbit in Ri . The construction will depend on the choice of coset representatives of Hi
and particular orbits in each Ri . However, all induced actions constructed in this way will be
equivalent; finiteness of the collection of subgroups will be essentially used in proving this.
To define the induced action formally, we fix a collection of base vertices
B = {b1 , . . . , bn },
where bi ∈ V (Ri ). For each i we also fix a collection Ti of representatives of left cosets of Hi
in G and denote by ti : G → Ti the map assigning to an element g ∈ G, the representative of
the coset gHi . Without loss of generality, we can (and will) assume that
ti (h) = 1 ∀ h ∈ Hi .

(17)

Let
T = {t1 , . . . , tn }.
We call T the transversal of G with respect to {H1 , . . . , Hn }.
For each i = 1, . . . , n, let
Yi = G/Hi × Ri .

(18)

We think of Yi as a graph, which is a disjoint union of copies {gHi } × Ri of Ri , for all
gHi ∈ G/Hi . We endow every Yi with the combinatorial metric (which may take infinite
values as Yi is not connected unless Hi = G), so every Yi becomes an extended metric space.
We first want to define an action of G on each Yi such that:
(A1 ) It extends the action of Hi on Ri (Ri is identified with {Hi } × Ri ) in the set theoretic
sense, i.e., h(Hi , r) = (Hi , hr) for all h ∈ Hi and r ∈ Ri .
(A2 ) G permutes subsets {aHi } × Ri according to the rule g({aHi } × Ri ) = {gaHi } × Ri for
all g, a ∈ G.
18

It is fairly easy to see that there is a unique way to define such an action of G. For every
i ∈ {1, . . . , n}, r ∈ Ri , and g, a ∈ G, we let
g(aHi , r) = (gaHi , αi (g, a)r)

(19)

αi (g, a) = (ti (ga))−1 gti (a).

(20)

where
Note that αi (g, a)r ∈ Ri is well-defined since αi (g, a) ∈ Hi .
Lemma 3.19. Formulas (19) and (20) define an isometric action of G on each Yi satisfying
conditions (A1 ) and (A2 ). The restriction of this action to each vertex set V (Yi ) is free.
Proof. Throughout the proof we fix some i ∈ {1, . . . , n}. First let us check that the identity
element acts trivially. We obviously have αi (1, a) = (ti (a))−1 ti (a) = 1 and hence 1(aHi , r) =
(aHi , αi (1, a)r) = (aHi , r). Further, using (20) we obtain
αi (f g, a) = (ti (f ga))−1 f gti (a) = (ti (f ga))−1 f ti (ga)((ti (ga))−1 gti (a)) =
(21)
αi (f, ga)αi (g, a)
for all f, g, a ∈ G. Therefore,
(f g)(aHi , r) = (f gaHi , αi (f g, a)r) = (f gaHi , αi (f, ga)αi (g, a)r) =
f (gaHi , αi (g, a)r) = f (g(aHi , r)).
Thus formulas (19) and (20) indeed define an action of G.
That the restriction of the action to V (Yi ) is free easily follows from our assumption that
the action Hi y V (Ri ) is free. Indeed assume that we have
g(aHi , r) = (aHi , r)

(22)

for some (aHi , r) ∈ V (Yi ). Without loss of generality we can assume that a ∈ Ti , i.e., ti (a) = a.
Comparing (22) to (19), we obtain ga ∈ aHi and hence ti (ga) = a. Together with (20), this
implies αi (g, a) = a−1 ga. Again combining (22) and (19), we obtain a−1 gar = r. Since the
action of G on V (Ri ) is free, we have a−1 ga = 1 and hence g = 1.
If h ∈ Hi , we obtain αi (h, 1) = (ti (h))−1 hti (1) = h using (17). This implies (A1 ). Condition
(A2 ) follows from (19) immediately.
It remains to show that G y Yi is isometric. Let y1 = (a1 Hi , r1 ) and y2 = (a2 Hi , r2 ) be
two points of Yi . First assume that a1 Hi 6= a2 Hi . Then dYi (y1 , y2 ) = ∞. By (19) we also have
dYi (gy1 , gy2 ) = ∞ as ga1 Hi 6= ga2 Hi in this case. Further if a1 = a2 = a, then
dYi (gy1 , gy2 ) = dRi (αi (g, a)r1 , αi (g, a)r2 ) = dRi (r1 , r2 ) = dYi (y1 , y2 )
by (19) and the definition of the extended metric on Yi .

19

Definition 3.20 (The induced action). Let SX,T ,B,A denote the graph obtained by gluing Yi ,
i = 1, . . . , n, to Γ(G, X) by identifying vertices g(Hi , bi ) ∈ V (Yi ) and g ∈ V (Γ(G, X)) for all
g ∈ G and i = 1, . . . , n. We call the graph SX,T ,B,A the space of the induced action.
Since SX,T ,B,A is obtained by gluing, vertices of SX,T ,B,A are, formally speaking, equivalence
classes. By abuse of notation, we will use representatives of these equivalence classes to denote
vertices of SX,T ,B,A . Thus we think of a vertex of SX,T ,B,A is a pair (gHi , v), where g ∈ G
and v ∈ V (Ri ) for some i. If v does not belong to the Hi -orbit of bi , then the equivalence
class of (gHi , v) consists of a single pair. Otherwise the corresponding vertex has exactly n
representatives. Indeed, in the graph SX,T ,B,A , we have
(gHi , ti (g)−1 gbi ) = g(Hi , bi ) = g(Hj , bj ) = (gHj , t−1
j (g)gbj )

(23)

for all i, j ∈ {1, . . . , n}.
Since the gluing maps used to construct SX,T ,B,A are G-equivariant, the actions of G on
graphs Yi and Γ(G, X) induce an action of G on SX,T ,B,A , denoted by IndX,T ,B (A).
Since the actions Hi y V (Yi ) are free by Lemma 3.19, the natural (G-equivariant) maps
Yi → SX,T ,B,A and Γ(G, X) → SX,T ,B,A are injective. Henceforth, we will think of Γ(G, X)
and Yi , i = 1, . . . , n, as subgraphs of SX,T ,B,A .
Lemma 3.21. The graph SX,T ,B,A is connected.
Proof. By construction, every vertex (gHi , v) in SX,T ,B,A can be connected to the vertex g =
g(Hi , bi ) of Γ(G, X) ⊆ SX,T ,B,A inside {gHi } × Ri ⊆ Yi . Thus it suffices to show that every
two vertices of Γ(G, X) can be connected by a path in SX,T ,B,A
S
Let y = y(Hi , bi ) and z = z(Hi , bi ) be any two vertices of Γ(G, X). Since X ∪ ( ni=1 Hi )
generates G, we have
z = yx1 h1 x2 h2 . . . xm hm ,
where xj ∈ X ∪ {1} and hj ∈ Hi(j) for each j. By construction, any two vertices of the
form g = g(Hi , bi ) and gxj = gxj (Hi , bi ) are connected by an edge of Γ(G, X), while any two
vertices of the form g = g(Hi , bi ) = g(Hi(j) , bi(j) ) and ghj = ghj (Hi , bi ) = ghj (Hi(j) , bi(j) ) are
connected by a path in the subgraph {gHi(j) } × Ri(j) of SX,T ,B,A . Thus, there is a path in
SX,T ,B,A connecting y to z.
We are now ready to prove the main result of this section.
Proposition 3.22. Let G be a group and let {H1 , . . . , Hn } be a collection of subgroups of G.
Let X, X 0 be generating sets of G relative to {H1 , . . . , Hn } such that |X 4 X 0 | < ∞. Let
T = {t1 , . . . , tn } and T 0 = {t01 , . . . , t0n }
be transversals of G with respect to {H1 , . . . , Hn }. Let
A = {H1 y R1 , . . . , Hn y Rn } and A0 = {H1 y R10 , . . . , Hn y Rn0 }

20

be collections of actions on graphs Ri and Ri0 such that the restrictions of these actions to
vertex sets are free and Hi y Ri ∼ Hi y Ri0 for all i. Finally let
B = {b1 , . . . , bn } and B 0 = {b01 , . . . , b0n }
be collections of base vertices in graphs Ri and Ri0 , respectively. Then
IndX,T ,B (A) ∼ IndX 0 ,T 0 ,B0 (A0 ).
Proof. We fix coarsely Hi -equivariant quasi-isometries ρi : Ri → Ri0 . Since changing ρi by
a bounded function does not violate the property of being a coarsely Hi -equivariant quasiisometry, we can assume that ρi maps V (Ri ) to V (Ri0 ) and
ρi (bi ) = b0i

(24)

without loss of generality. We also fix a constant C such that the following inequalities hold
for all i = 1, . . . , n:
(25)
sup dRi0 (ρi (hbi ), hρi (bi )) ≤ C,
h∈Hi

and
dRi0 (ρi (x), ρi (y)) ≤ CdRi (x, y) + C.

(26)

To prove the proposition, we will construct coarsely G–equivariant quasi-isometries between
the vertex sets
ϕ1

ϕ2

ϕ3

V1 = V (SX,T ,B,A ) −→ V2 = V (SX,T 0 ,B,A ) −→ V3 = V (SX,T 0 ,B0 ,A0 ) −→ V4 = V (SX 0 ,T 0 ,B0 ,A0 ).
We assume that every Vi is equipped by the metric induced from the corresponding graph; this
metric is denoted by di .
To change the transversals we use the map defined by
ϕ1 (aHi , v) = (aHi , t0i (a)−1 ti (a)v)
for all a ∈ G, i = 1, . . . , n, and v ∈ V (Ri ). Notice that this map is well-defined. Indeed if a
point of SX,T ,B,A has more than one representative of type (aHi , v), then these representatives
must be of the form (23) and we have
ϕ1 (gHi , ti (g)−1 gbi ) = (gHi , t0i (g)−1 gbi ) = (gHj , t0j (g)−1 gbj ) = ϕ1 (gHj , tj (g)−1 gbj )
for all i, j ∈ {1, . . . , n}.
To change the actions and basepoints, we define

(aHi , αi0 (a, 1)b0i ), if v = αi0 (a, 1)bi ,
ϕ2 (aHi , v) =
(aHi , ρi (v)),
otherwise,
where αi0 is defined in the same way as αi using T 0 instead of T ; i.e., αi0 (a, b) = t0i (ab)−1 at0i (b).
Finally, to change relative generating sets we use the map defined by
ϕ3 (aHi , v) = (aHi , v).
As above, it is straightforward to verify that ϕ2 and ϕ3 are well-defined.
The proof of the proposition is based on two lemmas.
21

Lemma 3.23. The maps ϕ1 , ϕ2 , and ϕ3 are coarsely G-equivariant.
Proof. Indeed, ϕ1 is G-equivariant:
g(ϕ1 (aHi , v)) = g(aHi , t0i (a)−1 ti (a)v) = (gaHi , αi0 (g, a)t0i (a)−1 ti (a)v)
= (gaHi , t0i (ga)−1 gti (a)v) = ϕT (g(aHi , v)).
We next check ϕ2 . Fix some g ∈ G. First suppose that x = a(Hi , bi ) for some a ∈ G and
i ∈ {1, . . . , n}; then using (21) we obtain
ϕ2 (gx) = ϕ2 (gaHi , α0 (ga, 1)bi ) = (gaHi , α0 (ga, 1)b0i ) = g(aHi , α0 (a, 1)b0i ).
In particular, ϕ2 is equivariant on vertices of Γ(G, X). If x ∈
/ V (Γ(G, X)), then we have
0
0
gϕ2 (x) = (gaHi , αi (g, a)ρi (v)) and ϕ2 (gx) = (gaHi , ρi (αi (g, a)(v))). Hence
sup d3 (ϕ2 (gx), gϕ2 (x)) ≤ sup dRi (αi0 (g, a)ρi (r), ρi (αi0 (g, a)v))) < ∞

g∈G

g∈G

since the maps ρi are coarsely Hi -equivariant.
Finally, for ϕ3 there is nothing to prove, its G-equivariance is obvious.
Lemma 3.24. The maps ϕ1 , ϕ2 , and ϕ3 are Lipschitz.
Proof. In each case it suffices to verify that there exists a constant K such that if two vertices
x, y ∈ Vj span an edge in the corresponding graph, then
dj+1 (ϕj (x), ϕj (y)) ≤ K

(27)

for j = 1, 2, 3.
We consider two cases.
Case 1. Assume that x and y are connected by an edge of Γ(G, X) ⊆ Sj . Then we have
x = f (Hi , bi ), y = g(Hi , bi ), and f −1 g ∈ X. It is easy to see that K = 1 works for i = 1, 2 by
equivariance and for i = 3 we can take
K = sup d4 (f (Hi , bi ), f x(Hi , bi )) = sup d4 ((Hi , bi ), x(Hi , bi )) < ∞
x∈X

x∈X

as d4 ((Hi , bi ), x(Hi , bi )) ≤ 1 for all x ∈ X 0 and |X 4 X 0 | < ∞.
Case 2. Next we assume that x and y are connected by an edge of Yi for some i. Then
x = (aHi , u), y = (aHi , v) and u, v span an edge of Ri . In this case it is straightforward
to see that K = 1 works for i = 1, 3. Let us now consider the case i = 2. If none of x, y
is of the form (aHi , αi0 (a, 1)bi ), then we can take K = 2C by (26). Further, we can assume
that the first line in the definition of ϕ2 applies to at most one of x, y (otherwise Case 1
applies). Thus it suffices to consider the case u = αi0 (a, 1)bi , ϕ2 (aHi , u) = (aHi , αi0 (a, 1)b0i ) and
ϕ2 (aHi , v) = (aHi , ρi (v)). Combining (24), (25), and (26) we obtain
d3 (ϕ2 (x), ϕ2 (y)) ≤ dRi0 (αi0 (a, 1)b0i , ρi (v)) = dRi0 (αi0 (a, 1)ρi (bi ), ρi (v))
≤ dRi0 (αi0 (a, 1)ρi (bi ), ρi (u)) + dRi0 (ρi (u), ρi (v)) ≤ 3C.

22

Let us now return to the proof of Proposition 3.22. It is easy to see that ϕ1 and ϕ3 are
bijective on vertex sets and the inverse maps are obtained by reversing the roles of T and T 0
(respectively X and X 0 ) in the construction. Thus Lemmas 3.23 and 3.24 apply to ϕ−1
1 and
ϕ−1
as
well.
This
easily
implies
that
ϕ
and
ϕ
are
quasi-isometries.
1
3
3
Further, recall that a map f : R → S between two metric spaces is called coarsely surjective
if there exists a constant ε such that S coincides with the closed ε-neighborhood of f (R). The
map ϕ2 is coarsely surjective because so are all ρi . To find a coarse inverse of ϕ2 , choose a
coarsely Hi -invariant coarse inverse ρ0i of each ρi (see Definition 3.11 and Lemma 3.12). The
map ϕ02 defined in the same way as ϕ2 but reversing the roles of (A, B) and (A0 , B 0 ) and using
the quasi-isometries ρ0i , we see that ϕ02 (ϕ2 (y)) = y whenever y = a(Hi , bi ) for some a ∈ G,
otherwise for y = (Hi , v) we have
dS (y, ϕ02 (ϕ2 (y))) ≤ dRi (r, ρ0i (ρi (r)))
which is uniformly bounded. Thus ϕ02 is indeed a (coarsely G-equivariant) coarse inverse of
ϕ2 . It is straightforward to check that the existence of such a map implies that ϕ2 is also a
quasi-isometry.
To complete the proof of the proposition, it remains to note that for every group G acting
on graphs R, S, every coarsely G-equivariant quasi-isometry V (R) → V (S) can be extended to
a coarsely G-equivariant quasi-isometry R → S. Thus the actions of G on the spaces SX,T ,B,A ,
SX,T 0 ,B,A , SX,T 0 ,B0 ,A0 , and SX 0 ,T 0 ,B0 ,A0 are equivalent.
Recall that A(G) denotes the set of all equivalence classes of actions of a group G on
geodesic metric spaces (of cardinality at most c). Proposition 3.22 allows us to formulate the
following.
Definition 3.25 (Induced action). Let
A = ([H1 y R1 ], . . . , [Hn y Rn ]) ∈ A(H1 ) × · · · × A(Hn ).

(28)

By Proposition 3.14, we can assume that every Ri is a graph and the action of Hi on V (Ri ) is
free. We define the induced action IndX (A) ∈ A(G) by the formula
IndX (A) = [IndX,T ,B (A)],
where T is any transversal of G with respect to {H1 , . . . , Hn }, B = {b1 , . . . , bn } is any collection
of base vertices bi ∈ V (Ri ), and
A = {H1 y R1 , . . . , Hn y Rn }.
In the situation where G is finitely generated relative to H1 , . . . , Hn the induced action
does not depend on the choice of finite relative generating set by Proposition 3.22, so we may
also define a map
Ind : A(H1 ) × · · · × A(Hn ) → A(G)
by the formula
Ind(A) = [IndX,T ,B (A)],
for every A as in (28), where X is any finite relative generating set.
23

Notice that properness and coboundedness of a group action of a metric space is invariant
under equivalence. Thus it makes sense to define proper and cobounded elements of A(G).
The following proposition summarizes some elementary properties of the induced action, which
follow immediately from our construction.
Proposition 3.26. Let G be a group, {H1 , . . . , Hn } a collection of subgroups of G, X a
generating set of G relative to {H1 , . . . , Hn }. Let A = (A1 , . . . , An ) ∈ A(H1 ) × · · · × A(Hn ).
(a) Assume that Ai is cobounded for all i. Then so is IndX (A).
S
(b) Suppose that Hi is generated by a set Yi and let Y = ni=1 Yi . If Ai = [Hi y Γ(Hi , Yi )],
then IndX (A) = [G y Γ(G, X ∪ Y )].
(c) If G is finitely generated relative to {H1 , . . . , Hn } and Ai is proper for all i, then Ind(A)
is proper.
In order to better understand the construction of the induced actions, we also recommend
the reader to consider the following.
Example 3.27. Let G be the fundamental group of a finite graph of groups with vertex groups
{Gv }v∈V . Let A = (Av )v∈V be the collection of equivalence classes of trivial actions Av =
[Gv y {pt}]. Then G is finitely generated relative to {Gv }v∈V and Ind(A) is the equivalence
class of the action of G on the associated Bass-Serre tree.

3.4

Incompressible subgroups

Our next goal is to introduce the notion of an incompressible collection of subgroups and
to prove Theorem 1.11. The reader is encouraged to review Section 2.4 before reading the
following.
Definition 3.28 (Incompressible subgroups). Let G be a group, {Hλ }λ∈Λ a (possibly infinite)
collection of subgroups of G, and let X be a generating set of G relative to {Hλ }λ∈Λ . We say
that the collection {Hλ }λ∈Λ is incompressible in G with respect to X if for every collection
C = {dHλ }λ∈Λ of left invariant metrics dHλ ∈ M(Hλ ), the inclusion map Hλ → G gives rise
to a quasi-isometric embedding (Hλ , dHλ ) → (G, dC,X ) for every λ ∈ Λ, where dC,X is the
corresponding induced metric on G (see Definition 2.18).
Further, if G is finitely generated with respect to {Hλ }λ∈Λ , we say that the collection of
subgroups {Hλ }λ∈Λ is incompressible in G if it is incompressible with respect to some finite
generating set X of G relative to {Hλ }λ∈Λ . In particular, this definition makes sense if G is
finitely generated.
We could replace “some finite generating set” with “any finite generating set” in the definition above. Moreover, we have the following.
Lemma 3.29. Let X, Y be two generating sets of G with respect to {Hλ }λ∈Λ such that the
symmetric difference X4Y is finite. Then {Hλ }λ∈Λ is incompressible with respect to X if and
only if it is incompressible with respect to Y .
24

Proof. It is easy to see using the definition of the induced metric that for every collection
C = {dHλ }λ∈Λ of left invariant metrics dHλ ∈ M(Hλ ), the identity map (G, dC,X ) → (G, dC,Y )
is Lipschitz with the Lipschitz constant maxx∈X {|x|Y }; the maximum exists since X4Y is
finite.
In particular, Lemma 3.29 holds true if both X and Y are finite.
The main result of this section is the following (Theorem 1.11 is clearly a particular case
of it).
Theorem 3.30. Let G be a group, {H1 , . . . , Hn } a collection of subgroups of G. Suppose that
G is finitely generated relative to {H1 , . . . , Hn }. Then the following conditions are equivalent.
(a) The extension problem for {H1 , . . . , Hn } and G is solvable.
(b) {H1 , . . . , Hn } is incompressible in G.
(c) For every A ∈ A(H1 ) × · · · × A(Hn ), Ind(A) is an extension of A.
We break the proof into two lemmas.
Lemma 3.31. Let G be a group, {H1 , . . . , Hn } a collection of subgroups of G. Suppose that
G is finitely generated modulo {H1 , . . . , Hn } and the extension problem for {H1 , . . . , Hn } and
G is solvable. Then {H1 , . . . , Hn } is incompressible in G.
Proof. Given a collection C = {dH1 , . . . , dHn } of metrics dHi ∈ M(Hi ), let dC,X be the metric
on G induced by C and a finite relative generating set X (see Definition 2.18). For each i, let
ϕi : (Hi , dHi ) → (G, dC,X ) be the map induced by inclusion. Our goal is to prove that it is a
quasi-isometric embedding. It is clear that ϕi is Lipschitz. Thus we only need to prove that
ϕi cannot “compress points too much,” i.e., that it satisfies the left inequality in (2).
Our proof will be based on the following observation, which is straightforward to verify
using the definitions: if a composition of two Lipschitz maps is a quasi-isometric embedding,
then each of these maps is a quasi-isometric embedding.
Let G y S be an extension of {H1 y (H1 , dH1 ), . . . , Hn y (Hn , dHn )}. By the definition of an extension, for every i there is a coarsely Hi -equivariant quasi-isometric embedding
βi : (Hi , dHi ) → S. We fix s ∈ S. Without loss of generality we can assume that βi (1) = s for
all i.
Assume that each βi satisfies the definition of a quasi-isometric embedding (2) and the
property of being coarsely Hi -equivariant (1) with a constant K. Since
sup {dS (hs, βi (h))} = sup {dS (hβ(1), βi (h1))} ≤ K,
h∈Hi

h∈Hi

s : (H , d ) → (H s, d ) is also a quasi-isometric embedding. It is clear that
the orbit map OH
i Hi
i
S
i
s
s
OH
= OG
◦ ϕi .
i
s is Lipschitz.
As we remarked above, it suffices to show that OG

25

Let f, g ∈ G and let f1 . . . fk be a geodesic decomposition of f −1 g; that is,
!
n
k
[
X
f1 , . . . , fk ∈ X ∪
Hi \ {1} and dC,X (f, g) =
wC,X (fj ).
i=1

j=1

If fj ∈ Hi for some i and wC,X (fj ) = dHi (1, fj ), then
dS (s, fj s) = dS (βi (1), βi (fj )) + dS (βi (fj ), fj βi (1)) ≤ KdHi (1, fj ) + 2K ≤ KwC,X (fj ) + 2K.
As fj 6= 1, it follows that wC,X (fj ) ≥ 1, and thus dS (s, fj s) ≤ 3KwC,X (fj ). Further, if fj ∈ X,
then dS (s, fj s) ≤ M = maxx∈X dS (s, xs). Hence, we have
dS (f s, gs) ≤

k
X

dS (s, fj s) ≤

j=1

k
X
(3K + M )wC,X (fj ) = (3K + M )dC,X (f, g).
j=1

s is Lipschitz, as required.
Thus, OG

Lemma 3.32. Let G be a group, {H1 , . . . , Hn } a collection of subgroups of G, X a (not
necessarily finite) generating set of G modulo {H1 , . . . , Hn }. Suppose that {H1 , . . . , Hn } is
incompressible in G with respect to X. Then for every A ∈ A(H1 ) × · · · × A(Hn ), IndX (A) is
an extension of A.
Proof. Let A = (A1 , . . . , An ). By Proposition 3.14, we can choose Hi y Ri ∈ Ai for each i
so that Ri is a graph and the action of Hi restricted to the vertex set of Ri is free. We fix a
transversal T = {t1 , . . . , tn } and a collection of basepoints B = {b1 , . . . , bn } as in Section 3.3.
Let S = SX,T ,B,A , where A = {H1 y R1 , . . . , Hn y Rn }, be the space of the induced action
(see Definition 3.20).
By Proposition 3.18, it suffices to prove that, for each i, the natural inclusion
ψi : V (Ri ) → V (S)

given by ψi (r) = (Hi , r)

is an Hi -equivariant quasi-isometry. The Hi -equivariance follows immediately from condition
(A1 ) in the definition of the induced action. Note that ψi extends to an embedding of Ri as a
subgraph of S, so each ψi is 1-Lipschitz. Thus we only need to prove that ψi satisfies the left
inequality in the definition of a quasi-isometric embedding (2).
We define left invariant metrics dHi on subgroups Hi by
dHi (g1 , g2 ) = dRi (g1 bi , g2 bi )

(29)

for all g1 , g2 ∈ Hi , and set C = {dH1 , . . . , dHn }. Since {H1 , . . . , Hn } is incompressible, there
exists a constant D ≥ 1 such that
dHi (g1 , g2 ) ≤ DdC,X (g1 , g2 )

(30)

dC,X (g1 , g2 ) ≤ dS (g1 (Hi , bi ), g2 (Hi , bi ))

(31)

for all g1 , g2 ∈ Hi .
We claim that

26

for any g1 , g2 ∈ G. (In fact, this is an equality, but the inequality is sufficient for our goal.)
Indeed let p be a geodesic path in the graph S connecting g1 (Hi , bi ) to g2 (Hi , bi ). Let
v1 = g1 (Hi , bi ), v2 , . . . , vk+1 = g2 (Hi , bi ) be consecutive vertices of p that belong to G(Hi , bi ),
the set of vertices of Γ(G, X) considered as a subgraph of S. That is, for every j = 1, . . . , k + 1,
we have vj = uj (Hi , bi ) for some uj ∈ G (recall that these are independent of the choice of i,
see (23)), where u1 = g1 and uk+1 = g2 . Let fj = u−1
j uj+1 . For every j = 1, . . . , k, the vertices
vj , vj+1 either span an edge of Γ(G, X) or simultaneously belong to some
Zj = {uj Hi(j) } × Ri(j) = {uj+1 Hi(j) } × Ri(j) .
In the former case, we have fj ∈ X and therefore
wC,X (fj ) = 1 = dS (vj , vj+1 ).
In the latter case, we have fj ∈ Hi(j) and vj+1 = uj fj (Hi(j) , bi(j )) = uj (Hi(j) , fj bi(j )) by
property (A1 ) (see Lemma 3.19). Hence
wC,X (fj ) ≤ dHi(j) (1, fj ) = dRi(j) (bi(j) , fj bi(j) ) = dYi(j) ((Hi(j) , bi(j) ), (Hi(j) , fj bi(j) ))
= dYi(j) (uj (Hi(j) , bi(j) ), uj (Hi(j) , fj bi(j) )) = dYi(j) (vj , vj+1 ) ≤ dS (vj , vj+1 ),
where Yj is defined as in Section 3.3, see (18). Therefore,
dC,X (g1 , g2 ) = dC,X (g1 , g1 f1 · · · fk ) ≤

k
X

wC,X (fj ) ≤

j=1

k
X

dS (vj , vj+1 )

j=1

= dS (g1 (Hi , bi ), g2 (Hi , bi )).
This finishes the proof of (31).
Now let i be fixed and let r1 , r2 be vertices of Ri . Let q be a geodesic path from (Hi , r1 ) to
(Hi , r2 ) in S. If this path does not intersect G(Hi , bi ), then dRi (r1 , r2 ) = dS (ψi (r1 ), ψi (r2 )) and
we are done. Otherwise, let g1 (Hi , bi ) and g2 (Hi , bi ) be the first and last vertices in q∩G(Hi , bi ).
Clearly g1 , g2 ∈ Hi . Using (31), (30), and (29) we obtain
dS (ψi (r1 ), ψi (r2 )) = dS ((Hi , r1 ), (Hi , g1 bi )) + dS ((Hi , g1 bi ), (Hi , g2 bi )) + dS ((Hi , g2 bi ), (Hi , r2 ))
≥ dRi (r1 , g1 bi ) + dC,X (g1 , g2 ) + dRi (g2 bi , r2 )
≥ dRi (r1 , g1 bi ) + D−1 dHi (g1 , g2 ) + dRi (g2 bi , r2 )
= dRi (r1 , g1 bi ) + D−1 dRi (g1 bi , g2 bi ) + dRi (g2 bi , r2 )
≥ D−1 dRi (r1 , r2 ).
Proof of Theorem 3.30. That (a) implies (b) follows from Lemma 3.31. Further (b) implies (c)
by Lemma 3.32. Finally, (c) implies (a) by Proposition 3.10.

4
4.1

Proofs of the main results
The extension problem for elementary amenable groups

Recall that the class of amenable groups is closed under the following four operations:
27

(S) Taking subgroups.
(Q) Taking quotient groups.
(E) Group extensions.
(U) Directed unions.
As in [10], let EG denote the class of elementary amenable groups, that is, the smallest
class which contains all abelian and finite groups and is closed under the operations (S)–(U).
In particular, EG contains all solvable and locally finite groups.
To prove Theorem 1.7 we will need several elementary facts. The first one is well-known;
we provide a brief outline of the proof for convenience of the reader.
Lemma 4.1. Every subgroup of a finitely generated virtually abelian group is undistorted. That
is, if G is a virtually abelian group generated by a finite set X and H ≤ G is generated by a
finite set Y , then the natural map (H, dY ) → (G, dX ) is a quasi-isometric embedding.
Proof. Let A ≤ G be a free abelian subgroup of finite index in G. It is easy to show that
H is undistorted in G if (and only if) H ∩ A is undistorted in A. The latter result follows
from the well-known (and easy to prove) facts that every subgroup of a finitely generated free
abelian group A is a retract of a finite index subgroup of A and that retracts and finite index
subgroups of finitely generated groups are undistorted.
The following lemma can be found in [15].
Lemma 4.2. Let G be an elementary amenable group such that every subgroup of G is finitely
generated. Then G is virtually polycyclic.
The next result is also well-known (see, for example, the proof of Proposition 4.17 in [5]).
Lemma 4.3. Let G be a virtually polycyclic group. If every subgroup of G is undistorted, then
G is virtually abelian.
We are now ready to prove Theorem 1.7.
Proof of Theorem 1.7. Assume first that G is a finitely generated virtually abelian group that
splits as described in Theorem 1.7. We want to show that the extension problem is solvable for
all H ≤ G. By Lemma 3.9, we can assume that H ≤ A without loss of generality. Furthermore,
by Theorem 1.11 it suffices to show that every H ≤ A is incompressible in G.
Let X be a finite generating set of G. Let C = {dH }, where dH ∈ M(H) (see Definition
2.16). Let wC,X and dC,X denote the corresponding weight function and the induced metric
on G (see Definition 2.18). Finally let Y be a finite generating set of H. By Lemma 4.1 there
exists a constant D such that
|h|Y ≤ D|h|X
(32)
for every h ∈ H.
28

Let
g = f1 . . . fk

(33)

be a geodesic decomposition of an element g ∈ H; that is, we have fi ∈ X ∪ H for all i and
dC,X (1, g) =

k
X

wC,X (fi ).

(34)

i=1

By our assumption, for every a ∈ A and every g ∈ G, we have
ga = a±1 g.

(35)

In particular, this is true for every a ∈ H since H ≤ A. We can rearrange the multiples in (33)
using (35) so that f1 , . . . , fn ∈ X \ {1} and fn+1 , . . . , fk ∈ H \ (X ∪ {1}) for some 0 ≤ n ≤ k.
Let f = f1 · · · fn and h = fn+1 · · · fk ; we assume here that f = 1 (respectively, h = 1) if n = 0
(respectively, n = k). Thus g = f h. Since g, h ∈ H, we have f ∈ H. Since Y is finite, there
exists M = maxy∈Y dH (1, y). Using (32) we obtain
dH (1, f ) ≤ M |f |Y ≤ M D|f |X ≤ M Dn = M D

n
X

wC,X (fi )

i=1

and
dH (1, h) ≤

k
X

k
X

dH (1, fi ) =

i=n+1

wC,X (fi ).

i=n+1

Taking these two inequalities together and using (34), we obtain
dH (1, g) ≤ dH (1, f ) + dH (1, h) ≤ (M D + 1)

k
X

wC,X (fi ) = (M D + 1)dC,X (1, g).

i=1

Hence dH (a, b) ≤ (M D+1)dC,X (a, b) for all a, b ∈ H. This completes the proof of the backward
implication in Theorem 1.7.
To prove the forward implication, we first note that if the extension problem is solvable
for all H ≤ G, then G must be virtually abelian. Indeed by part (a) of Proposition 3.7 and
Lemma 4.2 the group G must be virtually polycyclic and hence it is virtually abelian by part
(b) of Proposition 3.7 and Lemma 4.3. Let A C G be a finite index free abelian subgroup of G.
We will first show that (35) holds for all a ∈ A and g ∈ G. (Note that this is a priori weaker
than the assumption that the action of Q = G/A on A factors through the action of Z/2Z
by inversion as the exponent of a in the right side of (35) may depend on a.) Since A is free
abelian, it suffices to prove (35) under the assumption that a is not a proper power, i.e., hai is
a maximal cyclic subgroup of A.
Arguing by contradiction, assume that gag −1 = b 6= a±1 for some a ∈ A and g ∈ G. If
ha, bi is cyclic, then b = an for some n ∈ Z since hai is a maximal cyclic subgroup of A. Hence
we have gag −1 = an and n 6= ±1. Since A is of finite index in G, there exists k ∈ N such that
g k ∈ A and we have
k
a = g k ag −k = an .
29

This contradicts the assumption that A is torsion free.
Thus the subgroup H = ha, bi has rank 2 and therefore is naturally isomorphic to hai⊕hbi ∼
=
Let H y R be the action of H such that a acts trivially and b acts by translation: bx = x+1
for all x ∈ R. Then for any extension G y S of H y R, the subgroup hai will have bounded
orbits in S while the orbits of hbi will be unbounded. However this is impossible as these
subgroups are conjugate in G. This means that the action H y R does not extend to an
action of G.
Z2 .

Thus we have (35) for all a ∈ A and g ∈ G. To complete the proof it remains to show that
the choice of the exponent in the right side of (35) depends only on g. Assume that there are
−1
−1 = (a a )±1 .
a1 , a2 ∈ A\{1} such that ga1 g −1 = a1 and ga2 g −1 = a−1
1 2
2 . Then a1 a2 = ga1 a2 g
−1
±1
However the equality a1 a2 = (a1 a2 ) is impossible for non-trivial elements a1 , a2 of a free
abelian group. This contradiction completes the proof.

4.2

Lipschitz retractions to hyperbolically embedded subgroups

Our next goal is to show that hyperbolically embedded collections of subgroups are incompressible with respect to suitable relative generating sets. We prove a slightly stronger result,
Proposition 4.6, which seems to be of independent interest and may have other applications.
Throughout this section we fix a group G, a (possibly infinite) collection of subgroups
{Hλ }λ∈Λ of G, and a generating set X of G relative to {Hλ }λ∈Λ . Until Proposition 4.6 we do
not assume that {Hλ }λ∈Λ is hyperbolically embedded in G.
Definition 4.4 (Equivariant nearest point projection). For every g ∈ G and λ ∈ Λ, let
πλ : G → Hλ be a map satisfying
dX∪H (g, πλ (g)) = min dX∪H (g, h).
h∈Hλ

(36)

Then we call πλ a nearest point projection of G to Hλ . If, in addition, πλ (hg) = hπλ (g) for
all h ∈ Hλ and g ∈ G, we say that πλ is an equivariant nearest point projection. (Note that
“equivariant” refers to “Hλ –equivariant”, not “G–equivariant” here.)
Lemma 4.5. For every λ ∈ Λ, an equivariant nearest point projection πλ : G → Hλ exists.
Proof. Let Tλ be the set of representatives of right cosets of Hλ in G. For every t ∈ Tλ , define
πλ (t) to be an arbitrary element of H satisfying (36) and for every g ∈ Hλ t define
πλ (g) = gt−1 πλ (t).
Then we have
dX∪H (g, πλ (g)) = dX∪H (g, gt−1 πλ (t)) = dX∪H (t, πλ (t)).
On the other hand, we obtain
dX∪H (g, Hλ ) = dX∪H (tg −1 g, tg −1 Hλ ) = dX∪H (t, Hλ ) = dX∪H (t, πλ (t))

30

(37)

since tg −1 ∈ Hλ . Therefore, dX∪H (g, πλ (g)) = dX∪H (g, Hλ ), i.e., πλ is indeed a nearest point
projection. It remains to note that for every t ∈ Tλ , h ∈ Hλ , and g ∈ Hλ t, we have hg ∈ Hλ t.
Using (37) we obtain
πλ (hg) = hgt−1 πλ (t) = hπλ (g),
i.e., πλ is equivariant.
The main result of this section is the following.
Proposition 4.6. Suppose that {Hλ }λ∈Λ ,→h (G, X) and let C = {dHλ }λ∈Λ be a collection
of metrics dHλ ∈ M(Hλ ). Let dC,X denote the corresponding induced metric on G. Then for
every λ ∈ Λ, every nearest point projection πλ : G → Hλ induces a Lipschitz map (G, dC,X ) →
(Hλ , dHλ ).
Proof. Throughout this proof, we fix λ ∈ Λ. Let K be a positive integer satisfying
K ≥ max{dHλ (1, h) | h ∈ Hλ , dλ (1, h) ≤ 4D},

(38)

where D is the constant from Lemma 2.12. Note that the maximum is taken over a finite set
since {Hλ }λ∈Λ ,→h (G, X). We will show that for every f, g ∈ G,
dHλ (πλ (f ), πλ (g)) ≤ KdC,X (f, g).

(39)

Let f −1 g = f1 · · · fk be a geodesic decomposition. That is, f1 , . . . , fk are elements of the
set
!
[
Y =X∪
Hλ
λ∈Λ

and
dC,X (f, g) =

k
X

wC,X (fi ).

(40)

i=1

To prove (39) we first show that
dHλ (πλ (a), πλ (at)) ≤ KwC,X (t)

(41)

for every a ∈ G and t ∈ Y .
To this end we denote by u (respectively w) a geodesic path in Γ(G, X t H) that goes from
πλ (a) to a (respectively, from at to πλ (at)). Without loss of generality, we can assume that
t 6= 1. Let e be the edge labeled by an element of Hλ that connects πλ (at) to πλ (a). Finally,
let v denote the edge of Γ(G, X t H) starting at a and labeled by t (see Fig. 1).
If e is connected to an edge e0 of u labeled by an element of Hλ , then e0+ ∈ Hλ and
dX∪H (a, e0+ ) < dX∪H (a, πλ (a)), which contradicts our assumption that πλ is a nearest point
projection. Similarly e cannot be connected to an edge of w labeled by an element of Hλ . Thus
e is an isolated Hλ –component of the path weu.
We now consider two cases.
31

at

v

a

w

u
e

πλ (at)

πλ (a)

Hλ

Figure 1: The proof of Proposition 4.6.
Case 1. First assume that e is isolated in the geodesic quadrilateral p = weuv. Then
dλ (πλ (a), πλ (at)) = dλ (e− , e+ ) ≤ 4D
by Lemma 2.12. Therefore, dHλ (πλ (a), πλ (at)) ≤ K by (38). In particular, (41) holds since
wC,X takes values in N ∪ {0} and wC,X (t) = 0 only if t = 1.
Case 2. Suppose now that e is not isolated in p. Since e is isolated in weu, it can only be
connected to v. In particular, we have t ∈ Hλ . Without loss of generality, we can also assume
that
wC,X (t) = dHλ (1, t)
(42)
in this case. Indeed, recall that labels of edges of Γ(G, X t H) are taken from the disjoint
union of X and subgroups Hµ , µ ∈ Λ. Thus if we also have t ∈ Hµ for some µ 6= λ, we can
simply replace v with another edge of Γ(G, X t H) with the same endpoints as v and the label
t ∈ Hµ , which makes Case 2 impossible. Similarly we rule out the case t ∈ X. Thus, without
loss of generality, we can assume that t ∈
/ X and Λ(t) = {λ} in the notation of Definition 2.18;
hence (42) holds by the definition of wC,X .
Since e is connected to v, we have a = v− ∈ Hλ and at = v+ ∈ Hλ . Since πλ is a nearest
point projection, it is identical on Hλ . In particular, we have πλ (a) = a and πλ (at) = at.
Therefore, by (42) we have
dHλ (πλ (a), πλ (at)) = dHλ (a, at) = dHλ (1, t) = wC,X (t)
in this case. This completes the proof of (41).
Now let
h0 = πλ (f ), h1 = πλ (f f1 ), . . . , hk = πλ (f f1 · · · fk ) = πλ (g).
Using the triangle inequality, (40) and (41), we obtain
dHλ (πλ (f ), πλ (g)) ≤

k
X

dHλ (hi−1 , hi ) ≤

i=1

k
X
i=1

32

KwC,X (fi ) = KdC,X (f, g).

Since every nearest point projection G → Hλ is the identity on Hλ , we obtain the following
corollary of Lemma 4.5 and Proposition 4.6.
Corollary 4.7. Assume that {Hλ }λ∈Λ ,→h (G, X). Then for every collection of left invariant
metrics C = {dHλ }, where dHλ ∈ M(Hλ ), and every λ ∈ Λ, there exists an Hλ –equivariant
Lipschitz map (G, dC,X ) → (Hλ , dHλ ) whose restriction to Hλ is the identity map.
Finally, we pass from the property of being a Lipschitz retract to the property of being
undistorted. This is a fairly standard argument.
Corollary 4.8. Suppose that {Hλ }λ∈Λ ,→h (G, X). Then {Hλ }λ∈Λ is incompressible in G
with respect to X. In particular, if G is finitely generated and hyperbolic relative to a collection
of subgroups {H1 , . . . , Hn }, then {H1 , . . . , Hn } is incompressible in G.
Proof. Fix any λ ∈ Λ. Let C = {dHλ }, where dHλ ∈ M(Hλ ). Let πλ : (G, dC,X ) → (Hλ , dHλ )
be a Lipschitz map whose restriction to Hλ is the identity map. Let L be the corresponding
Lipschitz constant. Then for every g, h ∈ Hλ , we have
dHλ (g, h) = dHλ (πλ (g), πλ (h)) ≤ LdC,X (g, h).
The opposite inequality dC,X (g, h) ≤ dHλ (g, h) for all g, h ∈ H is obvious from the definition
of the induced metric. Thus the inclusion Hλ → G induces a quasi-isometric embedding
(Hλ , dHλ ) → (G, dC,X ). To derive the claim about relatively hyperbolic groups we only need
to refer to Proposition 2.9.

4.3

Extending actions of hyperbolically embedded subgroups

Recall that an equivalence class A ∈ A(G) is called hyperbolic if it consists of G-actions on
hyperbolic metric spaces. The main goal of this section is to prove the following result; Theorem
1.8 follows from it immediately via Proposition 2.9.
Theorem 4.9. Let G be a group, {H1 , . . . , Hn } a collection of subgroups of G, and let X a be
relative generating set of G with respect to {H1 , . . . , Hn }.
(a) Suppose that {H1 , . . . , Hn } ,→h (G, X). Then for every collection
A = (A1 , . . . , An ) ∈ A(H1 ) × · · · × A(Hn ),

(43)

the induced action IndX (A) is an extension of A; if, in addition, each Ai is hyperbolic,
then so is IndX (A).
(b) Conversely, suppose that H1 , . . . , Hn are countable and for every collection (43), where
each Ai is hyperbolic, the induced action IndX (A) is a hyperbolic extension of A. Then
{H1 , . . . , Hn } ,→h (G, X).
Proof. (a) That IndX (A) is an extension of A follows from Corollary 4.8 and Lemma 3.32.
Thus we only need to prove hyperbolicity. The proof is fairly standard; it essentially repeats
the proof of [9, Lemma 6.45] with obvious adjustments.
33

Z1
b1
(b1 )+

(b1 )−

Z2

e1
Zk
c

c0

b2

(b2 )−

bk
(bk )−

(bk )+

. . .

(b2 )+
e2

. . .

ek
Γ(G, X)

S

Γ(G, X t H)

Figure 2: Constructing the loop c0
For details about van Kampen diagrams, isoperimetric functions, etc., we refer to Section 2.2 and [9]. Given a (combinatorial) path p in a van Kampen diagram ∆ over a group
presentation, we denote by Lab(p) the label of p.
Theorem 2.15 provides us with a strongly bounded relative presentation
G = hX, H | S ∪ Ri

(44)

with linear relative isoperimetric function.
Let A = {H1 y R1 , . . . , Hn y Rn }, where for each i we have Hi y Ri ∈ Ai , Ri is a
hyperbolic graph, and the action of Hi restricted to the vertex set of Ri is free. We fix some
transversal T and collection of base points B as in Section 3.3 and let G y S = IndX,T ,B (A) ∈
IndX (A). In what follows we naturally think of Γ(G, X) as a subgraph of both Γ(G, X t H)
and S.
We will show that condition (b) from Proposition 2.6 holds for S. Let c be a loop in S.
Without loss of generality we can assume that c has at least one vertex in Γ(G, X) as otherwise
it is contained in a copy of a hyperbolic graph Ri and the isoperimetric inequality follows.
(Notice that we need finiteness of the collection {H1 , . . . , Hn } here to ensure uniformness of
the isoperimetric constants.)
To every such c we associate a loop in Γ(G, X t H) as follows. Let b1 , . . . , bk be the set of
all maximal subpaths of c such that each bi belongs to some {gi Hj(i) } × Rj(i) . We replace each
bi with the corresponding edge ei in Γ(G, X t H) connecting (bi )− to (bi )+ and labeled by an
element of Hj(i) . This naturally defines a loop c0 in Γ(G, X t H).
Consider a van Kampen diagram ∆ over (44) such that:
1) The boundary label of ∆ is Lab(c0 ).
2) ∆ has minimal number of R-cells among all diagrams satisfying 1).
34

3) Every edge of ∂∆ labeled by a letter from H belongs to an S-cell.
4) ∆ has minimal number of S-cells among all diagrams satisfying 1)–3).
Note that we can always ensure 3) by gluing cells labeled by hh−1 where h ∈ H (the so-called
0-cells) to the boundary of ∆. In what follows we identify ∂∆ with c0 .
The maps ei 7→ bi naturally induce a continuous map ϕ from c0 to S whose image is c.
Observe that 4) implies that no edge of ∆ can belong to two S-cells, for otherwise these S-cells
could be replaced with a single cell. Thus every internal edge e of ∆ belongs to an R-cell
and hence Lab(e) ∈ X ∪ Y , where Y is the set of all letters from H that appear in relations
R ∈ R. Since (44) is strongly bounded, Y is finite. By Proposition 3.22 and Lemma 2.10, we
can assume without loss of generality that for every y ∈ Y there exists xy ∈ X such that xy
and y represent the same element of G. This allows us to extend ϕ to the 1-skeleton of ∆ by
mapping every internal edge e of ∆ to the corresponding edge of Γ(G, X) ⊆ S (edges labeled
by y ∈ Y are mapped to the corresponding edges labeled by xy ∈ X).
Let f (n) = Cn be a relative isoperimetric function of (44) and let M = maxR∈R kRk. Note
that M < ∞ as (44) is bounded. The map ϕ : Sk (1) (∆) → S gives rise to a decomposition of
[c] into the sum of at most C`(c) homotopy classes of loops of length at most M corresponding
to R-cells of ∆ (here we use the fact that no ei belongs to the boundary of an R-cell, which
is ensured by 3)) plus [s1 ] + · · · + [sm ], where all si are images of boundaries of S-cells of ∆.
Clearly the total length of all loops [si ] does not exceed 2 times the total number of internal
edges of ∆ plus `(c). Again taking into account that every internal edge e of ∆ belongs to an
R-cell and using the isoperimetric inequality we obtain
m
X

`(si ) ≤ 2M C`(c0 ) + `(c) ≤ (2M C + 1)`(c).

(45)

i=1

Note that every si is a loop in {gi Hj(i) } × Rj(i) for some gi ∈ G and j(i) ∈ {1, . . . , n}, which
is an isometric copy of the (hyperbolic) graph Rj(i) . Therefore there exist constants A, B such
that every [si ] decomposes into the sum of at most A`(si ) homotopy classes of loops of length
at most B. Consequently [c] decomposes into the sum of at most (C + A(2M C + 1))`(c)
homotopy classes of loops of length at most max{M, B}. This completes the proof of (a).
(b)Applying Proposition 3.26 to the collection A = (A1 , . . . , An ), where Ai = [Hi y
Γ(Hi , Hi )], we obtain that IndX (A) = [G y Γ(G, X t H)]. It follows from our assumption that
the relative Cayley graph Γ(G, X t H) is hyperbolic. Thus it remains to verify condition (b)
from Definition 2.8.
Arguing by contradiction, assume that condition (b) from the definition of a hyperbolically
embedded collection of subgroups does not hold. We are going to construct a specific collection
of actions of subgroups Hi on hyperbolic spaces such that the corresponding induced action of
G is not a hyperbolic extension.
By the Higman-Neumann-Neumann theorem, every countable group embeds in a finitely
generated group. We embed Hi into a finitely generated group Ki . Let Zi be a finite generating
set of Ki and let | · |Zi denote the corresponding word length. That is, for some j ∈ {1, . . . , n},
35

some C ∈ N, and every N ∈ N, there exist a path q of length at most C in
∆j = Γ(G, X t H) \ E(Γ(Hj , Hj ))
connecting 1 to a vertex h ∈ Hj such that |h|Zj ≥ N . Let Ri = H(Γ(Ki , Zi )) for all i (see
Definition 2.4) and let A = (H1 y R1 , . . . , Hn y Rn ).
We fix some transversal T and a collection of base points B as in Section 3.3. By Lemma
2.5, all Ri are hyperbolic. Therefore, so is the space S = SX,T ,B,A of the induced action.
Let u = qr, where r is the edge of Γ(G, X t H) labeled by h−1 ∈ Hj and connecting h to 1.
Then r is an isolated Hj -component of u. Let v be a loop in S obtained from u by replacing
all Hi -components of u (for all i) with geodesics in corresponding graphs {gHi } × Ri . We
call these subpaths of v Ri -components. Since IndX (A) is an extension of A, geodesics in the
graphs {gHi } × Ri are (λ, c)-quasi-geodesics in S for some λ ≥ 1, c > 0.
Let s denote the subpath of v corresponding to the Hj -component r of u; thus s is a geodesic
in {1Hj } × Rj and therefore it is a (λ, c)-quasi-geodesic in S. If N is large enough compared to
C, the hyperbolicity constant δ of S, and the constant κ = κ(δ, λ, c) provided by Lemma 2.2,
the combination of Lemma 2.2 and Lemma 2.3 provide us with a subpath s0 of s and a subpath
t0 of some other Ri -component t of v such that s0 and t0 belong to (2κ + 15δ)-neighborhoods
of each other and `(s0 ) ≥ 2(2κ + 15δ) + 4. Note that t cannot belong to {1Hj } × Rj as r is
an isolated Hj -component of u. Since s0 is a geodesic in {1Hj } × Rj , which is an isometric
copy of the combinatorial horoball Rj , it must contain a vertical subsegment of length greater
than 2κ + 15δ by Lemma 2.5. Thus s0 cannot belong to the closed (2κ + 15δ)-neighborhood of
Γ(G, X). This contradicts the fact that s0 belongs to a (2κ + 15δ)-neighborhood of t0 . Indeed
t0 belongs to some {gHi } × Ri 6= {1Hj } × Rj and every path in S originating in {1Hj } × Rj
and terminating in {gHi } × Ri must intersect Γ(G, X). This contradiction completes the proof
of part (b) of Definition 2.8 and the theorem.
The following immediate corollary is a generalization of Theorem 1.8.
Corollary 4.10. Let G be a group, {H1 , . . . , Hn } a collection of hyperbolically embedded subgroups of G. Then the extension problem for {H1 , . . . , Hn } and G is solvable.
Yet another corollary follows immediately from Theorem 4.9 and Proposition 2.9.
Corollary 4.11. Let G be a group, {H1 , . . . , Hn } a collection of subgroups of G.
(a) Suppose that G is hyperbolic relative to {H1 , . . . , Hn }. Then for every collection (43),
the induced action Ind(A) is an extension of A and if each Ai is hyperbolic, then so is
Ind(A).
(b) Conversely, suppose that G is finitely generated relative to {H1 , . . . , Hn }, the subgroups
H1 , . . . , Hn are countable, and for every collection (43), where each Ai is hyperbolic,
Ind(A) is a hyperbolic extension of A. Then G is hyperbolic relative to {H1 , . . . , Hn }.
We consider a couple of examples illustrating that part (b) of Theorem 1.12 can fail under
certain weaker assumptions.
36

Example 4.12. Let H = Sym(N) and let G = H ×Z/2Z. As we already mentioned, all actions of
H on metric spaces have bounded orbits. This easily implies that for every A ∈ A(H), Ind(A)
is an extension of A. In addition, if [H y R] is hyperbolic then the space of the induced
action is quasi-isometric to two copies of R glued along a bounded subset; in particular, this
space is also hyperbolic. Thus all assumptions of part (b) of Theorem 4.9 are satisfied except
countability of H. However the conclusion fails: G is not hyperbolic relative to H as peripheral
subgroups of relatively hyperbolic groups must be almost malnormal.
Example 4.13. Let G = Z × Z/2Z. Then H = Z × {1} ≤ G is a retract of G, hence every
action of H on a hyperbolic space extends to an action of G on the same hyperbolic space, see
Example 1.4 (b). However G is not hyperbolic relative to H. This shows that the condition
that Ind(A) is a hyperbolic extension of A for every hyperbolic A ∈ A(H) cannot be replaced
with the assumption that every action of H on a hyperbolic space extends to an action of G
on a hyperbolic space.
We now turn to the proof of Corollary 1.9 from the introduction. Recall that two elements
a, b of infinite order of a group H are called commensurable in H if some non-trivial powers of
a and b are conjugate in H. We will need the following.
Definition 4.14. We say that a group embedding H ≤ G is commensurability preserving if
infinite order elements of H are commensurable in H whenever they are commensurable in G.
Example 4.15. If H is almost malnormal in G, then the embedding H ≤ G is commensurability
preserving. Indeed suppose that a, b ∈ H are infinite order elements and are commensurable
in G. Then there exists t ∈ G and m, n ∈ N such that t−1 am t = b±n . In particular, the
intersection H ∩ t−1 Ht contains hbn i and therefore it is infinite. By almost malnormality,
we get t ∈ H, which means that a and b are commensurable in H. Note, however, that
malnormality is a strictly stronger condition. (Hint: consider the embedding 2Z ≤ Z.)
We will need two more lemmas.
Lemma 4.16. Let H be a subgroup of a hyperbolic group G and let a, b ∈ H be two noncommensurable (in H) elements of infinite order. Then there exists an action of H on a
metric space such that the orbits of hai are bounded while the orbits of hbi are unbounded.
Proof. By [9, Theorem 6.8] applied to the action of H on the Cayley graph of G with respect
to a finite generating set, a and b are contained in virtually cyclic subgroups E(a), E(b) of H
such that {E(a), E(b)} ,→h (H, X) for some X ⊆ H. Let A ∈ A(E(a)) be the equivalence class
of a geometric action of E(a) and let B ∈ A(E(b)) be the the equivalence class of the trivial
action on the point. By Theorem 4.9, there is an extension of the pair (A, B) to an H–action
C ∈ A(H). Clearly C satisfies the required conditions.
Lemma 4.17. Let G be a virtually cyclic group. Then the extension problem is solvable for
all subgroups of G.
Proof. Without loss of generality we can assume that G is infinite. Let A ∼
= Z be a normal cyclic
subgroup of finite index in G. It is clear that Theorem 1.7 applies to G since Aut(Z) ∼
= Z/2Z.
Thus the extension problem is solvable for all subgroups of G.
37

Corollary 1.9 is a simplified version of the following.
Corollary 4.18. Let G be a hyperbolic group.
(a) Suppose that H is quasiconvex in G and either virtually cyclic or almost malnormal.
Then the extension problem is solvable for H ≤ G.
(b) Conversely, if the extension problem is solvable for a subgroup H ≤ G, then H is quasiconvex and the embedding H ≤ G is commensurability preserving.
Proof. (a) Assume first that H is quasiconvex and almost malnormal. Then by a result of
Bowditch [3], G is hyperbolic relative to H and hence the extension problem for H ≤ G is
solvable by Theorem 3.30. Now assume that H is virtually cyclic. If H is finite the claim is
obvious, so we assume that H is infinite. Then there exists a maximal virtually cyclic subgroup
E of G containing H. By Lemma 4.17, every H-action on a metric space extends to an Eaction. By [9, Theorem 6.8] we have E ,→h G and therefore every E-action in turn extends to
a G–action. Thus the extension problem is solvable for H ≤ G in this case as well.
(b) Assume that the extension problem is solvable for H. Then H is finitely generated and
undistorted in G by Proposition 3.7. This is well-known to be equivalent to quasiconvexity.
To prove that the embedding H ≤ G is commensurability preserving we argue by contradiction. Assume that there are elements a, b ∈ H of infinite order that are commensurable in
G but not in H. By Lemma 4.16, there exists an H–action such that the orbits of hai are
bounded while the orbits of hbi are not. Since the extension problem is solvable for H ≤ G,
there is an action of G with the same property. However, this contradicts the fact that a and
b are commensurable in G.
The following example shows that the sufficient condition for the extension problem to be
solvable from part (a) is not necessary.
Example 4.19. Let G = H × K, where K is a non-trivial finite group and H is a finitely
generated non-cyclic free group. Then G is hyperbolic and H is neither virtually cyclic nor
malnormal. However the extension problem for H ≤ G is solvable since H is a retract of G.
We conclude with few open problems.
Problem 4.20. Is the sufficient condition for the extension problem to be solvable from Corollary 4.18 (a) necessary in case G is torsion free?
The negative answer can likely be obtained by studying the case of G = F (a, b), the free
group with basis {a, b}, and H = ha2 , b2 i ≤ G.
Problem 4.21. Is the necessary condition for the extension problem to be solvable from Corollary 4.18 (b) sufficient?
It would be also interesting to address the extension problem for individual subgroups in
other classes of groups.
Problem 4.22. Describe incompressible subgroups of finitely generated nilpotent groups, polycyclic groups, free solvable groups, etc.
38

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Carolyn Abbott: Department of Mathematics, University of Wisconsin-Madison, Madison
53706, USA.
E-mail: abbott@math.wisc.edu
David Hume: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2
6GG, UK.
E-mail: david.hume@maths.ox.ac.uk
Denis Osin: Department of Mathematics, Vanderbilt University, Nashville 37240, USA.
E-mail: denis.v.osin@vanderbilt.edu

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