Acylindrical group actions on quasi-trees

arXiv:1602.03941v3 [] 26 Mar 2016

S. Balasubramanya
Abstract
A group G is acylindrically hyperbolic if it admits a non-elementary acylindrical
action on a hyperbolic space. We prove that every acylindrically hyperbolic group G
has a generating set X such that the corresponding Cayley graph Γ is a (non-elementary)
quasi-tree and the action of G on Γ is acylindrical. Our proof utilizes the notions of
hyperbolically embedded subgroups and projection complexes. As an application, we
obtain some new results about hyperbolically embedded subgroups and quasi-convex
subgroups of acylindrically hyperbolic groups.

1

Introduction

Recall that an isometric action of a group G on a metric space (S, d) is acylindrical if for
every ε > 0 there exist R, N > 0 such that for every two points x, y with d(x, y) ≥ R,
there are at most N elements g ∈ G satisfying
d(x, gx) ≤ ε and d(y, gy) ≤ ε.
Obvious examples are provided by geometric (i.e., proper and cobounded) actions; note,
however, that acylindricity is a much weaker condition.
A group G is called acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. Over the last few years, the class of acylindrically hyperbolic groups has received considerable attention. It is broad enough to include many
examples of interest, e.g., non-elementary hyperbolic and relatively hyperbolic groups,
all but finitely many mapping class groups of punctured closed surfaces, Out(Fn ) for
n ≥ 2, most 3-manifold groups, and finitely presented groups of deficiency at least 2.
On the other hand, the existence of a non-elementary acylindrical action on a hyperbolic space is a rather strong assumption, which allows one to prove non-trivial results.
In particular, acylindrically hyperbolic groups share many interesting properties with
non-elementary hyperbolic and relatively hyperbolic groups. For details we refer to
[5, 10, 11, 12] and references therein.
The main goal of this paper is to answer the following.
Question 1.1. Which groups admit non-elementary cobounded acylindrical actions on
quasi-trees?
In this paper, by a quasi-tree we mean a connected graph which is quasi-isometric to
a tree. Quasi-trees form a very particular subclass of the class of all hyperbolic spaces.
From the asymptotic point of view, quasi-trees are exactly “1-dimensional hyperbolic
spaces”.

1

The motivation behind our question comes from the following observation. If instead
of cobounded acylindrical actions we consider cobounded proper (i.e., geometric) ones,
then there is a crucial difference between the groups acting on hyperbolic spaces and
quasi-trees. Indeed a group G acts geometrically on a hyperbolic space if and only if G is
a hyperbolic group. On the other hand, Stallings theorem on groups with infinitely many
ends and Dunwoodys accessibility theorem implies that groups admitting geometric
actions on quasi-trees are exactly virtually free groups. Yet another related observation
is that acylindrical actions on unbounded locally finite graphs are necessarily proper.
Thus if we restrict to quasi-trees of bounded valence in Question 1.1, we again obtain the
class of virtually free groups. Other known examples of groups having non-elementary,
acylindrical and cobounded actions on quasi-trees include groups associated with special
cube complexes and right angled artin groups (see [1], [6], [8]).
Thus one could expect that the answer to Question 1.1 would produce a proper
subclass of the class of all acylindrically hyperbolic groups, which generalizes virtually
free groups in the same sense as acylindrically hyperbolic groups generalize hyperbolic
groups. Our main result shows that this does not happen.
Theorem 1.2. Every acylindrically hyperbolic group admits a non-elementary
cobounded acylindrical action on a quasi-tree.
In other words, being acylindrically hyperbolic is equivalent to admitting a nonelementary acylindrical action on a quasi-tree are equivalent. Although this result does
not produce any new class of groups, it can be useful in the study of acylindrically hyperbolic groups and their subgroups. In this paper we concentrate on proving Theorem
1.2 and leave applications for future papers to explore (for some applications, see [10]).
It was known before that every acylindrically hyperbolic group admits a nonelementary cobounded action on a quasi-tree satisfying the so-called weak proper discontinuity property, which is weaker than acylindricity. Such a quasi-tree can be produced
by using projection complexes introduced by Bestvina-Bromberg-Fujiwara in [2]. To
the best of our knowledge, whether the corresponding action is acylindrical is an open
question. The main idea of the proof of Theorem 1.2 is to combine the BestvinaBromberg-Fujiwara approach with an ‘acylindrification’ construction from [11] in order
to make the action acylindrical. An essential role in this process is played by the notion
of a hyperbolically embedded subgroup introduced in [5] - this fact is of independent
interest since it provides a new setting for the application of the Bestvina-BrombergFujiwara construction.
The above mentioned construction has been applied in the setting of geometrically
separated subgroups (see [5]), but hyperbolically embedded subgroups do not necessarily
satisfy this condition. Nevertheless, it is possible to employ them in this construction,
possibly with interesting applications. If fact, we prove much stronger results in terms
of hyperbolically embedded subgroups (see Theorem 3.1) of which Theorem 1.2 is an
easy consequence, and derive an application in this paper which is stated below (see
Corollary 3.23).
Corollary 1.3. Let G be a group. If H ≤ K ≤ G , H is countable and H is hyperbolically embedded in G, then H is hyperbolically embedded in K.
We would like to note that the above result continues to hold even when we have
a finite collection {H1 , H2 , ..., Hn } of hyperbolically embedded subgroups in G such
that Hi ≤ K for all i = 1, 2, ..., n. Interestingly, A.Sisto obtains a similar result in [14],

2

Corollary 6.10. His result does not require H to be countable, but under the assumption
that H ∩ K is a virtual retract of K, it states that H ∩ K ,→h K. Although similar,
these two theorems are distinct in the sense that neither follows from the other.
Another application of Theorem 3.1 is to the case of finitely generated subgroups,
as stated below (see Corollary 3.26).
Corollary 1.4. Let H be a finitely generated subgroup of an acylindrically hyperbolic
group G. Then there exists a subset X ⊂ G such that
(a) Γ(G, X) is hyperbolic, non-elementary and acylindrical
(b) H is quasi-convex in Γ(G, X)
The above result indicates that in order to develop a theory of quasi-convex subgroups in acylindrically hyperbolic groups, the notion of quasi-convexity is not sufficient,
i.e., a stronger set of conditions is necessary in order to prove results similar to those
known for quasi-convex subgroups in hyperbolic groups. For example, using Rips’ construction from [13] and the above corollary, one can easily construct an example of an
infinite, infinite index, normal subgroup in an acylindrically hyperbolic group, which is
quasi-convex with respect to some non-elementary acylindrical action.
Acknowledgements: My heartfelt gratitude to my advisor Denis Osin for his guidance and support, and to Jason Behrstock and Yago Antolin Pichel for their remarks.
My sincere thanks to Bryan Jacobson for his thorough proof-reading and comments on
this paper.

2

Preliminaries

We recall some definitions and theorems which we will need to refer to.

2.1

Relative Metrics on subgroups

Definition 2.1 (Relative metric). Let G be a group and {Hλ }λ∈Λ a fixed collection of
subgroups of G. LetFX ⊂ G such that G is generated by X along with the union of all
{Hλ }λ∈Λ . Let H = λ∈Λ Hλ . We denote the corresponding Cayley graph of G (whose
edges are labeled by elements of X t H) by Γ(G, X t H).
Remark 2.2. It is important that the union in the definition above is disjoint. This
disjoint union leads to the following observation : for every h ∈ Hi ∩ Hj , the alphabet
H will have two letters representing h in G, one from Hi and another from Hj . It may
also be the case that a letter from H and a letter from X represent the same element of
the group G. In this situation, the corresponding Cayley graph Γ(G, X t H) has bigons
(or multiple edges in general) between the identity and the element, one corresponding
to each of these letters.
We think of Γ(Hλ , Hλ ) as a complete subgraph in Γ(G, X tH). A path p in Γ(G, X t
H) is said to be λ-admissible if it contains no edges of the subgraph Γ(Hλ , Hλ ). In other
words, the path p does not travel through Hλ in the Cayley graph. Using this notion,
we can define a metric, known as the relative metric dbλ : Hλ × Hλ → [0, +∞] by setting
dbλ (h, k) for h, k ∈ Hλ to be the length of the shortest admissible path in Γ(G, X t H)
that connects h to k. If no such path exists, we define dbλ (h, k) = +∞. It is easy to
check that dbλ satisfies all the conditions to be a metric function.

3

Definition 2.3. Let q be a path in the Cayley graph of Γ(G, XtH) and p be a nontrivial
subpath of q. p is said to be an Hλ - subpath if the label of p (denoted Lab(p)) is a
word in the alphabet Hλ . Such a subpath is further called an Hλ -component if it is not
contained in a longer Hλ -subpath of q. If q is a loop, we must also have that p is not
contained in a longer Hλ -subpath of any cyclic shift of q. We refer to an Hλ -component
of q (for some λ ∈ Λ) simply by calling it a component of q. We note that on a geodesic,
Hλ - components must be single Hλ -edges.
Let p1 , p2 be two Hλ components of a path q for some λ ∈ Λ. p1 and p2 are said
to be connected if there exists a path p in Γ(G, X t H) such that Lab(p) is a word
consisting only of letters from Hλ , and p connects some vertex of p1 to some vertex of
p2 . In algebraic terms, this means that all vertices of p1 and p2 belong to the same
(left) coset of Hλ . We refer to a component of a path q as isolated if it is not connected
to any other component of q.
If p is a path, we denote its initial point by p− and its terminating point by p+ .
Lemma 2.4 ([5], Proposition 4.13). Let G be a group and {Hλ }λ∈Λ a fixed collection
of subgroups in G. Let X ⊂ G such that G is generated by X together with the union
of all {Hλ }λ∈Λ . Then there exists a constant C > 0 such that for any n-gon p with
geodesic sides in Γ(G, X t H), any λ ∈ Λ, and any isolated Hλ component a of p,
dbλ (a− , a+ ) ≤ Cn.

2.2

Hyperbolically embedded subgroups

Hyperbolically embedded subgroups will be our main tool in constructing the quasi-tree.
The notion has been taken from [5]. We recall the definition here.
Definition 2.5 (Hyperbolically embedded subgroups). Let G be a group. Let X be a
(not necessarily finite) subset of G and let {Hλ }λ∈Λ be a collection of subgroups of G.
We say that {Hλ }λ∈Λ is hyperbolically embedded in G with respect to X (denoted by
{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 }λ∈Λ
F
(b) The Cayley graph Γ(G, X t H) is hyperbolic, where H = λ∈Λ Hλ .
(c) For every λ ∈ Λ, the metric space (Hλ , dbλ ) is proper, i.e., every ball of finite radius
has finite cardinality.
Further we say that {Hλ }λ∈Λ is hyperbolically embedded in G (denoted by
{Hλ }λ∈Λ ,→h G) if {Hλ }λ∈Λ ,→h (G, X) for some X ⊆ G. The set X is called a
relative generating set.
Since the notion of a hyperbolically embedded subgroup plays a crucial role in this
paper, we include two examples borrowed from [5].
Example 2.6. Let G = H × Z and Z = hxi. Let X = {x}. Then Γ(G, X t H) is
quasi-isometric to a line and is hence hyperbolic. The corresponding relative metric
satisfies the following inequality which is easy to see from the Cayley graph (see Fig.1)
b 1 , h2 ) ≤ 3 for every h1 , h2 ∈ H. Indeed if ΓH denotes the Cayley graph Γ(H, H),
: d(h
then in its shifted copy xΓH , there is an edge e connecting xh1 to xh2 (labeled by
h−1
1 h2 ∈ H) . There is thus an admissible path of length 3 connecting h1 to h2 . We
conclude that if H is infinite, then H is not hyperbolically embedded in (G, X), since
the relative metric will not be proper. In this example, one can also note that the

4

...

...

...

...

...

xΓH
xh1

e

xh2
...

x

x

x
x

ΓH
h1
x

x

x

h2
x

...

xΓH

ΓH

1

x

x
x−1 ΓH

x

x

...

...
x−1 ΓH

...

...

...

...
...

Figure 1: H × Z

Figure 2: H ∗ Z

admissible path from h1 to h2 contains an H-subpath, namely the edge e, which is also
an H-component of this path.
Example 2.7. Let G = H ∗ Z and Z = hxi. As in the previous example, let X = {x}.
In this case Γ(G, X t H) is quasi-isometric to a tree (see Fig.2) and it is easy to see
b 1 , h2 ) = ∞ unless h1 = h2 . This means that every ball of finite radius in the
that d(h
relative metric has cardinality 1. We can thus conclude that H ,→h (G, X).

2.3

A slight modification to the relative metric

The aim of this section is to modify the relative metric on countable subgroups that
are hyperbolically embedded, so that the resulting metric takes values only in R, i.e., is
finite valued. This will be of importance in section 3. The main result of this section is
the following.
Theorem 2.8. Let G be a group. Let H < G be countable, such that H ,→h G. Then
there exists a metric de: H × H → R, such that
(a) de ≤ db
(b) de is proper, i.e., every ball of finite radius has finitely many elements.
Proof. There exists aScollection of finite, symmetric (closed under inverses) subsets {Fi }
∞
of H such that H = i=1 Fi and 1 ⊆ F1 ⊆ F2 ⊆ ...
b h) < ∞}.
Let db be the relative metric on H. Let H0 = {h ∈ H | d(1,
Define a function w : H → N as

b h)
d(1,
, if h ∈ H0
w(h) =
min {i | h ∈ Fi } , otherwise

5

Since Fi ’s are symmetric, w(h) = w(h−1 ) for all h ∈ H. Define a function l on H as
follows- for every word u = x1 x2 ...xk in the elements of H, set
l(u) =

k
X

w(xi ).

i=1

Set a length function on H as
|g|w = min{l(u) | u is a word in the elements of H that represents g},
for each g in H. We can now define a metric dw : H × H → N as
dw (g, h) = |g −1 h|w .
It is easy to check that dw is a (finite valued) well-defined metric, which satisfies
dw (1, h) ≤ w(h) for all h ∈ H.
It remains to show that dw is proper. Let N ∈ N. Suppose h ∈ H such that
b h) ≤ N which implies that there are finitely many
w(h) ≤ N . If h ∈ H0 , then d(1,
b
choices for h, since d is proper. If h ∈
/ H0 , then h ∈ Fi for some minimal i. But each Fi
is a finite set, so there are finitely many choices for h. Thus |{h ∈ H | w(h) ≤ N }| < ∞
for all N ∈ N. This implies dw is proper.
Indeed, if y 6= 1 is such that |y|w ≤ n, then there exists a word u, written without
the identity element (which
Pr has weight zero), representing y in the alphabet H such
that u = x1 x2 ...xr and i=1 w(xi ) ≤ n. Since w(xi ) ≥ 1 for every xi 6= 1, r ≤ n.
Further, w(xi ) ≤ n for all i. Thus xi ∈ {x ∈ H | w(x) ≤ n} for all i. So there only
finitely many choices for each xi , which implies there are finitely many choices for y.
b So we can set de = dw .
By definition, dw ≤ d.

2.4

Acylindrically Hyperbolic Groups

In the following theorem ∂ represents the Gromov boundary.
Theorem 2.9. For any group G, the following are equivalent.
(AH1 ) There exists a generating set X of G such that the corresponding Cayley graph
Γ(G,X) is hyperbolic, |∂Γ(G, X)| ≥ 2, and the natural action of G on Γ(G,X) is
acylindrical.
(AH2 ) G admits a non-elementary acylindrical action on a hyperbolic space.
(AH3 ) G contains a proper infinite hyperbolically embedded subgroup.
It follows from the definitions that (AH1 ) ⇒ (AH2 ). The implication (AH2 ) ⇒
(AH3 ) is non-trivial and was proved in [5]. The implication (AH3 ) ⇒ (AH1 ) was
proved in [11].
Definition 2.10. We call a group G acylindrically hyperbolic if it satisfies any of the
equivalent conditions (AH1 )-(AH3 ) from Theorem 2.9.
Lemma 2.11 ([5], Corollary 4.27). Let G be a group, {Hλ }λ∈Λ a collection of subgroups
of G, and X1 and X2 be relative generating sets. Suppose that |X1 ∆X2 | < ∞. Then
{Hλ }λ∈Λ ,→h (G, X1 ) if and only if {Hλ }λ∈Λ ,→h (G, X2 ).

6

Theorem 2.12 ([11], Theorem 5.4). Let G be a group, {Hλ }λ∈Λ a finite collection of
subgroups of G, X a subset of G. Suppose that {Hλ }λ∈Λ ,→h (G, X). Then there exists
Y ⊂ G such that the following conditions hold.
(a) X ⊂ Y
(b) {Hλ }λ∈Λ ,→h (G, Y ). In particular, the Cayley graph Γ(G, Y t H) is hyperbolic.
(c) The action of G on Γ(G, Y t H) is acylindrical.
Definition 2.13. Let (X, dX ) and (Y, dY ) be two metric spaces. A map φ : X → Y is
said to be a (λ,C)-quasi-isometry if there exist constants λ > 1, C > 0 such that
(a)

1
λ dX (a, b)

− C ≤ dY (φ(a), φ(b)) ≤ λdX (a, b) + C, for all a, b ∈ X and

(b) Y is contained in the C-neighborhood of φ(X).
The spaces X and Y are said to be quasi-isometric if such a map φ : X → Y exists.
It is easy to check that being quasi-isometric is an equivalence relation. If the map φ
satisfies only condition (a), then it is said to be a (λ,C)-quasi-isometric embedding.
Definition 2.14. A graph Γ with the combinatorial metric dΓ is said to be a quasi-tree
if it is quasi-isometric to a tree T .
Definition 2.15. A quasi-geodesic is a quasi-isometric embedding of an interval
(bounded or unbounded) I ⊆ R into a metric space X. Note that geodesics are (1, 0)quasi-geodesics. By slight abuse of notation, we may identify the map that defines a
quasi-geodesic with its image in the space.
Theorem 2.16 ([9], Theorem 4.6, Bottleneck property). Let Y be a geodesic metric
space. The following are equivalent.
(a) Y is quasi-isometric to some simplicial tree Γ
(b) There is some µ > 0 so that for all x, y ∈ Y, there is a midpoint m = m(x, y) with
d(x,m) = d(y,m) = 21 d(x,y) and the property that any path from x to y must pass
within less than µ of the point m.
We remark that if m is replaced with any point p on a geodesic between x and y,
then the property that any path from x to y passes within less than µ of the point p still
follows from (a), as proved below in Lemma 2.18. We will need the following lemma.
Lemma 2.17. [[4], Proposition 3.1] For all λ ≥ 1, C ≥ 0, δ ≥ 0, there exists an
R = R(δ, λ, C) such that if X is a δ-hyperbolic space, γ is a (λ, C)-quasi-geodesic in
X, and γ 0 is a geodesic segment with the same end points, then γ 0 and γ are Hausdorff
distance less than R from each other.
Lemma 2.18. If Y is a quasi-tree, then there exists µ > 0 such that for any point z
on a geodesic connecting two points, any other path between the same end points passes
within µ of z.
Proof. Let T be a tree and q : Y → T be the (λ, C) quasi-isometry. Let dY and dT denote
the metrics in the spaces Y and T respectively. Note that since T is 0-hyperbolic, Y is
δ-hyperbolic for some δ.
Let x, y be two points in Y , joined by a geodesic γ. Let z be any point of γ, and
let α be another path from x to y. Let V denote the vertex set of α, ordered according
to the geodesic γ. Take its image q(V ) and connect consecutive points by geodesics (of

7

w
q
≤k

≤k
p1
x

p2
z1

z2

z

y

Figure 3: Corresponding to Lemma 2.20
length at most λ + C) to get a path β in T from q(x) to q(y). Then the unique geodesic
σ in T must be a subset of β. Since q(V ) ⊂ q ◦ α, we get that any point of σ s at most
λ + C from q ◦ α. Also, q ◦ γ is a (λ, C)-quasi-isometric embedding of an interval, and
hence a (λ, C)-quasi-geodesic. Thus, by Lemma 2.17 the distance from q(z) to σ is less
than R = R(0, λ, C).
Let p be the point on σ closest to q(z). There is a point w ∈ Y on α such that
d(q(w), p) ≤ λ + C. Since d(p, q(z)) < R, we have d(q(w), q(z)) ≤ λ + C + R. Thus
d(z, w) ≤ λ2 + 2λC + Rλ.
Thus α must pass within µ = λ2 + 2λC + Rλ of the point z.

2.5

A modified version of Bowditch’s lemma

In this section, Nk (x) denotes the closed k-neighborhood of a point x in a metric space.
The following theorem will be used in Section 5. Part (a) is a simplified form of a
result taken from [7], which is infact derived from a hyperbolicity criterion developed
by Bowditch in [3].
Theorem 2.19. Let Σ be a hyperbolic graph, and ∆ be a graph obtained from Σ by
adding edges.
(a) [3] Suppose there exists M > 0 such that for all vertices x, y ∈ Σ joined by an
edge in ∆ and for all geodesics p in Σ between x and y, all vertices of p lie in an
M-neighborhood of x, i.e., p ⊆ NM (x) in ∆. Then ∆ is also hyperbolic, and there
exists a constant k such that for all vertices x,y ∈ Σ, every geodesic q between x
and y in Σ lies in a k-neighborhood in ∆ of every geodesic in ∆ between x and y.
(b) If, under the assumptions of (a), we additionally assume that Σ is a quasi-tree,
then ∆ is also a quasi-tree.
Lemma 2.20. Let p,q be two paths in a metric space S between points x and y, such
that p is a geodesic and q ⊆ Nk (p). Then p ⊆ N2k (q).
Proof. Let z be any point on p. Let p1 , p2 denote the segments of the geodesic p with
end points x, z and z, y respectively.

8

s
z in Case(2)

s0

≤k
z in Case(1)
≤

≤ µ0

µ0

r

m
x

y
≤ 2k
n

Figure 4: Corresponding to Theorem 2.19
Define a function f : q → R as f (s) = d(s, p1 ) − d(s, p2 ). Then f is a continuous
function. Further, f (x) < 0 and f (y) > 0. By the intermediate value theorem, there
exists a point w on q such that f (w) = 0. Thus d(w, p1 ) = d(w, p2 ) (see Fig.3). Let z1
(resp. z2 ) be a point of p1 (resp. p2 ) such that d(pi , w) = d(zi , w) for i = 1, 2. Then
d(z1 , w) = d(z2 , w). By the hypothesis, d(w, p) = min{d(w, p1 ), d(w, p2 )} ≤ k. So we
get that d(w, p1 ) = d(w, p2 ) ≤ k. Thus d(z1 , z2 ) ≤ 2k, which implies d(z, w) ≤ 2k.
Proof of Theorem 2.19. We proceed with the proof of part (b).
We prove that ∆ is a quasi-tree by verifying the bottleneck property from Theorem
2.16. Let dΣ (resp. d∆ ) denote the distance in the graph Σ (resp. ∆). Note that the
vertex sets of the two graphs are equal.
Let x, y be two vertices. Let m be the midpoint of a geodesic r in ∆ connecting
them. Let s be any path from x to y in ∆. The path s consists of edges of two types
(i) edges of the graph Σ;
(ii) edges added in transforming Σ to ∆ (marked as bold edges on Fig.4).
Let p be a geodesic in Σ between x and y. By Part (a), there exists k such that p is in
the k-neighborhood of r in ∆. Applying Lemma 2.20 , we get a point n on p such that
d∆ (m, n) ≤ 2k.
Let s0 be the path in Σ between x and y, obtained from s by replacing every edge e
of type (ii) by a geodesic path t(e) in Σ between its end points (marked by dotted lines
in Fig.4). Since Σ is a quasi-tree, by Lemma 2.18, there exists µ0 > 0 and a point z on
s0 such that
dΣ (z, n) ≤ µ0 .
Case 1: If z lies on an edge of s of type (i) , then
d∆ (z, m) ≤ d∆ (z, n) + d∆ (n, m) ≤ dΣ (z, n) + d∆ (n, m) ≤ µ0 + 2k.

9

Case 2: If z lies on a path t(e) that replaced an edge e of type (ii), then by Part (a),
d∆ (e− , m) ≤ d∆ (e− , z) + d∆ (z, n) + d∆ (n, m) ≤ k + µ0 + 2k = µ0 + 3k.
Thus the bottleneck property holds for µ = µ0 + 3k > 0.

3

Proof of the main result

Our main result is the following theorem, from which Theorem 1.2 and other corollaries
stated in the introduction can be easily derived (see Section 3.5).
Theorem 3.1. Let {H1 , H2 , ..., Hn } be a finite collection of countable subgroups of a
group G such that {H1 , H2 , ..., Hn } ,→h (G, Z) for some Z ⊂ G. Let K be a subgroup
of G such that Hi ≤ K for all i. Then there exists a subset Y ⊂ K such that:
(a) {H1 , H2 , ..., Hn } ,→h (K, Y )
(b) Γ(K, Y t H) is a quasi-tree, where H =

Fn

i=1

Hi

(c) The action of K on Γ(K, Y t H) is acylindrical
(d) Z ∩ K ⊂ Y

3.1

Outline of the proof

Step 1: In order to prove Theorem 3.1, we first prove the following proposition. It is
distinct from Theorem 3.1 since it does not require the action of K on the Cayley graph
Γ(K, X t H) to be acylindrical.
Proposition 3.2. Let {H1 , H2 , ..., Hn } be a finite collection of countable subgroups of
a group G such that {H1 , H2 , ..., Hn } ,→h G with respect to a relative generating set Z.
Let K be a subgroup of G such that Hi ≤ K for all i. Then there exists X ⊂ K such
that
(a) {H1 , H2 , ..., Hn } ,→h (K, X)
(b) Γ(K, X t H) is a quasi-tree, where H =

Fn

1=1 {Hi }

(c) Z ∩ K ⊂ X
Step 2: Once we have proved Proposition 3.2, we will utilize an ’acylindrification’
construction from [11] to make the action acylindrical, which will prove Theorem 3.1.
The details of this step are as follows.
Proof. By Proposition 3.2, there exists X ⊆ K such that
(a) {H1 , H2 , ..., Hn } ,→h (K, X)
(b) Γ(K, X t H) is a quasi-tree
(c) Z ∩ K ⊂ X
By applying Theorem 2.12 to the above, we get that there exists Y ⊂ K such that
(a) X ⊆ Y
(b) {H1 , H2 , ..., Hn } ,→h (K, Y ). In particular, the Cayley Graph Γ(K, Y t H) is
hyperbolic

10

(c) The action of K on Γ(K, Y t H) is acylindrical.
From the proof of Theorem 2.12 (see [11] for details), it is easy to see that the
Cayley graph Γ(G, Y t H) is obtained from Γ(G, X t H) in a manner that satisfies the
assumptions of Theorem 2.19, with M = 1. Thus by Theorem 2.19, Γ(K, Y t H) is also
a quasi-tree. Further
K ∩ Z ⊂ X ⊂ Y.
Thus Y is the required relative generating set.
We will thus now focus on proving Proposition 3.2. In order to prove this proposition,
will use a construction introduced by Bestvina, Bromberg and Fujiwara in [2]. We
describe the construction below.

3.2

The projection complex

Definition 3.3. Let Y be a set. Suppose that for each Y ∈ Y we have a function
dπY : (Y\{Y } × Y\{Y }) → [0, ∞)
called a projection on Y , and a constant ξ > 0 that satisfy the following axioms for all
Y and all A, B, C ∈ Y\{Y } :
(A1) dπY (A, B) = dπY (B, A)
(A2) dπY (A, B) + dπY (B, C) ≥ dπY (A, C)
(A3) min {dπY (A, B), dπB (A, Y )} < ξ
(A4) #{Y | dπY (A, B) ≥ ξ} is finite
Let J be a positive constant. Then associated to this data we have the projection
complex PJ (Y), which is a graph constructed in the following manner : the set of
vertices of PJ (Y) is the set Y. To specify the set of edges, one first defines a new function
dY : (Y\{Y } × Y\{Y }) → [0, ∞), which can be thought of as a small perturbation of
dπY . The exact definition of dY can be found in [2]. An essential property of the new
function is the following inequality, which is an immediate corollary of [2], Proposition
3.2.
For every Y ∈ Y and every A, B ∈ Y\{Y }, we have
|dπY (A, B) − dY (A, B)| ≤ 2ξ.

(1)

The set of edge of the graph PJ (Y) can now be described as follows : two vertices
A, B ∈ Y are connected by an edge if and only if for every Y ∈ Y\{A, B}, dY (A, B) ≤ J.
This construction strongly depends on the constant J. Complexes corresponding to
different J are not isometric in general.
We would like to mention that if Y is endowed with an action of a group G that
preserves projections, i.e., dπg(Y ) (g(A), g(B)) = dπY (A, B) ), then the action of G can be
extended to an action on PJ (Y). We also mention the following proposition, which has
been proved under the assumptions of Definition 3.3.
Proposition 3.4 ([2], Theorem 3.16). For a sufficiently large J > 0, PJ (Y) is connected
and quasi-isometric to a tree.

11

Definition 3.5. [Nearest point projection] In a metric space (S, d), given a set Y and
a point a ∈ S, we define the nearest point projection as
projY (a) = {y ∈ Y | d(Y, a) = d(y, a)}.
If A, Y are two sets in S, then
projY (A) =

[

projY (a).

a∈A

We note that in our case, since elements of Y will come from a Cayley graph, which is
a combinatorial graph, the nearest point projection will exist. This is because distances
on a combinatorial graph take discrete values in N ∪ {0}. Since this set is bounded
below, we cannot have an infinite strictly decreasing sequence of distances. This may
not be the case if we have a real tree. For example, on the real line, the point 0 has no
nearest point projection to the open interval (0, 1).
We make all geometric considerations in the Cayley graph Γ(G, Z t H). Let dZtH
denote the metric on this graph. Since {H1 , H2 , ..., Hn } ,→h G under the assumptions
of Proposition 3.2, by Remark 4.26 of [5], Hi ,→h G for all i = 1, 2, ..., n. By Theorem
2.8, we can define a finite valued, proper metric dei on Hi , for all i = 1, 2, ..., n, satisfying
dei (x, y) ≤ dbi (x, y) for all x, y ∈ Hi and for all i = 1, 2, ..., n

(2)

We can extend both dbi and dei to all cosets gHi of Hi by setting dei (gx, gy) = dei (x, y)
[ (resp. diam)
] denote the diameter
and dbi (gx, gy) = dbi (x, y) for all x, y ∈ Hi . Let diam
of a subset of Hi or a coset of Hi with respect to the dbi (resp. dei ) metric.
Let
Y = {kHi | k ∈ K, i = 1, 2, ..., n}
be the set of cosets of all Hi in K. We think of cosets of Hi as a subset of vertices of
Γ(G, Z t H).
For each Y ∈ Y, and A, B ∈ Y\{Y }, define
]
dπY (A, B) = diam(proj
Y (A) ∪ projY (B)),

(3)

where projY (A) is defined as in Definition 3.5. The fact that (3) is well-defined will
follow from Lemma 3.6 and Lemma 3.7, which are proved below. We will also proceed
to verify the axioms (A1) − (A4) of the Bestvina-Bromberg-Fujiwara construction in
the above setting.
]
Lemma 3.6. For any Y ∈ Y and any x ∈ G, diam(proj
Y (x)) is bounded.
0
[
Proof. By (2), it suffices to prove that diam(proj
Y (x)) is bounded. Let y, y ∈ projY (x).
0
Then dZtH (x, y) = dZtH (x, y ) = dZtH (x, Y ). Without loss of generality, x ∈
/ Y , else
the diameter is zero.
Let Y = gHi . Let e denote the edge connecting y and y 0 , and labelled by an element
of Hi . Let p and q denote the geodesics between the points x and y, and x and y 0
respectively. (see Fig.5)

12

x
q
p

y

e

y0
Y = gHi

Figure 5: The bold red edge denotes a single edge labelled by an element of H
Consider the geodesic triangle T with sides e, p, q. Since p and q are geodesics
between the point x and Y , e is an isolated component in T , i.e., e cannot be connected
to either p or q. Indeed if e is connected to, say, a component of p, then that would
imply that e+ and e− are in Y , i.e., the geodesic p passes through a point of Y before y.
But then y is not the nearest point from Y to x, which is a contradiction. By Lemma
2.4, dbi (y, y 0 ) ≤ 3C. Hence
[
diam(proj
Y (x)) ≤ 3C.

[
Lemma 3.7. For every pair of distinct elements A, Y ∈ Y, diam(proj
Y (A)) ≤ 4C,
]
where C is the constant as in Lemma 2.4. As a consequence, diam(projY (A)) is
bounded.
Proof. let Y = gHi and A = f Hj . Let y1 , y2 ∈ projY (A). Then there exist a1 , a2 ∈ A
such that dZtH (a1 , y1 ) = dZtH (a1 , Y ) and dZtH (a2 , y2 ) = dZtH (a2 , Y ). Now y1 and
y2 are connected by a single edge e, labelled by an element of Hi , and similarly, a1 and
a2 are connected by an edge f , labelled by an element of Hj (see Fig.6). Let p and q
denote geodesics that connect y1 , a1 and y2 , a2 respectively. We note that p and/or q
may be trivial paths (consisting of a single point), but this does not alter the proof.
Consider e in the quadrilateral Q with sides p, f, q, e. As argued in the previous
lemma, since p (respectively q) is a path of minimal length between the points a1
(respectively a2 ) and Y, e cannot be connected to a component of p or q.
If i = j, then e cannot be connected to f since A 6= Y . If i 6= j, then obviously e
and f cannot be connected. Thus e is isolated in this quadrilateral Q. By Lemma 2.4,
dbi (y1 , y2 ) ≤ 4C. Thus
[
diam(proj
Y (A)) ≤ 4C.

Corollary 3.8. The function dπY defined by (3) is well-defined.
Proof. Since the dei metric takes finite values for all i = 1, 2, ..., n, using Lemma 3.7, we
have that dπY also takes only finite values.

13

a2

A = f Hj
f
a1

q
p
e
y1

y2
Y = gHi

Figure 6: Lemma 3.7
Lemma 3.9. The function dπY defined by (3) satisfies conditions (A1) and (A2) in
Definition 3.3
Proof. (A1) is obviously satisfied. For any Y ∈ Y and any A, B, C ∈ Y\{Y }, by the
triangle inequality, we have that
]
dπY (A, C) = diam(proj
Y (A) ∪ projY (C))
]
]
≤ diam(proj
Y (A) ∪ projY (B)) + diam(projY (B) ∪ projY (C))
= dπY (A, B) + dπY (B, C).
Thus (A2) also holds.
Lemma 3.10. The function dπY from (3) satisfies condition (A3) in Definition 3.3 for
any ξ > 14C, where C is the constant from Lemma 2.4
Proof. By (2), it suffices to prove that
[
[
min{diam(proj
Y (A) ∪ projY (B)), diam(projB (A) ∪ projB (Y ))} < ξ.
Let A, B ∈ Y\{Y } be distinct. Let Y = gHi , A = f Hj and B = tHk .
[
diam(proj
Y (A) ∪ projY (B)) ≤ 14C, then we are done. So let
[
diam(proj
Y (A) ∪ projY (B))) > 14C.

If

(4)

Choose a ∈ A, b ∈ B, and x, y ∈ Y such that dZtH (A, Y ) = dZtH (a, x) and
dZtH (B, Y ) = dZtH (b, y). In particular,
x ∈ projY (A), y ∈ projY (B)

(5)

and b ∈ projB (Y ). Let p, q denote the geodesics connecting a, x and b, y respectively.
Let h1 denote the edge connecting x and y, which is labelled by an element of Hi .

14

a0

b0

r
h0
u
v

A = f Hj

B = tHk

r0

h3

h2
b

s

a

q

p
Hi edges
h1
x

y
Y = gHi

Figure 7: Condition (A3)
By (5), we have that
b
[
[
[
diam(proj
Y (A) ∪ projY (B)) ≤ diam(projY (A)) + diam(projY (B)) + di (x, y).
Combining this with (4) and Lemma 3.7, we get
[
[
[
dbi (x, y) ≥ diam(proj
Y (A) ∪ projY (B)) − diam(projY (A)) − diam(projY (B))
> 14C − 8C = 6C.
Choose any a0 ∈ A and b0 ∈ projB (a0 ), i.e.,: dZtH (a0 , B) = dZtH (a0 , b0 ); (see Fig.7).
(Note that if a0 = a, the following arguments still hold). Let h2 and h3 denote the edges

≤ 4C

≤ 4C

b0

b
projB (A)

c
projB (Y )
B = tHk

Figure 8: Estimating the distance between arbitrary points b and c of projB (A) and
projB (Y ) resp.
15

Y0
x0

A = f Hj

e

y0

a0

b0
r

h0
u

h2

B = tHk
h3

v

b
a
p

q

h1
y

x

Y = gHi
Figure 9: Condition (A4)
connecting a, a0 and b, b0 ; which are labelled by elements of Hj and Hk respectively.
Let r denote the geodesic connecting a0 and b0 . Consider the geodesic hexagon W with
sides p, h1 , q, h3 , r, h2 . Then h1 is not isolated in W , else by Lemma 2.4, dbi (x, y) ≤ 6C,
a contradiction.
Thus h1 is connected to another Hi -component in W . Arguing as in Lemma 3.7,
h1 cannot be connected to a component of p or q. Since A, B, Y are all distinct, h1
cannot be connected to h2 or h3 . So h1 must be connected to an Hi -component on the
geodesic r. Let this edge be h0 with end points u and v as shown in Fig 7. Let s denote
the edge (labeled by an element of Hi ), that connects y, v. Let r0 denote the segment
of r that connects v to b0 . Then r0 is also a geodesic.
Consider the quadrilateral Q with sides r0 , h3 , q, s. As argued before, h3 cannot be
connected to r0 , q or s. Thus h3 is isolated in Q. By Lemma 2.4,
dbk (b, b0 ) ≤ 4C.
Since the above argument holds for any a0 ∈ A and for b0 ∈ projB (A), we have that
ck (b, b0 ) ≤ 4C. Using Lemma 3.7 (see Fig.8), we get that
d
[
diam(proj
B (Y ) ∪ projB (A)) ≤ 4C + 4C = 8C < ξ.

Lemma 3.11. The function dπY defined by (3) satisfies condition (A4) in Definition
3.3, for ξ > 14C, where C is the constant from Lemma 2.4

16

π
[
Proof. If dπY (A, B) ≥ ξ, then by (2), diam(proj
Y (A) ∪ projY (B)) ≥ dY (A, B) ≥ ξ.
Thus it suffices to prove that the number of elements Y ∈ Y satisfying

[
diam(proj
Y (A) ∪ projY (B)) ≥ ξ

(6)

is finite. Let A, B ∈ Y, A = f Hj and B = tHk . Let Y ∈ Y\{A, B}, Y = gHi . Let
[
a0 ∈ A, b0 ∈ projB (a0 ). Arguing as in Lemma 3.10, if is such that diam(proj
Y (A) ∪
projY (B)) ≥ ξ, then for any a ∈ A, b ∈ B, x ∈ projY (a), y ∈ projY (b), we have that
dbi (x, y) > 6C.
Let h1 denote the edge connecting x, y, which is labelled by an element of Hi (see
Fig.9). Let h2 denote the edge connecting a, a0 , which is labelled by an element of Hj
and h3 denote the edge connecting b, b0 , which is labelled by an element of Hk . Let
p be a geodesic between a, x, let q be a geodesic between b, y, and let r be a geodesic
between a0 , b0 . As argued in Lemma 3.10, we can show that h1 cannot be isolated in
the hexagon W with sides p, h1 , q, h2 , r, h3 and must be connected to an Hi -component
of r, say the edge h0 .
We claim that the edge h0 uniquely identifies Y . Indeed, let Y 0 be a member of Y,
with elements x0 , y 0 connected by an edge e (labelled by an element of the corresponding
subgroup). Suppose that e is connected to h0 . Then we must have that Y 0 is also a
coset of Hi . But cosets of a subgroup are either disjoint or equal, so Y = Y 0 . Thus, the
number of Y ∈ Y satisfying (6) is bounded by the number of distinct Hi -components
of r, which is finite.

3.3

Choosing a relative generating set

We now have the necessary details to choose a relative generating set X which will satisfy
conditions (a) and (b) of Proposition 3.2. This set will later be altered slightly to obtain
another relative generating set which will satisfy all three conditions of Proposition 3.2.
We will repeat arguments
Fnsimilar to those from pages 60-63 of [5].
Recall that H =
i=1 Hi , and Z is the relative generating set such that
{H1 , H2 , ..., Hn } ,→h (G, Z). Let PJ (Y) be the projection complex corresponding to
the vertex set Y as specified in section 3.2 and the constant J is as in Proposition
3.4, i.e., PJ (Y) is connected and a quasi-tree. Let dP denote the combinatorial metric
onPJ (Y). Our definition of projections is K- equivariant and hence the action of K on
Y extends to a cobounded action of K on PJ (Y).
In what follows, by considering Hi to be vertices of the projection complex PJ (Y),
we denote by star(Hi ), the set
{kHj ∈ Y |dP (Hi , kHj ) = 1 }.
We choose the set X in the following manner. For all i = 1, 2, ..., n and each edge
e in star(Hi ) in PJ (Y) that connects Hi to kHj , choose an element xe ∈ Hi kHj such
that
dZtH (1, xe ) = dZtH (1, Hi kHj ).
We say that such an xe has type (i, j). Since Hi ≤ K for all i, xe ∈ K. We observe
the following:

17

(a) For each xe of type (i, j) as above, there is an edge in PJ (Y) connecting Hi and
xe Hj . Indeed if xe = h1 kh2 , for h1 ∈ Hi , h2 ∈ Hj , then
dP (Hi , xe Hj ) = dP (Hi , h1 kh2 Hj ) = dP (Hi , h1 kHj )
= dP (h−1
1 Hi , kHj ) = dP (Hi , kHj ) = 1.
(b) For each edge e connecting Hi and kHj , there is a dual edge f connecting Hj
and k −1 Hi . We will choose the elements xe and xf to be mutually inverse. In
particular, the set given by
X = {xe 6= 1|e ∈ star(Hi ), i = 1, 2, ..., n}

(7)

is symmetric, i.e., closed under taking inverses. Obviously, X ⊂ K.
(c) If xe ∈ X is of type (i, j), then xe is not an element of Hi or Hj . Indeed if
xe = h1 kh2 ∈ Hi for some h1 ∈ Hi and some h2 ∈ Hj , then k = hf for some
h ∈ Hi and some f ∈ Hj . Consequently
dZtH (1, Hi kHj ) = dZtH (1, Hi Hj ) = 0 = dZtH (1, xe ),
which implies xe = 1, which is a contradiction to (7).
Lemma 3.12 (cf. Lemma 4.49 in [5]). The subgroup K is generated by X together
with the union of all Hi ’s. Further, the Cayley graph Γ(K, X t H) is quasi-isometric to
PJ (Y), and hence a quasi-tree.
Proof. Let Σ = {H1 , H2 , ..., Hn } ⊆ Y. Let diam(Σ) denote the diameter of the set Σ
in the combinatorial metric dP . Since Σ is a finite set, diam(Σ) is finite. Define
φ : K → Y as φ(k) = kH1
By Property (a) above, if xe ∈ X is of type (i, j),
dP (xe H1 , H1 ) ≤ dP (xe H1 , xe Hj ) + dP (xe Hj , Hi ) + dP (Hi , H1 )
= dP (H1 , Hj ) + 1 + dP (Hi , H1 ) ≤ 2diam(Σ) + 1.
Further, for h ∈ Hi ,
dP (hH1 , H1 ) ≤ dP (hH1 , hHi ) + dP (hHi , H1 )
= dP (H1 , Hi ) + dP (Hi , H1 ) ≤ 2diam(Σ).
Thus for all g ∈ hX ∪ H1 ∪ H2 ... ∪ Hn i, we have
dP (φ(1), φ(g)) ≤ (2diam(Σ) + 1)|g|XtH ,

(8)

where |g|XtH denotes the length of g in the generating set X ∪ H1 ∪ H2 ... ∪ Hn . (We
use this notation for the sake of uniformity).
Now let g ∈ K and suppose dP (φ(1), φ(g)) = r, i.e., dP (H1 , gH1 ) = r. If r = 0, then
H1 = gH1 , thus g ∈ H1 and |g|XtH ≤ 1. If r > 0, consider the geodesic p in PJ (Y)
connecting H1 and gH1 . Let
v0 = H1 = g0 H1 (g0 = 1), v1 = g1 Hλ1 , v2 = g2 Hλ2 , ..., vr−1 = gr−1 Hλr−1 , vr = gH1

18

v2

v1

. . .
v0 = H1

vr = gH1

p

vr−1

Figure 10: The geodesic p
be the sequence of vertices of p, for some λj ∈ {1, 2, ..., n}, and some gi ∈ K (see Fig.10).
Now gi Hλi is connected by a single edge to gi+1 Hλi+1 . Thus dP (gi Hλi , gi+1 Hλi+1 ) =
1, which implies dP (Hλi , gi−1 gi+1 Hλi+1 ) = 1. Then there exists x ∈ X such that
x ∈ Hλi gi−1 gi+1 Hλi+1
and
dZtH (1, x) = dZtH (1, Hλi gi−1 gi+1 Hλi+1 ).
Thus x = hgi−1 gi+1 k for some h ∈ Hλi and some k ∈ Hλi+1 which implies gi−1 gi+1 =
h−1 xk −1 . So |gi−1 gi+1 |XtH ≤ 3, which implies
|g|XtH =

r
Y
i=1

−1
gi−1
gi

≤
XtH

r
X

−1
gi−1
gi

XtH

≤ 3r = 3dP (φ(1), φ(g))

(9)

i=1

The above argument also provides a representation for every element g ∈ K as a
product of elements from X ∪ H1 ∪ H2 ... ∪ Hn . Thus K is generated by the union of
X and all Hi ’s. By (8) and (9), φ is a quasi-isometric embedding of (K, |.|XtH ) into
(PJ (Y), dP ) satisfying
1
|g|XtH ≤ dP (φ(1), φ(g)) ≤ (2diam(Σ) + 1)|g|XtH .
3
Since Y is contained in the closed diam(Σ)-neighborhood of φ(K), φ is a quasi-isometry.
This implies that Γ(K, X t H) is a quasi-tree.
Let dei denote the modified relative metric on Hi associated with the Cayley graph
X denote the relative metric on H associated with
Γ(G, Z tH) from Theorem 2.8. Let dc
i
i
c
X
the Cayley graph Γ(K, X t H). We will now show that d is proper for all i = 1, 2, ..., n.
i

X.
We will use the fact that dei is proper and derive a relation between dei and dc
i

Lemma 3.13 (cf. Lemma 4.50 in [5]). There exists a constant α such that for any
Y ∈ Y and any x ∈ X t H, if
]
diam(proj
Y {1, x}) > α,
then x ∈ Hj and Y = Hj for some j.

19

Proof. We prove the result for
α = max{J + 2ξ, 6C}.
]
Suppose that diam(proj
Y {1, x}) > α and x ∈ X has type (k, l), i.e., there exists an
edge connecting Hk and gHl in PJ (Y), where g ∈ K. We consider three possible cases
and arrive at a contradiction in each case.
Case 1: Hk 6= Y 6= xHl . Then
π
]
diam(proj
Y {1, x}) ≤ dY (Hk , xHl ) ≤ dY (Hk , xHl ) + 2ξ ≤ J + 2ξ ≤ α,

using (1) and the fact that Hk and xHl are connected by an edge in PJ (Y), which
is a contradiction.
Case 2: Hk = Y . Since x ∈
/ Hk , let y ∈ projY (x), i.e., dZtH (x, y) = dZtH (x, Hk ) =
dZtH (x, Y ).

x

1
Hk = Y

y

Figure 11: Case 2
By Lemma 3.6, if dbk (1, y) ≤ 3C, then
b
[
[
[
diam(proj
Y {1, x}) ≤ diam(projY (1)) + diam(projY (x)) + dk (projY (1), projY (x))
≤ 0 + 3C + dbk (1, y) ≤ 6C ≤ α.
Then by (2), we have
]
diam(proj
Y {1, x}) ≤ α,
which is a contradiction. Thus dbk (1, y) > 3C. This implies that 1 ∈
/ projY (x) (see
Fig.11). By definition of the nearest point projection, dZtH (1, x) > dZtH (y, x),
which implies dZtH (1, x) > dZtH (1, y −1 x). Since y −1 x ∈ Hk gHl , we obtain
dZtH (1, x) > dZtH (1, Hk gHl ), which is a contradiction to the choice of x.
Case 3: Y = xHl , Hk 6= Y . This case reduces to Case 2, since we can translate everything
by x−1 .
Thus we must have x ∈ Hj for some j. Suppose that Hj 6= Y . But then
]
]
diam(proj
Y {1, x}) ≤ diam(projY (Hj )) ≤ 4C ≤ α, by Lemma 3.7; which is a contradiction.

20

e

v0 = 1
x1
v1

h = vr
xr
vr−1

x2
v2

x3

...

v3

...

Figure 12: The cycle ep
Lemma 3.14 (cf. Lemma 4.45 in [5]). If Hi = f Hj , then Hi = Hj and f ∈ Hi .
Consequently, if gHi = f Hj , then Hi = Hj and g −1 f ∈ Hi .
Proof. If Hi = f Hj , then 1 = f k for some k ∈ Hj . Then f = k −1 ∈ Hj , which implies
Hi = Hj .
Lemma 3.15 (cf. Theorem 4.42 in [5]). For all i = 1, 2, ..., n and any h ∈ Hi , we have
X (1, h) ≥ d
ei (1, h),
αdc
i
X is proper.
where α is the constant from Lemma 3.13. Thus dc
i
X (1, h) = r. Let e denote the H -edge in the Cayley
Proof. Let h ∈ Hi such that dc
i
i
graph Γ(K, X t H) connecting h to 1, labeled by h−1 . Let p be an admissible (see
Definition 2.1) geodesic path of length r in Γ(K, X t H) connecting 1 and h. Then ep
forms a cycle. Since p is admissible, e is isolated in this cycle.
Let Lab(p) = x1 x2 ...xr for some x1 , x2 , ..., xr ∈ X t H. Let

v0 = 1, v1 = x1 , v2 = x1 x2 , ..., vr = x1 x2 ...xr = h.
Since these are also elements of G, for all k = 1, 2, ..., r we have
]
]
diam(proj
Hi {vk−1 , vk }) = diam(projHi {x1 x2 ...xk−1 , x1 x2 ...xk−1 xk })
]
= diam(proj
Y {1, xk },
where Y = (x1 x2 ...xk−1 )−1 Hi .
]
If diam(proj
Y {1, xk }) > α for some k, then by Lemma 3.13, xk ∈ Hj and Y = Hj
for some j. By Lemma 3.14, Hi = Hj and x1 x2 ...xk−1 ∈ Hj . But then e is not isolated
in the cycle ep, which is a contradiction.
Hence
]
diam(proj
Hi {vk−1 , vk }) ≤ α
for all k = 1, 2, ..., r, which implies
]
dei (1, h) ≤ diam(proj
Hi {v0 , vr }) ≤

r
X

c
X
]
diam(proj
Hi {vj−1 , vj } ≤ rα = αd i (1, h).

j=1

21

1

Hi

e

zHi

z
t

s

f

h
q

p

x

h2

y
Y = gHj

Figure 13: Dealing with elements of Z ∩ K that represent elements of H

3.4

Proof of Proposition 3.2

The goal of this section is to alter our relative generating set X from Section 3.3, so that
we obtain another relative generating set that satisfies all the conditions of Proposition
3.2. To do so, we need to establish a relation between the set X and the set Z. We will
need the following obvious lemma.
Lemma 3.16. Let X and Y be generating sets of G, and supx∈X |x|Y < ∞ and
supy∈Y |y|X < ∞. Then Γ(G, X) is quasi-isometric to Γ(G, Y ). In particular Γ(G, X)
is a quasi-tree if and only if Γ(G, Y ) is a quasi-tree.
Remark 3.17. The above lemma implies that if we change a generating set by adding
finitely many elements, then the property that the Cayley graph is a quasi-tree still
holds.
We also need to note that from (1) in Definition 3.3, it easily follows that
dY (A, B) ≤ dπY (A, B) + 2ξ.

(10)

Lemma 3.18. For a large enough J, the set X constructed in Section 3.3 satisfies
the following property : If z ∈ Z ∩ K does not represent any element of Hi for all
i = 1, 2, ..., n, then z ∈ X.
Proof. Recall that dZtH denotes the combinatorial metric on Γ(G, Z t H). Let z ∈
Z ∩ K be as in the statement of the lemma. Then z ∈ Hi zHi for all i and 1 ∈
/ Hi zHi .
Thus
dZtH (1, Hi zHi ) ≥ 1 = dZtH (1, z) ≥ dZtH (1, Hi zHi ),
which implies
dZtH (1, Hi zHi ) = dZtH (1, z) for all i.
In order to prove z ∈ X, we must show that Hi and zHi are connected by an edge
in PJ (Y). By Definition 3.3, this is true if
dY (Hi , zHi ) ≤ J for all Y 6= Hi , zHi .

22

z
1

z
h1

Figure 14: Bigons in the Cayley graph
In view of (10), we will estimate dπY (Hi , zHi ).
Let dZtH (h, x) = dZtH (Hi , Y ) and dZtH (f, y) = dZtH (zHi , Y ) for some h ∈
Hi , f ∈ zHi and for some x, y ∈ Y = gHj . Let p be a geodesic connecting h and x;
and q be a geodesic connecting y and f . Let h2 denote the edge connecting x and y,
labelled by an element of Hj . Similarly, let s, t denote the edges connecting h, 1 and
z, f respectively, that are labelled by elements of Hi . Let e denote the edge connecting
1 and z, labelled by z. Consider the geodesic hexagon W with sides p, h2 , q, t, e, s (see
Fig.13).
Arguing as in Lemma 3.10), we can show that h2 cannot be connected to q , p, s or
t. Since z does not represent any element of Hi for all i, h2 cannot be connected to e.
Thus, h2 is isolated in W . By Lemma 2.4, dbj (x, y) ≤ 6C. By Lemma 3.7,
dY (Hi , zHi ) ≤ dπY (Hi , zHi ) ≤ 14C + 2ξ.
So we conclude that by taking the constant J to be sufficiently large so that Proposition
3.4 holds and J exceeds 14C + 2ξ, we can ensure that z ∈ X and the arguments of the
previous section still hold.
Lemma 3.19. There are only finitely many elements of Z ∩ K that can represent an
element of Hi for some i ∈ {1, 2, ..., n}.
Proof. Let z ∈ Z ∩ K represent an element of Hi for some i = 1, 2, ..., n. Then in the
Cayley graph Γ(G, Z t H), we have a bigon between the elements 1 and h, where one
edge is labelled by z, and the other edge is labelled by an element of Hi , say h1 (see
Rem. 2.2 and Fig.14).
fi (1, 1), i.e., the ball of
This implies that dbi (1, z) ≤ 1, so dei (1, z) ≤ 1. But then z ∈ B
radius 1 in the subgroup Hi in the relative metric, centered at the identity. But this is
a finite ball. Take
n
[
fi (1, 1) .
ρ=
B
i=1

Then z has at most ρ choices, which is finite.
By Lemma 3.19 and by selecting the constant J as specified in Lemma 3.18, we
conclude that the set X from Section 3.3 does not contain at most finitely many elements
of Z ∩ K. By adding these finitely many remaining elements of Z ∩ K to X, we
obtain a new relative generating set X 0 such that |X 0 ∆X| < ∞. By Proposition 2.11,
{H1 , H2 , ..., Hn } ,→h (K, X 0 ) and Z ∩ K ⊂ X 0 . By Remark 3.17, Γ(K, X 0 t H) is also
a quasi-tree. Thus X 0 is the required set in the statement of Proposition 3.2, which
completes the proof.

23

3.5

Applications of Theorem 3.1

In order to prove Theorem 1.2, we first need to recall the following definitions.
Definition 3.20 (Loxodromic element). Let G be a group acting on a hyperbolic space
S. An element g ∈ G is called loxodromic if the map Z → S defined by n → g n s is a
quasi-isometric embedding for some (equivalently, any) s ∈ S.
Definition 3.21 (Elementary subgroup, Lemma 6.5 in [5]). Let G be a group acting
acylindrically on a hyperbolic space S, g ∈ G a loxodromic element. Then g is contained
in a unique maximal elementary subgroup E(g) of G given by
E(g) = {h ∈ G |dHau (l, h(l)) < ∞ },
where l is a quasi-geodesic axis of g in S.
Corollary 3.22. A group G is acylindrically hyperbolic if and only if G has an acylindrical and non-elementary action on a quasi-tree.
Proof. If G has an acylindrical and non-elementary action on a quasi-tree, by Theorem
2.9, G is acylindrically hyperbolic. Conversely, let G be acylindrically hyperbolic, with
an acylindrical non-elementary action on a hyperbolic space X. Let g be a loxodromic
element for this action. By Lemma 6.5 of [5] the elementary subgroup E(g) is virtually
cyclic and thus countable. By Theorem 6.8 of [5], E(g) is hyperbolically embedded in G.
Taking K = G and E(g) to be the hyperbolically embedded subgroup in the statement
of Theorem 3.1 now gives us the result. Since E(g) is non-degenerate, by [11], Lemma
5.12, the resulting action of G on the associated Cayley graph Γ(G, X t E(g)) is also
non-elementary.
The following corollary is an immediate consequence of Theorem 3.1.
Corollary 3.23. Let {H1 , H2 , ..., Hn } be a finite collection of countable subgroups of a
group G such that {H1, H2 , ..., Hn } ,→h G. Let K be a subgroup of G. If Hi ≤ K for
all i = 1, 2, ..., n, then {H1 , H2 , ..., Hn } ,→h K.
Definition 3.24. Let (M, d) be a geodesic metric space, and  > 0 a fixed constant. A
subset S ⊂ M is said to be -coarsely connected if there for any two points x, y in S,
there exist points x0 = x, x1 , x2 , ..., xn−1 , xn = y in S such that for all i = 0, ..., n − 1,
d(xi , xi+1 ) ≤ .
Further we say that S is coarsely connected if it is -coarsely connected for some  > 0.
Recall that we denote the closed σ neighborhood of S by S +σ .
Definition 3.25. Let (M, d) be a geodesic metric space, and σ > 0 a fixed constant. A
subset S ⊂ M is said to be σ-quasi-convex if for any two points x, y in S, any geodesic
connecting x and y is contained in S +σ . Further, we say that S is quasi-convex if it is
σ-quasi-convex for some σ > 0.
Corollary 3.26. Let H be a finitely generated subgroup of an acylindrically hyperbolic
group G. Then there exists a subset X ⊂ G such that
(a) Γ(G, X) is hyperbolic, non-elementary and acylindrical

24

(b) H is quasi-convex in Γ(G, X)
To prove the above corollary, we need the following two lemmas.
Lemma 3.27. Let T be a tree, and let Q ⊂ T be -coarsely connected. Then Q is
-quasi-convex.
Proof. Let  > 0 be the constant from Definition 3.24. Let x, y be two points in Q, and
p be any geodesic between them. Then there exist points x0 = x, x1 , x2 , ..., xn−1 , xn = y
in Q such that for all i = 0, ..., n−1, d(xi , xi+1 ) ≤ . Let pi denote the geodesic segments
between xi and xi+1 for all i = 0, 1, ..., n − 1. Since T is a tree, we must have that
p⊆

n−1
[

pi .

i=0

By definition, for all i = 0, 1, ..., n − 1, pi ⊆ B(xi , ), the ball of radius  centered at xi .
Since xi ∈ Q for all i = 0, 1, ..., n − 1, we obtain
pi ⊆ Q+ .
This implies p ⊆ Q+ .
Lemma 3.28. Let Γ be a quasi-tree, and S ⊂ Γ be coarsely connected. Then S is
quasi-convex.
Proof. Let T be a tree, and dΓ and dT denote distances in Γ and T respectively. Let
δ > 0 be the hyperbolicity constant of Γ. Let q : T → Γ be a (λ, C)-quasi-isometry. i.e.,
−C +

1
dT (a, b) ≤ dΓ (q(a), q(b)) ≤ λdT (a, b) + C.
λ

Let  > 0 be the constant from Definition 3.24 for S. Set Q = q −1 (S). Then Q ⊂ T . It
is easy to check that Q is ρ-coarsely connected with constant ρ = λ( + C). By Lemma
3.27, Q is ρ-quasi-convex.
Let x, y be two points in S, and p be a geodesic between them. Choose points a, b in
Q such that q(a) = x and q(b) = y. Let r denote the (unique) geodesic in T between a
and b. Since Q is ρ-quasi-convex, we have
r ⊆ Q+ρ .
Set σ = λρ + C. Then
q(r) ⊆ S +σ .
Further q ◦ r is a quasi-geodesic between x and y. By Lemma 2.17, there exists a
constant R(= R(λ, C, δ)) such that q(r) and p are Hausdorff distance less than R from
each other. This implies that p ⊆ S +(R+σ) . Thus S is quasi-convex.
Proof of Corollary 3.26. By Corollary 3.22, there exists a generating set X of G such
that Γ(G, X) is a quasi-tree (hence hyperbolic), and the action of G on Γ(G, X) is
acylindrical and non-elementary. Let dX denote the metric on Γ(G, X) induced by the
generating set X. Let H = hx1 , x2 , ..., xn i. Set
 = max{dX (1, x±1
i ) | i = 1, 2, ..., n}.

25

We claim that HQis coarsely connected with constant . Indeed if u, v are elements of
k
±1
H, then u−1 v = j=1 wj , where wj ∈ {x±1
1 , ..., xn }. Set
z0 = u, z1 = uw1 , ..., zk−1 = uw1 w2 ...wk−1 , zk = v.
Clearly zi ∈ H for all i = 0, 2, ..., k − 1. Further
dX (zi , zi+1 ) = dX (1, wi+1 ) ≤ 
for all i = 0, 1, 2, ..., k − 1. By Lemma 3.28, H is quasi-convex in Γ(G, X).

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Sahana Balasubramanya
Department of Mathematics, Vanderbilt University
Nashville, TN -37240, U.S.A
Email : sahana.balasubramanya@vanderbilt.edu

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