1 Introduction
A group G is acylindrically hyperbolic if it admits a nonelementary, acylindrical action on a hyperbolic space. An alternative characterization is that G is acylindrically hyperbolic if and only if G contains a hyperbolically embedded subgroup H, denoted
$H\hookrightarrow _h G$
, and we will give a characterization from [Reference Martínez-Pedroza and Rashid13] in Proposition 3.1.
The class of acylindrically hyperbolic groups generalizes the classes of nonelementary hyperbolic and relatively hyperbolic groups while sharing many similar properties [Reference Minasyan and Osin17]. In spite of this, there are still foundational questions that remain open, for instance, it is known that a group being hyperbolic or relatively hyperbolic is invariant under quasi-isometry [Reference Druţu8, Reference Gromov9], but the corresponding question for acylindrical hyperbolicity is still open.
Question 1.1 [Reference Osin17, Question 2.20(a)]
Is the class of finitely generated acylindrically hyperbolic groups closed under quasi-isometry?
Some partial results are known, for instance, acylindrical hyperbolicity passes to finite-index subgroups and is preserved by quotienting out a finite normal subgroup [Reference Minasyan and Osin15]. If the group is
${\mathcal {A}}{\mathcal {H}}$
-accessible, then acylindrical hyperbolicity can be passed to finite extensions [Reference Minasyan and Osin16]. The property of being
${\mathcal {A}}{\mathcal {H}}$
-accessible also passes to finite-index overgroups [Reference Balasubramanya3]. However, not every finitely presented acylindrically hyperbolic group is
${\mathcal {A}}{\mathcal {H}}$
-accessible [Reference Abbott, Balasubramanya and Osin1, Theorem 2.18]. Some experts in the field do not expect a complete positive answer to Question 1.1.
This article relies on the notion of quasi-isometry of pairs, and our results provide technical conditions to ensure that a quasi-isometry of pairs carries the property of being a hyperbolically embedded subgroup.
Definition 1.1 (Quasi-isometry of pairs)
Let X and Y be metric spaces, and let
$\mathcal {A}$
and
$\mathcal {B}$
be collections of subspaces of X and Y, respectively. A quasi-isometry
$q\colon X\to Y$
is a quasi-isometry of pairs
$q\colon (X,\mathcal {A}) \to (Y,\mathcal {B})$
if there is
$M>0$
:
-
(1) For any
$A\in \mathcal {A}$
, the set
$\{ B\in \mathcal {B} \colon \operatorname {\mathrm {\mathsf {hdist}}}_Y(q(A), B) <M \}$
is nonempty. -
(2) For any
$B\in \mathcal {B}$
, the set
$\{ A\in \mathcal {A} \colon \operatorname {\mathrm {\mathsf {hdist}}}_Y(q(A), B) <M \}$
is nonempty.
In this case, if
$q\colon X \to Y$
is an
$(L,C)$
-quasi-isometry, then
$q\colon (X, \mathcal {A})\to (Y,\mathcal {B})$
is called an
$(L,C,M)$
-quasi-isometry. If there is a quasi-isometry of pairs
$(X,\mathcal {A}) \to (Y,\mathcal {B})$
, we say that
$(X,\mathcal {A})$
and
$(Y,\mathcal {B})$
are quasi-isometric pairs.
We specialize the previous definition to the case of finitely generated groups with finite collections of subgroups as follows.
Definition 1.2 (Quasi-isometry of group pairs)
Consider two pairs
$(G, \mathcal {P})$
and
$(H, \mathcal {Q})$
where G and H are finitely generated groups with chosen word metrics
$\operatorname {\mathrm {\mathsf {dist}}}_G$
and
$\operatorname {\mathrm {\mathsf {dist}}}_H$
. Denote the Hausdorff distance between subsets of H by
$\operatorname {\mathrm {\mathsf {hdist}}}_H$
. An
$(L,C)$
-quasi-isometry
$q\colon G \to H$
is an
$(L,C,M)$
-quasi-isometry of pairs
$q\colon (G, \mathcal {P})\to (H, \mathcal {Q})$
if the relation
$$ \begin{align*} \dot{q} = \{ (A, B)\in G/\mathcal{P} \times H/\mathcal{Q} \colon \operatorname{\mathrm{\mathsf{hdist}}}_H(q(A), B) <M \} \end{align*} $$
satisfies that the projections into
$G/\mathcal {P}$
and
$H/\mathcal {Q}$
are surjective.
Example 1.1 (Quasi-isometry of pairs and finite extensions)
Let H be a finite index normal subgroup of finitely generated group G, and let
${\mathcal {Q}}$
be a finite collection of subgroups of H. Then the inclusion
$(H,{\mathcal {Q}}) \hookrightarrow (G,{\mathcal {Q}})$
is a quasi-isometry of pairs if the collection
$\{hQh^{-1} \colon h\in H \text { and } Q\in {\mathcal {Q}}\}$
is invariant under conjugation by G (see Proposition 4.1).
Recall that the commensurator of a subgroup P of a group G is the subgroup
$$\begin{align*}Comm_G(P)=\{g\in G \colon P\cap gPg^{-1} \text{ is a finite index subgroup of } P \text{ and } gPg^{-1}\}. \end{align*}$$
Definition 1.3 (Refinements)
Let
${\mathcal {P}}$
be a collection of subgroups of group G. A refinement
${\mathcal {P}}^\ast $
of
${\mathcal {P}}$
is a set of representatives of conjugacy classes of the collection of subgroups
$$\begin{align*}\{\operatorname{{\mathrm{Comm}}}_G(gPg^{-1}) \colon P\in{\mathcal{P}} \text{ and } g\in G \}. \end{align*}$$
Example 1.2 (Refinements and qi of pairs)
Let
${\mathcal {Q}}$
be a finite collection of subgroups of a finitely generated group H, and let
${\mathcal {Q}}^\ast $
be a refinement. If each
$Q\in {\mathcal {Q}}$
is finite index in
$\operatorname {{\mathrm {Comm}}}_H(Q)$
, then the identity map on G is a quasi-isometry of pairs
$(H,{\mathcal {Q}})\to (H,{\mathcal {Q}}^\ast )$
.
Example 1.3 (Refinements and finite extensions)
Let A be a group, let
$\mathcal {H}$
be an almost malnormal collection of infinite subgroups, and let
$F\leq \mathsf {Aut}(A)$
be a finite subgroup. If F acts freely on
$\mathcal {H}$
and
$\mathcal {H}_F$
is a collection of representatives of F-orbits in
$\mathcal {H}$
, then a refinement of
$\mathcal {H}$
in
$A\rtimes F$
is
$\mathcal {H}_F$
.
Definition 1.4 (Reduced collections)
A collection of subgroups
${\mathcal {P}}$
of a group G is reduced if for any
$P,Q\in {\mathcal {P}}$
and
$g\in G$
, if P and
$gQg^{-1}$
are commensurable, then
$P=Q$
and
$g\in P$
.
Our first result, Theorem A, describes a strategy to obtain positive results to Question 1.1. For a group G with a generating set S, let
$\Gamma (G,S)$
denote the corresponding Cayley graph (see Definition 2.3).
Theorem A (Theorem 3.1)
Let
$q\colon G\to H$
be a quasi-isometry of finitely generated groups, let
${\mathcal {P}}$
and
${\mathcal {Q}}$
be finite collections of subgroups of G and H, respectively, and let S and T be (not necessarily finite) generating sets of G and H, respectively. Suppose that:
-
(1)
$q\colon (G,{\mathcal {P}}) \to (H,{\mathcal {Q}})$
is a quasi-isometry of pairs and -
(2)
$q\colon \Gamma (G,S) \to \Gamma (H,T)$
is a quasi-isometry.
The following statements hold:
-
(1) If
${\mathcal {P}}$
and
${\mathcal {Q}}$
are reduced collections in G and H, respectively, then
${\mathcal {P}}\hookrightarrow _h (G,S)$
if and only if
${\mathcal {Q}}\hookrightarrow _h (G,T)$
. -
(2) If
${\mathcal {Q}}$
contains only infinite subgroups and
${\mathcal {Q}}\hookrightarrow _h (H,T)$
, then
${\mathcal {P}}^*\hookrightarrow _h (G,S)$
.
1.1 Qi-characteristic collections
The first numbered hypothesis of Theorem A raises the following general problem: Given a finite collection of subgroups
${\mathcal {Q}}$
of a group H and a quasi-isometry
$q\colon G\to H$
of finitely generated groups, is there a collection
${\mathcal {P}}$
of subgroups of G such that
$q\colon (G,{\mathcal {P}})\to (H,{\mathcal {Q}})$
is a quasi-isometry of pairs?
This problem was studied in [Reference Martínez-Pedroza and Sánchez Saldaña14] where the notion of qi-characteristic collection is introduced and it is proved that if the collection
${\mathcal {Q}}$
is qi-characteristic in H, then any quasi-isometry of finitely generated groups induces a collection
${\mathcal {P}}$
.
Definition 1.5 (Qi-characteristic [Reference Martínez-Pedroza and Sánchez Saldaña14])
A collection of subgroups
$\mathcal {P}$
of a finitely generated group G is quasi-isometrically characteristic (or shorter qi-characteristic) if
$\mathcal {P}$
is finite; each
$P\in \mathcal {P}$
has finite index in its commensurator; and for every
$L\geq 1$
and
$C\geq 0$
, there is
$M=M(G, \mathcal {P}, L,C)\geq 0$
such that every
$(L,C)$
-quasi-isometry
$q\colon G\to G$
is an
$(L,C,M)$
-quasi-isometry of pairs
$q\colon (G,\mathcal {P})\to (G,\mathcal {P})$
.
Example 1.4. The argument by Behrstock, Druţu, and Mosher proving quasi-isometric rigidity of relative hyperbolicity with respect to nonrelatively hyperbolic groups (NRH groups) shows that if H is hyperbolic group relative to a collection
${\mathcal {Q}}$
of NRH subgroups, then
${\mathcal {Q}}$
is qi-characteristic [Reference Behrstock, Druţu and Mosher4, Theorems 4.1 and 4.8]. Another example is provided by mapping class groups. Ruling out a few surfaces of low complexity, any self-quasi-isometry of the mapping class group is at uniform distance from left multiplication by an element of the group (see the work of Behrstock, Kleiner, Minsky, and Mosher [Reference Behrstock, Kleiner, Minsky and Mosher6, Theorem 1.1]). As a consequence, the hyperbolically embedded (virtually cyclic) subgroup generated by a pseudo-Anosov is qi-characteristic. More generally, any finite collection of subgroups of such mapping class groups are qi-characteristic.
Corollary B Let G and H be finitely generated groups, let T be a generating set of H, let
${\mathcal {Q}}$
be a finite collection of subgroups of H such that
${\mathcal {Q}}\hookrightarrow _h (H,T)$
, and let
$q\colon G \to H$
be a quasi-isometry. If:
-
(1)
${\mathcal {Q}}$
is a qi-characteristic collection of subgroups of H and -
(2) there is a generating set
$S\subset G$
such that
$q\colon \Gamma (G,S) \to \Gamma (H,T)$
is a quasi-isometry,
then there is a finite collection
${\mathcal {P}}$
of subgroups of G such that
${\mathcal {P}}\hookrightarrow _h (G,S)$
and
$q\colon (G,{\mathcal {P}}) \to (H,{\mathcal {Q}})$
is a quasi-isometry of pairs.
Proof Without loss of generality, assume that all subgroups in
${\mathcal {Q}}$
are proper infinite subgroups. Note that removing finite subgroups from
${\mathcal {Q}}$
preserves being qi-characteristic and that
${\mathcal {Q}}\hookrightarrow _h (H,T)$
. On the other hand, if
${\mathcal {Q}}$
contains H, then the theorem is trivial by taking
${\mathcal {P}}$
the collection that contains only G and S any finite generating set of G. Since
${\mathcal {Q}}$
is qi-characteristic, the quasi-isometry
$q\colon G\to H$
induces a finite collection
${\mathcal {P}}$
such that
$q\colon (G,{\mathcal {P}})\to (H,{\mathcal {Q}})$
is a quasi-isometry of pairs, and this is precisely [Reference Martínez-Pedroza and Sánchez Saldaña14, Theorem 1.1]. Then the second statement of Theorem A and
${\mathcal {Q}}\hookrightarrow _h (H,T)$
imply that
${\mathcal {P}}^\ast \hookrightarrow _h (G,S)$
.
1.2 Uniform quasi-actions
The second numbered hypothesis of Theorem A raises the problem: Given a group H with a generating set T and a quasi-isometry
$q\colon G\to H$
of finitely generated groups, is there a generating set
$S\subset G$
such that
$q\colon \Gamma (G,S) \to \Gamma (H,T)$
is a quasi-isometry of Cayley graphs?
We show that a positive answer to this question is equivalent to asking that the quasi-action of G on H induced by q is T-uniform in the following sense (see Proposition C).
Definition 1.6 (Uniform induced quasi-action)
Let G and H be finitely generated groups, and let
$q\colon G\to H$
be a quasi-isometry with quasi-inverse
$\bar q$
. Let
$T\subset H$
be a generating set (possibly infinite). We say that the quasi-action of G on H induced by q is uniform with respect to T if there are constants
$L\geq 1$
,
$C\geq 0$
such that, for each
${g\in G}$
, the function
$q_g\colon H\to H$
given by
$q_g(h)= q(g\cdot \bar q(h))$
is an
$(L,C)$
-quasi-isometry
$q_g\colon \Gamma (H,T)\to \Gamma (H,T)$
.
Example 1.5 (Uniform quasi-action and finite extensions)
Let H be a finite index normal subgroup of finitely generated group G, and let T be a generating set of H invariant under conjugation by G. The G-action by conjugation on H preserves the word metric induced by T. On the other hand, any transversal R of H in G induces a quasi-isometry
$q\colon G \to H$
given by
$q(hg)=h$
for
$h\in H$
and
$g\in R$
. In this case, the quasi-action of G on H induced by q is uniform with respect to T (see Lemma 2.1).
Proposition C (Proposition 2.1)
Let G and H be groups with finite generating sets
$S_0$
and
$T_0$
, and let
$q\colon \Gamma (G,S_0) \to \Gamma (H,T_0)$
be a quasi-isometry. Let
$T\subset H$
containing
$T_0$
. The following statements are equivalent:
-
(1) The quasi-action of G on H induced by q is uniform with respect to T.
-
(2) There is
$S\subset G$
containing
$S_0$
such that
$q\colon \Gamma (G,S) \to \Gamma (H,T)$
is a quasi-isometry.
Corollary D Let G and H be finitely generated groups with finite collections of infinite subgroups
${\mathcal {P}}$
and
${\mathcal {Q}}$
, respectively. Suppose that
$q\colon (G,{\mathcal {P}}) \to (H,{\mathcal {Q}})$
is a quasi-isometry of pairs inducing a T-uniform quasi-action of G on H. If
${\mathcal {Q}}\hookrightarrow _h(H,T)$
, then
${\mathcal {P}}^\ast \hookrightarrow _h G$
.
Proof Since the quasi-action of G on H induced by q is T-uniform, Proposition C implies that there is a generating set S of G such that
$q\colon \Gamma (G, S) \to \Gamma (H,T)$
is a quasi-isometry. Then the second statement of Theorem A and
${\mathcal {Q}}\hookrightarrow _h (H,T)$
imply that
${\mathcal {P}}^\ast \hookrightarrow _h (G,S)$
.
Let us remark that for this last corollary, in the case that T is finite, then there is a finite
$S\subset G$
such that
$\mathcal {P} \hookrightarrow _h (G, S)$
; this case is implied by the results on quasi-isometric rigidity of relative hyperbolicity in [Reference Behrstock, Druţu and Mosher4].
1.3 Finite extensions
The following application is a particular instance of Theorem 4.1 in the main body of the article.
Theorem E (Theorem 4.1)
Let H be a finite index normal subgroup of a finitely generated group G, and let
${\mathcal {Q}}$
be a finite collection of infinite subgroups of H such that
${\mathcal {Q}}\hookrightarrow _h (H,T)$
. Suppose that:
-
(1) The set T is invariant under conjugation by G.
-
(2) The collection
$\{hQh^{-1} \colon h\in H \text { and } Q\in {\mathcal {Q}}\}$
is invariant under conjugation by G.
If
${\mathcal {Q}}^*$
is a refinement of
${\mathcal {Q}}$
in G, then
${\mathcal {Q}}^*\hookrightarrow _h G$
.
Example 1.6. Let
$G= \langle a,b,t \colon tat^{-1}=b,\ t^2=1 \rangle \cong F_2\rtimes \mathbb {Z}_2$
, let
$H=\langle a,b\rangle $
, and let
${{\mathcal {Q}}=\{ \langle a\rangle , \langle b\rangle \}}$
. Note that
${\mathcal {Q}}\hookrightarrow _h H$
, and, for instance, one can take
${\mathcal {Q}}^\ast = \{\langle a \rangle \}$
and observe that
${\mathcal {Q}}^* \hookrightarrow _h G$
. In contrast, for
${\mathcal {Q}}_0=\{ \langle a \rangle \} \hookrightarrow _h H$
, the theorem does not apply since the conjugates of
$\langle a\rangle $
in H are not invariant under conjugation by elements of G.
The next result illustrates concrete examples where Theorem E applies.
Theorem F (Theorem 5.1)
Let A be a finitely generated group with a (not necessarily finite) generating set T, and let
$\mathcal {H}$
be a finite collection of infinite subgroups such that
$\mathcal {H}\hookrightarrow _h (A,T)$
. If
$F\leq \mathsf {Aut}(A)$
is finite, T and
$\mathcal {H}$
are F-invariant, and the F-action on
$\mathcal {H}$
is free, then
${\mathcal {H}}_F\hookrightarrow _h(A\rtimes F,T\cup F)$
where
${\mathcal {H}}_F$
is collection of representatives of F-orbits in
${\mathcal {H}}$
.
Example 1.7. Let
with each
$B_i$
isomorphic to a fixed finitely generated group B. Let
$F=\mathbb {Z}_n$
act on A by cyclically permuting the copies of B. Consider the generating set of A given by
$T=\bigcup _{i=1}^nB_i\backslash \{1\}$
, then T is F-invariant. Now, the collection
${\mathcal {H}}=\{B_1,\dots ,B_n\}$
is hyperbolically embedded into
$(A,T)$
and F acts freely by conjugation on
${\mathcal {H}}$
. All of the hypotheses of the previous theorem have been verified, so we conclude that
$B_1\hookrightarrow _h(A\rtimes F,T\cup F)$
.
1.4 Organization
The rest of the article is divided into five sections. Section 2 is on quasi-actions; it contains the proof of Proposition C as well as some corollaries. The proof of Theorem A is the content of Section 3. Then Sections 4 and 5 contain the proofs of Theorems E and F, respectively. Finally , Section 6 contains some questions and discussion about related to the results in this article and the definition of a quasi-isometry of pairs.
2 Uniform quasi-actions
Definition 2.1 (Uniform quasi-action)
Let G be a group, and let X be a metric space. Let
$\mathsf {QI(X)}$
denote the set of quasi-isometries
$X\to X$
. A function
$G\to \mathsf {QI(X)}$
,
$g\mapsto f_g$
, is a quasi-action if there is
$K\geq 0$
such that for any
$g_1,g_2\in G$
:
-
(1) the map
$f_{g_1g_2}$
is at distance at most K from the map
$f_{g_1}\circ f_{g_2}$
in the
$L_\infty $
-distance and -
(2) the map
$f_{g_1}\circ f_{g_1^{-1}}$
is at distance at most K from the identity.
The quasi-action
$G\to \mathsf {QI}(X)$
is uniform if there are constants
$L\geq 1$
and
$C\geq 0$
such that, for any
$g\in G$
, the map
$f_g$
is an
$(L,C)$
-quasi-isometry.
It is well known that a quasi-isometry
$q\colon G\to H$
of finitely generated groups induces a uniform quasi-action of G on H:
Definition 2.2 (Uniform quasi-action induced by a quasi-isometry)
Let G be a group with a word metric induced by a finite generating set, let X be a metric space, and let
$q\colon G \to X $
and
$\bar q\colon X \to G$
be
$(L_0,C_0)$
-quasi-isometries such that
$q\circ \bar q$
and
$\bar q\circ q$
are at distance less than
$C_0$
from the identity maps on X and G, respectively. For
$g\in G$
, let
$$\begin{align*}L_g\colon G\to G, \qquad x\mapsto gx, \end{align*}$$
and let
$$\begin{align*}q_g\colon X\to X,\qquad q_g= q \circ \mathbf{g}\circ\bar q. \end{align*}$$
It is an exercise to verify that there are constants
$L\geq 1$
and
$C\geq 0$
such that:
-
• For
$g\in G$
,
$q_g\colon X\to X$
is an
$(L,C)$
-quasi-isometry. -
• (G quasi-acts on X) For
$g_1,g_2\in G$
, the map
$q_{g_1g_2}$
is at distance at most C from the map
$q_{g_1}\circ q_{g_2}$
, and the map
$q_{g_1}\circ q_{g_1^{-1}}$
is at distance at most C from the identity. -
• (G acts
$C_0$
-transitively on X) For every
$x ,y\in X$
, there is
$g\in G$
such that
$\operatorname {\mathrm {\mathsf {dist}}}_G(x, q_g(y))\leq C$
.
The map
$G\to \mathsf {QI}(X)$
given by
$g\mapsto q_g$
is called the uniform quasi-action of G on X induced by q and
$\bar q$
.
Remark 2.1 (Equivalence of Definitions 1.6 and 2.2)
In the context of Definition 1.6, if the induced quasi-action of G on H is uniform with respect to T, then
$G\to \mathsf {QI}(\Gamma (H,T))$
given by
$g\mapsto q_q$
is a uniform quasi-action in the sense of Definition 2.2. Indeed, since T contains a finite generating set of H, there is
$M>0$
such that
$\operatorname {\mathrm {\mathsf {dist}}}_{(H,T)}\leq M\operatorname {\mathrm {\mathsf {dist}}}_{(H,T_0)}$
. Hence, if two functions
$H\to H$
are at finite
$L_\infty $
-distance with respect to
$\operatorname {\mathrm {\mathsf {dist}}}_{(H,T_0)}$
, then the same holds for
$\operatorname {\mathrm {\mathsf {dist}}}_{(H,T)}$
.
Definition 2.3 (Cayley graph)
Let G be a group with a generating set S. The Cayley graph
$\Gamma (G,S)$
of G with respect to S is the G-graph with vertex set G and edge set
$\{ \{g, gs\}\colon g\in G,\ s\in S \}$
.
Proposition 2.1 (Proposition C)
Let G and H be groups with finite generating sets
$S_0$
and
$T_0$
, and let
$q\colon \Gamma (G,S_0) \to \Gamma (H,T_0)$
be a quasi-isometry. Let
$T\subset H$
containing
$T_0$
. The following statements are equivalent:
-
(1) The quasi-action of G on H induced by q is uniform with respect to T.
-
(2) There is
$S\subset G$
containing
$S_0$
such that
$q\colon \Gamma (G,S) \to \Gamma (H,T)$
is a quasi-isometry.
Proof That the second statement implies the first one is immediate. Conversely, suppose that q and
$\bar q$
are
$(L_0,C_0)$
-quasi-isometries
$\Gamma (G,S_0)\to \Gamma (H,T_0)$
and
$\Gamma (H,T_0)\to \Gamma (G,S_0)$
, respectively. Without loss of generality, assume that
$q(e)=e$
and
$\bar q(e)=e$
where e denotes the identity in each corresponding group.
Let
$K_0=L_0+C_0+1$
and define
$$\begin{align*}\begin{aligned} S= \{ f^{-1}g \in G \colon & \text{ there are } h\in H \text{ and } t\in T \text{ such that } \\ & \operatorname{\mathrm{\mathsf{dist}}}_{(H, T_0)}(q(f),h)\leq K_0 \text{ and } \operatorname{\mathrm{\mathsf{dist}}}_{(H,T_0)}(q(g),ht)\leq K_0 \}. \end{aligned} \end{align*}$$
Note that
$S_0\subset S$
since
$q\colon \Gamma (G,S_0)\to \Gamma (H,T_0)$
is an
$(L_0,C_0)$
-quasi-isometry. In particular, S is a generating set of G.
Let
$L_1\geq 1$
and
$C_1\geq 0$
be such that the G-action on H induced by q is
$(L_1,C_1)$
-uniform with respect to T. In particular, for every
$g\in G$
, the function
$q_g\colon H\to H$
is an
$(L_1,C_1)$
-quasi-isometry
$\Gamma (H,T) \to \Gamma (H,T)$
.
Now, we prove that if the induced quasi-action of G on H is uniform with respect to T, then
$q\colon \Gamma (G,S) \to \Gamma (H,T)$
is a quasi-isometry. Observe that every vertex of
$\Gamma (H,T)$
is at distance at most
$C_0$
from
$q(G)$
with respect to
$\operatorname {\mathrm {\mathsf {dist}}}_{(H,T_0)}$
and hence with respect to
$\operatorname {\mathrm {\mathsf {dist}}}_{(H,T)}$
. Below, we prove inequalities (2.1) and (2.2), which will conclude proof.
Claim
There is constant
$\bar L$
such that
$$ \begin{align} \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q(a),q(b)) \leq \bar L\operatorname{\mathrm{\mathsf{dist}}}_{(G,S)}(a,b), \end{align} $$
for any
$a,b\in G$
.
Proof of claim
Let
$s\in S$
. Then there are
$f,g\in G$
,
$h\in H$
, and
$t\in T$
such that
$s=f^{-1}g$
and
$$\begin{align*}\operatorname{\mathrm{\mathsf{dist}}}_{(H,T_0)}(q(f),h)\leq K_0,\quad \operatorname{\mathrm{\mathsf{dist}}}_{(H,T_0)}(q(g),ht)\leq K_0.\end{align*}$$
It follows that
$$\begin{align*}\operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q_f(e),q_g(e))= \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q(f),q(g))\leq 2K_0+1.\end{align*}$$
Since the quasi-action of G on
$\Gamma (H,T)$
is
$(L_1,C_1)$
-uniform, the previous inequality implies that
$$\begin{align*}\begin{aligned} \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(e,q(s)) &= \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q_e(e),q_{f^{-1}g}(e)) \\ & \leq L_1\operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q_f\circ q_e(e),q_f\circ q_{f^{-1}g}(e))+ C_1\\ & \leq L_1\operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q_f(e), q_{g}(e))+ 3C_1\\ & \leq L_1(2K_0+1)+3C_1=:\bar L_0. \end{aligned} \end{align*}$$
For any
$g\in G$
and
$s\in S$
, we have that
$$ \begin{align*} \begin{aligned} \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q(g),q(gs)) & = \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q_{g}(e), q_{gs}(e)) \\ &\leq L_1 \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q_{g^{-1}}\circ q_{g}(e), q_{g^{-1}}\circ q_{gs}(e)) + C_1 \\ &\leq L_1\operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(e, q_{g^{-1}gs}(e)) +3C_1\\ & \leq L_1\operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q(e),q(s)) + 3C_1, \end{aligned} \end{align*} $$
and hence
$$\begin{align*}\operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q(g),q(gs)) \leq \operatorname{\mathrm{\mathsf{dist}}}_{(H,T_0)}(q(g),q(gs)) \leq \bar L, \end{align*}$$
where
$\bar L= L_1(\bar L_0)+3C_1$
. If
$a,b\in G$
and
$[u_0,\ldots , u_\ell ]$
is a geodesic in
$\Gamma (G,S)$
from a to b, then the triangle inequality implies inequality (2.1).
Claim
For any
$a,b\in G$
, we have
$$ \begin{align} \operatorname{\mathrm{\mathsf{dist}}}_{(G,S)}(a,b)\leq \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q(a),q(b)). \end{align} $$
Proof of claim
Suppose that
$[h_0,\ldots ,h_\ell ]$
is a geodesic in
$\Gamma (H,T)$
from
$q(a)$
to
$q(b)$
. Since
$q\colon \Gamma (G,S_0)\to \Gamma (H,T_0)$
is an
$(L_0,C_0)$
-quasi-isometry, for each i, there is
$g_i\in G$
such that
$\operatorname {\mathrm {\mathsf {dist}}}_{(H,T_0)}(q(g_i),h_i)\leq C_0$
. Let
$g_0=a$
and
$g_\ell =b$
. Observe that
$g_i^{-1}g_{i+1}\in S$
for
$0\leq i<\ell $
, and hence
$\operatorname {\mathrm {\mathsf {dist}}}_{(G,S)}(g_i,g_{i+1})\leq 1$
. Now,
$[g_0, \ldots , g_\ell ]$
is a path in
$\Gamma (G,S)$
from a to b and therefore
$\operatorname {\mathrm {\mathsf {dist}}}_{(G,S)}(a,b)\leq \operatorname {\mathrm {\mathsf {dist}}}_{(H,T)}(q(a),q(b))$
proving inequality (2.2).
Corollary 2.1 Let G and H be groups with finite generating sets
$S_0$
and
$T_0$
. Let
$q\colon G\to H$
be a group homomorphism which is also an
$(L_0,C_0)$
-quasi-isometry
$q\colon \Gamma (G,S_0) \to \Gamma (H,T_0)$
. If
$T\subset H$
contains
$T_0$
, then there is
$S\subset G$
containing
$S_0$
such that
$q\colon \Gamma (G,S) \to \Gamma (H,T)$
is a quasi-isometry.
Proof Let
$\bar q\colon H\to G$
be a quasi-inverse of q and, by increasing
$L_0$
and
$C_0$
if necessary, assume that
$\bar q\colon \Gamma (H,T_0) \to \Gamma (G, S_0)$
is an
$(L_0,C_0)$
-quasi-isometry. Moreover, suppose
$q\circ \bar q$
and
$\bar q\circ q$
are at distance at most
$C_0$
from the corresponding identity maps with respect to
$\operatorname {\mathrm {\mathsf {dist}}}_{(H,T_0)}$
and
$\operatorname {\mathrm {\mathsf {dist}}}_{(G,S_0)}$
. Note that for any
$g\in G$
,
$$\begin{align*}q_g (h) = q(g \cdot \bar q (h))=q(g) \cdot q (\bar q (h)). \end{align*}$$
Hence,
$q_g$
is a
$(1,C_0)$
-quasi-isometry since it is the composition of
$q \circ \bar q$
followed by the isometry given by multiplication on the left by
$q(g)$
. Then the proof concludes by invoking Proposition 2.1.
The following result is the particular case of Corollary 2.1 in which H is a finite index subgroup of G. In this case, one can give a more algebraic description of the generating set S. The proof follows the same lines as the previous argument modulo Lemma 2.1.
Proposition 2.2 Let H be a finite index normal subgroup of a finitely generated group G. Let T be a generating set of H, let R be a right transversal of H in G, and let
$S=T\cup R$
. If the G-action by conjugation on H is a uniform quasi-action on
$\Gamma (H,T)$
, then the inclusion
$\Gamma (H,T) \hookrightarrow \Gamma (G,S)$
is a quasi-isometry.
We divert the proof of the proposition after the following lemma.
Lemma 2.1 Let H be a finite index normal subgroup of a finitely generated group G. Let T be a generating set of H containing a finite generating set
$T_0$
, let R be transversal of H in G, let
$S_0$
be a finite generating set of G, and let
$q\colon \Gamma (G,S_0)\to \Gamma (H,T_0)$
be the quasi-isometry defined by
$q(hg)=h$
for
$h\in H$
and
$g\in R$
. The following statements are equivalent:
-
(1) The G-action by conjugation on H is a uniform quasi-action on
$\Gamma (H,T)$
. -
(2) The quasi-action of G on H induced by q is uniform with respect to T.
Proof Take as the quasi-inverse of q the inclusion
$H\hookrightarrow G$
. For
$h\in H$
, let
${L_h\colon H\to H}$
be given by
$L_h(x)=hx$
, i.e., multiplication on the left. Note that
$L_h\colon \Gamma (H,T)\to \Gamma (H,T)$
is an isometry for every
$h\in H$
.
Let
$g\in G$
and suppose that
$g=h_*g_*$
where
$h_*\in H$
and
$g_*\in R$
. Then
$$\begin{align*}q_g(h)=q(gh)=q(ghg^{-1}h_*g_*)=ghg^{-1}h_*=h_*g_*hg_*^{-1}h_*^{-1}h_*=h_*g_*hg_*^{-1}\end{align*}$$
and hence
$$\begin{align*}q_g = L_{h_*}\circ \textrm{Ad}({g_*}),\end{align*}$$
where
$\textrm {Ad}(g_*)$
is conjugation by
$g_*$
. It follows
$q_g\colon \Gamma (H,T)\to \Gamma (H,T)$
is an
$(L,C)$
-quasi-isometry for all
$g\in G$
if and only if
$\textrm {Ad}({g_*}) \colon \Gamma (H,T)\to \Gamma (H,T)$
is an
$(L,C)$
-quasi-isometry for all
$g_*\in R$
. In particular, the first statement implies the second by Remark 2.1, and the second statement implies the first since the constants L and C hold for all conjugations.
Proof [Proof of Proposition 2.2.]
Let
$T_0 \subset T$
be a finite generating set of H, and let
$S_0=T_0\cup R$
. Note that
$S_0$
is a finite generating set of G. Then
$q\colon \Gamma (G,S_0)\to \Gamma (H,T_0)$
is an
$(L_0,C_0)$
quasi-isometry for some
$L_0\geq 1$
and
$C_0\geq 0$
, and the quasi-inverse
$\bar q$
can be taken as the inclusion
$\Gamma (H,T_0)\hookrightarrow \Gamma (G,S_0)$
.
Observe that, in
$\Gamma (G,S)$
, the vertices
$g=hr$
and
$q(g)=h$
are adjacent since
${r\in S}$
. Therefore, if
$[v_0,\ldots ,v_\ell ]$
is a geodesic path in
$\Gamma (H,T)$
from
$q(a)$
to
$q(b)$
, then
$[a, v_0,\ldots ,v_\ell ,b]$
is a path in
$\Gamma (G,S)$
from a to b, and hence
$$\begin{align*}\operatorname{\mathrm{\mathsf{dist}}}_{(G,S)}(a,b) \leq \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q(a),q(b))+2. \end{align*}$$
We now prove the other inequality. Since the G-action on H by conjugation is a uniform quasi-action on
$\Gamma (H,T)$
, Lemma 2.1 implies that the quasi-action of G on H induced by q is
$(L_1,C_1)$
-uniform with respect to T, for some
$L_1\geq 1$
and
$C_1\geq 0$
.
Let
$K_0=L_0+C_0+1$
. Observe that
$$\begin{align*}\begin{aligned} S \subseteq \{ f^{-1}g \in G \colon & \text{ there are } h\in H \text{ and } t\in T \text{ such that } \\ & \operatorname{\mathrm{\mathsf{dist}}}_{(H, T_0)}(q(f),h)\leq K_0 \text{ and } \operatorname{\mathrm{\mathsf{dist}}}_{(H,T_0)}(q(g),ht)\leq K_0 \}. \end{aligned} \end{align*}$$
Indeed, let
$s\in S=T\cup R$
, and there are two cases. First, if
$s\in T$
, let
$f=h=e$
and
$g=t=s$
, and second, if
$s\in R$
, let
$f=h=e$
,
$g=s$
, and t be any element of
$T_0$
. Then, exactly as in the first claim in the proof of Proposition 2.1, one defines a constant
$\bar L=\bar L(L_1,C_1,K_0)$
and deduces the inequality
$$ \begin{align} \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q(a),q(b)) \leq \bar L\operatorname{\mathrm{\mathsf{dist}}}_{(G,S)}(a,b). \end{align} $$
It remains to show that
$$ \begin{align} \operatorname{\mathrm{\mathsf{dist}}}_{(G,S)}(a,b)\leq \operatorname{\mathrm{\mathsf{dist}}}_{(H,T)}(q(a),q(b))+2, \end{align} $$
for any
$a,b\in G$
, concluding the proof. This is clear since
$\Gamma (H,T)$
is a subgraph of
$\Gamma (G,T)$
and
$\operatorname {\mathrm {\mathsf {dist}}}_{G,S}(g,q(g))\leq 1$
for any
$g\in G$
.
The following example by Minasyan and Osin illustrates the need for the hypothesis relating to the conjugation action in Corollary 2.2.
Example 2.1 [Reference Minasyan and Osin16]
Let
$H=\langle a,b \rangle $
be the free group of rank 2, let
$G=\langle a,b,t \colon tat^{-1}\,{=}\,b, t^2=e \rangle $
, let
$T=\{b,a,a^{-1},a^2,a^{-2},\ldots \}$
, and let
$S=T\cup \{t\}$
. The inclusion
$\Gamma (H,T) \to \Gamma (G,S)$
is not a quasi-isometry. Indeed, in G, we have
$ta^nt^{-1}=b^n$
, and hence
$\operatorname {\mathrm {\mathsf {dist}}}_{(G,S)}(e,b^n)=3$
, but
$\operatorname {\mathrm {\mathsf {dist}}}_{(H,T)}(e,b^n)=n$
for every n. In particular, the map
$\Gamma (H,T)\to \Gamma (H,T)$
given by
$h\mapsto tht^{-1}$
is not a quasi-isometry, and hence the G-action on H by conjugation is not an action by quasi-isometries.
3 Quasi-isometries and hyperbolically embedded subgroups
In this section, we will prove Theorem A. The theorem is obtained by putting together a simple characterization of hyperbolically embedded subgroups in terms of coned-off Cayley graphs which appeared in the work of Rashid and the second author (see [Reference Martínez-Pedroza and Rashid13, Propositions 1.5 and 5.8]), some results about quasi-isometries of pairs from [Reference Hughes, Martínez-Pedroza and Sánchez Saldaña11], and some basic facts about hyperbolically embedded subgroups from [Reference Dahmani, Guirardel and Osin7]. Below, we state these results and then we discuss the proof of Theorem 3.1.
Definition 3.1 (Reduced collections)
A collection of subgroups
${\mathcal {Q}}$
of a group H is reduced if for any
$P,Q\in {\mathcal {Q}}$
and
$g\in H$
, if P and
$gQg^{-1}$
are commensurable subgroups, then
$P=Q$
and
$g\in P$
.
Remark 3.1. An almost malnormal collection is reduced.
Definition 3.2 (Fine)
Let
$\Gamma $
be a graph, and let v be a vertex of
$\Gamma $
. Let
$$ \begin{align*} T_v \Gamma = \{w \in V(\Gamma) \mid \{v,w\}\in E(\Gamma)\} \end{align*} $$
denote the set of the vertices adjacent to v. For
$x,y \in T_v \Gamma $
, the angle metric
$\angle _v (x,y)$
is the length of the shortest path in the graph
$\Gamma \setminus \{v\}$
between x and y, with
$\angle _v (x,y) = \infty $
if there is no such path. The graph
$\Gamma $
is fine at v if
$(T_v \Gamma ,\angle _v)$
is a locally finite metric space. The graph
$\Gamma $
is fine at
$C \subseteq V(\Gamma )$
if
$\Gamma $
is fine at v for all
$v \in C$
.
Definition 3.3 (Coned-off Cayley graph)
Let G be a group, let
${\mathcal {P}}$
be an arbitrary collection of subgroups of G, and let S be a subset of G. Denote by
$G/{\mathcal {P}}$
the set of all cosets
$gP$
with
$g\in G$
and
$P\in {\mathcal P}$
. The coned-off Cayley graph of G with respect to
${\mathcal {P}}$
is the graph
$\hat \Gamma (G,{\mathcal {P}},S)$
with vertex set
$G\cup G/{\mathcal {P}}$
and edges are of the following type:
-
•
$\{g,gs\}$
for
$s\in S$
and -
•
$\{x, gP\}$
for
$g\in G$
,
$P\in {\mathcal {P}}$
, and
$x\in gP$
.
We call vertices of the form
$gP$
cone points.
Proposition 3.1 [Reference Martínez-Pedroza and Rashid13]
Let
${\mathcal {P}}$
be a collection of infinite subgroups of G, and let S be a subset of G. Then
${\mathcal {P}} \hookrightarrow _h (G,S) $
if and only if the Coned-off Cayley graph
$\hat {\Gamma }(G,{\mathcal {P}},S)$
is a connected hyperbolic graph which is fine at every cone vertex.
Proposition 3.2 [Reference Hughes, Martínez-Pedroza and Sánchez Saldaña11, Proposition 5.6]
Let G and H be groups, let
$S\subset G$
and
$T\subset H$
, and let
$S_0\subset S$
and
$T_0\subset T$
be finite generating sets of G and H, respectively. Consider collections
${\mathcal P}$
and
$\mathcal Q$
of subgroups of G and H, respectively. Let
$q\colon G\to H$
be a function.
Suppose that q is a quasi-isometry
$\Gamma (G,S) \to \Gamma (H,T)$
, is a quasi-isometry of pairs
$(G, \mathcal {P},S_0)\to (H, \mathcal {Q},T_0)$
, and
$\dot {q}$
is a bijection
$G/{\mathcal {P}} \to H/{\mathcal {Q}}$
.
-
(1) Let
$\hat q = q\cup \dot {q}$
, then
$\hat q$
is a quasi-isometry
$\hat \Gamma (G,{\mathcal {P}}, S) \to \hat \Gamma (H,{\mathcal {Q}}, T)$
. -
(2) If
$\hat \Gamma (H,{\mathcal {Q}}, T)$
is fine at cone vertices, then
$\hat \Gamma (G,{\mathcal {P}}, S)$
is fine at cone vertices. -
(3) If
${\mathcal {Q}}\hookrightarrow _h(H,T)$
, then
${\mathcal {P}}\hookrightarrow _h(G,S)$
.
Items (1) and (2) of Proposition 3.2 are taken from [Reference Hughes, Martínez-Pedroza and Sánchez Saldaña11, Proposition 5.6], and the last item is a direct consequence of Proposition 3.1.
Proposition 3.3 [Reference Hughes, Martínez-Pedroza and Sánchez Saldaña11, Proposition 5.12]
Let
$q\colon (G, \mathcal {P})\to (H, \mathcal {Q})$
be an
$(L,C,M)$
-quasi-isometry of pairs. Then:
-
(1)
$\dot {q}$
is a surjective function
$G/{\mathcal {P}} \to H/{\mathcal {Q}}$
if
${\mathcal {Q}}$
is reduced. -
(2)
$\dot {q}$
is a bijection
$G/{\mathcal {P}} \to H/{\mathcal {Q}}$
if
${\mathcal {P}}$
and
${\mathcal {Q}}$
are reduced.
Proposition 3.4 [Reference Hughes, Martínez-Pedroza and Sánchez Saldaña11, Proposition 6.2]
Let
${\mathcal {P}}^\ast $
be a refinement of a finite collection of subgroups
${\mathcal {P}}$
of a finitely generated group G. If P is a finite index subgroup of
$\operatorname {{\mathrm {Comm}}}_G(P)$
for every
$P\in {\mathcal {P}}$
, then
$(G,{\mathcal {P}})$
and
$(G,{\mathcal {P}}^*)$
are quasi-isometric pairs via the identity map on G.
Proposition 3.5 [Reference Hughes, Martínez-Pedroza and Sánchez Saldaña11, Proposition 6.7]
Let
$q\colon (G, \mathcal {P})\to (H, \mathcal {Q})$
be a quasi-isometry of pairs. If
${\mathcal {Q}}$
is an almost malnormal finite collection of infinite subgroups and
${\mathcal {P}}$
is a finite collection, then any refinement
${\mathcal {P}}^*$
of
${\mathcal {P}}$
is almost malnormal.
Proposition 3.6 [Reference Dahmani, Guirardel and Osin7, Proposition 4.33]
Let
${\mathcal {P}}$
be a collection of subgroups of a group G. If
${\mathcal {P}}\hookrightarrow _h G$
, then
${\mathcal {P}}$
is an almost malnormal collection.
We are now ready to prove Theorem A.
Theorem 3.1 (Theorem A)
Let
$q\colon G\to H$
be a quasi-isometry of finitely generated groups, let
${\mathcal {P}}$
and
${\mathcal {Q}}$
be finite collections of subgroups of G and H, respectively, and let S and T be (not necessarily finite) generating sets of G and H, respectively. Suppose that:
-
(1)
$q\colon (G,{\mathcal {P}}) \to (H,{\mathcal {Q}})$
is a quasi-isometry of pairs and -
(2)
$q\colon \Gamma (G,S) \to \Gamma (H,T)$
is a quasi-isometry.
The following statements hold:
-
(1) If
${\mathcal {P}}$
and
${\mathcal {Q}}$
are reduced collections in G and H, respectively, then
${\mathcal {P}}\hookrightarrow _h (G,S)$
if and only if
${\mathcal {Q}}\hookrightarrow _h (H,T)$
. -
(2) If
${\mathcal {Q}}$
contains only infinite subgroups and
${\mathcal {Q}}\hookrightarrow _h (H,T)$
, then
${\mathcal {P}}^*\hookrightarrow _h (G,S)$
.
Proof For the first statement, since
${\mathcal {P}}$
and
${\mathcal {Q}}$
are reduced, Proposition 3.3 implies that
$\dot q\colon G/{\mathcal {P}} \to H/{\mathcal {Q}}$
is a bijection. Then Proposition 3.2 implies that
$\hat \Gamma (G,{\mathcal {P}},S)$
is hyperbolic and fine at cone vertices if and only if
$\hat \Gamma (H,{\mathcal {Q}},T)$
is hyperbolic and fine at cone vertices. Then Proposition 3.1 concludes the proof of the first statement.
The second statement is a consequence of the first statement as follows. That
${\mathcal {Q}}\hookrightarrow _h H$
implies that
${\mathcal {Q}}$
is an almost malnormal collection of subgroups in H (see Proposition 3.6). It follows that
${\mathcal {Q}}$
is reduced in H. Then, since
${\mathcal {Q}}$
contains only infinite subgroups, Proposition 3.5 implies that
${\mathcal {P}}^\ast $
is reduced. By Proposition 3.4,
$q\colon (G,{\mathcal {P}}^\ast )\to (H,{\mathcal {Q}})$
is a quasi-isometry of pairs. Then
${\mathcal {Q}}\hookrightarrow _h H$
and the first statement of the proposition imply that
${\mathcal {P}}^\ast \hookrightarrow (G,S)$
.
4 Hyperbolically embedded subgroups and commensurability
In this section, we prove Theorem E. The argument uses the following proposition, which is a strengthening of [Reference Martínez-Pedroza and Sánchez Saldaña14, Proposition 2.15]. It essentially follows from the proof in the cited article, but we have included the proof for the convenience of the reader.
Proposition 4.1 Let H be a finite index subgroup of a finitely generated group G, and let
$\mathcal {Q}$
be a finite collection of subgroups of H. The following statements are equivalent:
-
(1) The inclusion
$H\hookrightarrow G$
is a quasi-isometry of pairs
$(H,\mathcal {Q}) \hookrightarrow (G, \mathcal {Q})$
. -
(2) For any
$Q\in \mathcal {Q}$
and
$g\in G$
, there is
$Q'\in \mathcal {Q}$
and
$h\in H$
such that
${\operatorname {\mathrm {\mathsf {hdist}}}_G(gQ , hQ')<\infty} $
.
Proof That (1) implies (2) is trivial. Assume statement (2). Since H is a finite index subgroup of the finitely generated group G, assume that
$H\hookrightarrow G$
is an
$(L,C)$
quasi-isometry. Since H is finite index in G, and
$\mathcal {Q}$
is a finite collection, the H-action on
$G/\mathcal {Q}$
has finitely many orbits. For
$gQ\in G/\mathcal {Q}$
, let
$$\begin{align*}\operatorname{\mathrm{\mathsf{hdist}}}_G(gQ, H/\mathcal{Q}):= \min\left\{\operatorname{\mathrm{\mathsf{hdist}}}_G(gQ, hQ') \colon hQ'\in H/\mathcal{Q} \right\}\!. \end{align*}$$
Let
$\mathcal {R}$
be a finite collection of orbit representatives of the H-action on
$G/\mathcal {Q}$
. By hypothesis, for
$gQ \in \mathcal {R}$
, there is
$h Q'\in H/ \mathcal {Q}$
such that
$\operatorname {\mathrm {\mathsf {hdist}}}(gQ,hQ')<\infty $
and therefore
$$\begin{align*}M = \max\{\operatorname{\mathrm{\mathsf{hdist}}}_G(gQ,H/\mathcal{Q}) \colon gQ \in \mathcal{R}\} <\infty \end{align*}$$
is a well-defined integer since
$\mathcal {R}$
is a finite set. Since the subset
$H/\mathcal {Q}$
of
$G/\mathcal {Q}$
is H-invariant,
$$\begin{align*}\operatorname{\mathrm{\mathsf{hdist}}}_G(gQ, H/\mathcal{Q}) = \operatorname{\mathrm{\mathsf{hdist}}}_G(hgQ, H/\mathcal{Q}) \end{align*}$$
for every
$gQ\in \mathcal {R}$
and
$h\in H$
. Since
$\mathcal {R}$
is a collection of representatives of orbits of
$G/\mathcal {Q}$
,
$$\begin{align*}\operatorname{\mathrm{\mathsf{hdist}}}_G(gQ, H/\mathcal{Q}) \leq M \end{align*}$$
for every
$gQ \in G/ \mathcal {Q}$
. Hence,
$(H, \mathcal {Q}) \hookrightarrow (G, \mathcal {Q})$
is an
$(L,C,M)$
quasi-isometry of pairs.
Remark 4.1. Let G be a group, and let T and S generating sets with finite symmetric difference. Then the identity map on G is a quasi-isometry
$\Gamma (G,T) \to \Gamma (G,S)$
.
Theorem 4.1 (Theorem E)
Let H be a finite index normal subgroup of a finitely generated group G, and let
${\mathcal {Q}}$
be a finite collection of infinite subgroups of H such that
${\mathcal {Q}}\hookrightarrow _h (H,T)$
. Suppose that:
-
(1) The G-action by conjugation on H is a uniform quasi-action on
$\Gamma (H,T)$
. -
(2) The collection
$\{hQh^{-1} \colon h\in H \text { and } Q\in {\mathcal {Q}}\}$
is invariant under conjugation by G.
If
${\mathcal {Q}}^*$
is a refinement of
${\mathcal {Q}}$
in G and R is a transversal of H in G, then
${\mathcal {Q}}^*\hookrightarrow _h (G, T\cup R)$
.
Proof Since H is finitely generated, by adding a finitely many elements, we can assume that T generates H. Note that this preserves
${\mathcal {Q}}\hookrightarrow _h (H,T)$
by [Reference Dahmani, Guirardel and Osin7, Corollary 4.27], and the quasi-isometry type of
$\Gamma (H,T)$
by Remark 4.1. Under this assumption, the conclusion will follow from the second statement of Theorem 3.1 applied to the quasi-isometry of finitely generated groups given by the inclusion
$H\hookrightarrow G$
.
Since
${\mathcal {Q}}\hookrightarrow _h (H,T)$
,
${\mathcal {Q}}$
is an almost malnormal collection (see Proposition 3.6). The assumption that
${\mathcal {Q}}$
consists only of infinite subgroups implies that for any
$Q\in {\mathcal {Q}}$
,
$$\begin{align*}Q=\operatorname{{\mathrm{Comm}}}_H(Q)=\operatorname{{\mathrm{Comm}}}_G(Q)\cap H. \end{align*}$$
Since H is finite index in G, we have that Q is finite index in
$\operatorname {{\mathrm {Comm}}}_G(Q)$
. Then Proposition 3.4 implies that the identity map on G is a quasi-isometry of pairs
$(G,{\mathcal {Q}}) \xrightarrow {}(G,{\mathcal {Q}}^\ast )$
. On the other hand, since the collection
$\{hQh^{-1}\colon h\in H \text { and} Q\in {\mathcal {Q}}\}$
is invariant under conjugation by elements of G, we have for any
$g\in G$
and
$Q\in {\mathcal {Q}}$
there is
$h\in H$
such that
$gQg^{-1}=hQ'h^{-1}$
and hence
$$\begin{align*}\operatorname{\mathrm{\mathsf{hdist}}}_G(gQ,hQ')\leq \operatorname{\mathrm{\mathsf{hdist}}}_G(gQ,Q^g)+ \operatorname{\mathrm{\mathsf{hdist}}}_G(Q^g,(Q')^h)+ \operatorname{\mathrm{\mathsf{hdist}}}((Q')^h,hQ')<\infty.\end{align*}$$
Proposition 4.1 implies that
$H\hookrightarrow G$
is a quasi-isometry of pairs
$(H,{\mathcal {Q}})\to (G,{\mathcal {Q}})$
. It follows that
$H\hookrightarrow G$
is a quasi-isometry of pairs
$(H,{\mathcal {Q}}) \to (G,{\mathcal {Q}}^\ast )$
as it is the composition
$(H,{\mathcal {Q}})\hookrightarrow (G,{\mathcal {Q}}) \xrightarrow {}(G,{\mathcal {Q}}^\ast )$
. Let R be a transversal of H in G, and let
${S=T\cup R}$
. Since the G-action by conjugation on H is uniform on
$\Gamma (H,T)$
, Proposition 2.2 implies that
$H\hookrightarrow G$
is a quasi-isometry
$\Gamma (H,T) \to (G,S)$
. The hypothesis of Theorem 3.1 has been verified, and therefore
${\mathcal {Q}}\hookrightarrow _h (H,T)$
implies
${\mathcal {Q}}^*\hookrightarrow _h (G,S)$
.
5 Semidirect products and hyperbolically embedded subgroups
In this section, we will prove Theorem F about semidirect products. The hypothesis of the following proposition and theorem reflects the issues posed by the example of Minasyan and Osin (Example 2.1).
Proposition 5.1 Let A be a group with (not necessarily finite) generating set T, let
$\mathcal {H}$
be a collection of subgroups, and let
$F\leq \mathsf {Aut}(A)$
be a finite subgroup. Suppose that T and
$\mathcal {H}$
are F-invariant and that the F-action on
$\mathcal {H}$
is free. Let
${\mathcal {H}}_F$
be a collection of representatives of F-orbits in
${\mathcal {H}}$
. Then the inclusion
$A\hookrightarrow A\rtimes F$
induces:
-
(1) a quasi-isometry
$\Gamma (A,T) \to \Gamma (A\rtimes F, T\cup F)$
and -
(2) if A is finitely generated, a quasi-isometry of pairs
$(A,\mathcal {H}) \to (A\rtimes F,\mathcal {H}_F)$
.
Proof To prove the first statement, let
$S=T\cup F$
and let
$\operatorname {\mathrm {\mathsf {dist}}}_{T}$
and
$\operatorname {\mathrm {\mathsf {dist}}}_S$
be the word metrics on A and
$A\rtimes F$
induced by T and S, respectively. Let
$q \colon A \hookrightarrow A\rtimes F$
be the inclusion, and let
$\bar q\colon A\rtimes F \to A$
be such that for
$a\in A$
and
$f\in F$
,
$\bar q(af)=a$
. Note that
$\bar q$
is a well-defined A-equivariant map since each element of
$A\rtimes F$
can be expressed as a product
$af$
in a unique way. Observe that
$\bar q\circ q$
is the identity on A, and
$q\circ \bar q$
is at distance 1 from the identity map on
$A\rtimes F$
with respect to
$\operatorname {\mathrm {\mathsf {dist}}}_S$
. Since the Cayley graph
$\Gamma (A,T)$
is a subgraph of
$\Gamma (A\rtimes F, T\cup F)$
, it is immediate that for any
$u,v\in A$
,
$\operatorname {\mathrm {\mathsf {dist}}}_S(q(u),q(v))\leq \operatorname {\mathrm {\mathsf {dist}}}_T(u,v)$
. To conclude the proof of the statement, we show that for any
$u,v\in A\rtimes F$
,
$\operatorname {\mathrm {\mathsf {dist}}}_T(\bar q(u),\bar q(v))\leq \operatorname {\mathrm {\mathsf {dist}}}_S(u,v)$
. Note that it is enough to consider the case that
$\operatorname {\mathrm {\mathsf {dist}}}_S(u,v)=1$
. Let
$w_1,w_2\in A\rtimes F$
such that
$\operatorname {\mathrm {\mathsf {dist}}}_S(w_1,w_2)=1$
. Then
$w_1=a_1f_1$
and
$w_2=a_2f_2$
and
$\bar q(w_i)=a_i$
. It follows that
$g=(a_1f_1)^{-1}a_2f_2\in T\cup F$
. Observe that
$$\begin{align*}g= f_1^{-1}a_1^{-1}a_2f_2=(a_1^{-1})^{f_1^{-1}}f_1^{-1}a_2f_2=(a_1^{-1})^{f_1^{-1}}a_2^{f_1^{-1}}f_1^{-1} f_2=(a_1^{-1}a_2)^{f_1^{-1}}f_1^{-1}f_2\in T\cup F. \end{align*}$$
There are two cases, either
$g\in T$
or
$g\in F$
, since
$T\cap F=\emptyset $
. We regard
$T\cup F$
and F as a subset and a subgroup of
$A\rtimes F$
, respectively. If
$g\in T$
, then
$f_1=f_2$
and hence
$(a_1^{-1}a_2)^{f_1^{-1}} \in T$
; since T is F-invariant,
$a_1$
and
$a_2$
are adjacent in
$\Gamma $
, and hence
$\operatorname {\mathrm {\mathsf {dist}}}_T(\bar q(w_1), \bar q(w_2))=1$
. If
$g\in F$
, then
$a_1=a_2$
and hence
$\operatorname {\mathrm {\mathsf {dist}}}_T(\bar q(w_1), \bar q(w_2))=0$
.
For the second statement, suppose that A is finitely generated and let
$\operatorname {\mathrm {\mathsf {dist}}}$
denote word metric on
$A\rtimes F$
induced by finite generating set, and let
$\operatorname {\mathrm {\mathsf {hdist}}}_{A\rtimes F}$
be the induced Hausdorff distance. Let
$M=\max _{f\in F}\operatorname {\mathrm {\mathsf {dist}}}(1, f)$
. Since the inclusion
$A\hookrightarrow A\rtimes F$
is a quasi-isometry of finitely generated groups and
${\mathcal {H}}_F \subset {\mathcal {H}}$
, it is enough to prove that for any
$H\in {\mathcal {H}}$
, there is a left coset in
$(A\rtimes F)/{\mathcal {H}}_F$
at Haudorff distance at most M in
$A\rtimes F$
. Let
$H\in {\mathcal {H}}$
. Since the F-action on
${\mathcal {H}}$
by conjugation is free, there is a unique
$f\in F$
and a unique
$K\in {\mathcal {H}}_F$
such that
$H=fKf^{-1}$
. Observe that
$$\begin{align*}\operatorname{\mathrm{\mathsf{hdist}}}(H, fK) = \operatorname{\mathrm{\mathsf{hdist}}}(fKf^{-1}, fK) \leq \operatorname{\mathrm{\mathsf{dist}}}(1, f^{-1}) \leq M, \end{align*}$$
and this completes the proof.
Theorem 5.1 (Theorem F)
Let A be a finitely generated group with (not necessarily finite) generating set T, and let
${\mathcal {H}}$
be a finite collection of infinite subgroups such that
$\mathcal {H}\hookrightarrow _h (A,T)$
. If
$F\leq \mathsf {Aut}(A)$
is finite, T and
${\mathcal {H}}$
are F-invariant, and the F-action on
$\mathcal {H}$
is free, then
${\mathcal {H}}_F\hookrightarrow _h(A\rtimes F,T\cup F)$
, where
${\mathcal {H}}_F$
is collection of representatives of F-orbits in
${\mathcal {H}}$
.
Proof By Proposition 5.1, the inclusion
$A\hookrightarrow A\rtimes F$
induces a quasi-isometry
$\Gamma (A,T) \to \Gamma (A\rtimes F, T\cup F)$
, and a quasi-isometry of pairs
$(A,{\mathcal {H}}) \to (A\rtimes F, {\mathcal {H}}_F)$
. Since
${\mathcal {H}}\hookrightarrow _h A$
, the collection
${\mathcal {H}}$
is almost malnormal in A; then the assumption that F acts freely on
$\mathcal {H}$
implies that a refinement of
${\mathcal {H}}$
in
$A\rtimes F$
is
$\mathcal {H}_F$
, and this was observed in Example 1.3. Since
${\mathcal {H}}$
contains only infinite subgroups and
${\mathcal {H}}\hookrightarrow _h A$
, Theorem 3.1 implies that
${\mathcal {H}}_F \hookrightarrow _h (A\rtimes F, T\cup F)$
.
6 Concluding remarks
A positive answer to the following question would allow us to drop the first hypothesis of Theorem A for the relevant groups.
Question 6.1. Let G be a finitely generated NRH acylindrically hyperbolic group. Does G contain a qi-characteristic collection of hyperbolically embedded subgroups?
It is possible that
${\mathcal {A}}{\mathcal {H}}$
-accessibility as defined in [Reference Abbott, Balasubramanya and Osin1] may be necessary for a positive answer to Question 6.1. Note that this property does not always hold (see [Reference Abbott2]).
It is tempting to weaken the definition of a quasi-isometry of pairs
$q\colon (G,{\mathcal {P}})\to (H,{\mathcal {Q}})$
to remove the uniform constant M bounding the Hausdorff distances on the cosets and instead ask the relation
$$\begin{align*}\dot{q}=\{(A,B)\in G/{\mathcal{P}}\times H/{\mathcal{Q}} : \operatorname{\mathrm{\mathsf{hdist}}}_H(q(A),B)<\infty\}\end{align*}$$
satisfies that the projections into
$G/{\mathcal {P}}$
and
$H/{\mathcal {Q}}$
are surjective. We shall call the map q in this modified definition an almost quasi-isometry of pairs following [Reference Hughes, Martínez-Pedroza and Saldaña10, Section 5].
Indeed, there is work of Margolis [Reference Margolis12] where the main theorems do not require this additional hypothesis. However, Margolis shows that the hypotheses assumed in the main results of loc. cit. in fact imply that such a constant M exists (see [Reference Margolis12, Theorem 4.1]). Note that our results in this article rely on the existence of a constant M—primarily due to the use of [Reference Hughes, Martínez-Pedroza and Sánchez Saldaña11, Proposition 5.6]. Thus, we raise the following question.
Question 6.2. Let G and H be finitely generated groups with finite collections of subgroups
${\mathcal {P}}$
and
${\mathcal {Q}}$
, respectively. When is an
$(L,C)$
-almost quasi-isometry of pairs
$q\colon (G,{\mathcal {P}})\to (H,{\mathcal {Q}})$
an
$(L,C,M)$
-quasi-isometry of pairs?
Motivated by results of [Reference Behrstock, Hagen and Sisto5], the referee of the article suggested that it might be interesting to investigate other relaxations of the definition of a quasi-isometry of pairs (Defintion 1.1), for example, in the sense that the image of every element of the collection
$\mathcal {A}$
lies at uniform Hausdorff distance of the union of finitely many elements in the collection
$\mathcal {B}$
. Having a more general notion could allow a broader strategy toward tackling Question 1.1 based on the methods in this article.
Acknowledgment
The first author would like to thank his Ph.D. supervisor Professor Ian Leary. Both authors would like to thank Luis Jorge Sánchez Saldaña and the anonymous referee for their helpful comments.
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