aa r X i v : . [ m a t h . G R ] O c t Finite extensions of H− and AH -accessible groups S. H Balasubramanya
Abstract
We prove that the group properties of being H− accessible and AH -accessible arepreserved under finite extensions. We thus answer an open question from [1]. Introduction
An important open question related to the class of acylindrically hyperbolic groups is thefollowing (See [9, Question 2.20]).
Question 1.1.
Is the property of being acylindrically hyperbolic preserved under quasi-isometries for finitely generated groups? In other words, if G is a finitely generated acylin-drically hyperbolic group, and H is a finitely generated group that is quasi-isometric to G ,then is H also an acylindrically hyperbolic group ? A group G is called acylindrically hyperbolic if it admits a non-elementary, acylindricalaction on a hyperbolic space. The motivation behind the following question comes from theobservation that the class of acylindrically hyperbolic groups serves as a generalization tothe classes of non-elementary hyperbolic and relatively hyperbolic groups. The latter twoclasses are preserved under quasi-isometries (see [6]and [5, Theorem 5.12]), making Question1.1 a natural question to consider.An answer to this question seems currently out of reach given the lack of techniques to buildan action of H on a hyperbolic space simply starting from an action of G on a (possiblydifferent) hyperbolic space and a quasi-isometry between the two groups. Indeed, this ishard to do even in the case of actions on Cayley graphs that arise from finite generatingsets. By [10, Theorem 1.2], if G is acylindrically hyperbolic, then there exists a hyperbolicCayley graph, say Γ( G, A ), such that G y Γ( G, A ) is acylindrical and non-elementary. If G is acylindrically hyperbolic, but not a hyperbolic group, then A is necessarily infinite.Suppose that the groups G and H are quasi-isometric with respect to some finite generatingsets X and Y respectively. Then the lack of a relationship between G y Γ( G, X ) and G y Γ( G, A ) makes it unclear as to how we may transfer the desired properties to thegroup H .Since the answer to this general question is out of reach, we may consider some special casesof Question 1.1 instead. For instance, we may ask the following.1 uestion 1.2. Let
H < G be a finite index subgroup. If H is acylindrically hyperbolic,then is G acylindrically hyperbolic ? While it seems that this question is much simpler, in actuality it is also hard to answer.The primary obstruction here is the lack of any techniques that allow us to build an actionof the parent group G starting from the action of a finite index subgroup H while retainingthe properties of hyperbolicity and acylindricity. Indeed, [3] and sections of [1] explorethe possibility of extending group actions in certain settings, especially from peripheral andhyperbolically embedded subgroups, but those methods do not generalize to the finite indexcase. It is also worth noting, as shown in [3, Example 1.6], that the extension problem maynot be solvable even for finite index subgroups of finitely generated groups.However, the authors of [8] proved that the answer to Question 1.2 is affirmative in thespecial case when the finite index subgroup of G is normal and AH− accessible.
Proposition 1.3. [8, Lemma 6] Suppose that a group G contains a normal acylindricallyhyperbolic subgroup H of finite index which is AH -accessible. Then G is acylindricallyhyperbolic.A group G is said to be AH− accessible if the poset AH ( G ) contains the largest element.(Note that largest elements, if they exist, are unique.) We will show that under thehypothesis of Proposition 1.3, something even stronger holds true : the group G is also AH− accessible. Thus, we have the following.
Proposition 1.4.
Suppose that a group G contains a normal acylindrically hyperbolicsubgroup H of finite index which is AH -accessible. Then G is also AH− accessible.Note that the groups G and H in the above proposition are quasi-isometric. In particular,the above result shows that the property of being AH− accessible (and hence acylindricallyhyperbolic) is preserved in this very special situation of a quasi-isometry. We will also showthat the property of being H− accessible can be transferred under similar hypothesis. Proposition 1.5.
Suppose that a group G contains a normal, finite index subgroup H suchthat H is H− accessible. Then G is also H− accessible.Obviously, the Propositions 1.4 and 1.5 show that the properties of being H− accessible and AH− accessible are preserved under finite extensions, which answers an open question from[1] (See Problem 8.10).One important implication of being
AH− accessible is that the group admits a univer-sal action. i.e. there is an acylindrical action of the group on a hyperbolic space such thatevery generalized loxodromic acts loxodromically in this action (see [1, Section 7] for details).Another motivation for understanding whether a given group is H− or AH− accessible isthat when the answer is affirmative, it provides an obvious candidate for which group action2s the most informative in order to better understand certain properties of the group. Thelargest structure is also preserved by every automorphism of the group, which may be usefulfor studying the group (see [1, Section 2.3] for details). Preliminaries
We quickly recall some standard terminology and definitions from [1], before proceedingwith the proofs of the results.Throughout this paper, all group actions on metric spaces are assumed to be isometric.Given a metric space S , we denote by d S the distance function on S unless another notationis introduced explicitly. Given a point s ∈ S or a subset R ⊆ S and an element g ∈ G , wedenote by gs (respectively, gR ) the image of s (respectively R ) under the action of g . Givena group G y S and some s ∈ S , Gs denotes the G -orbit of s under the group action.Let X , Y be two generating sets of a group G . We say that X is dominated by Y , written X (cid:22) Y , if the identity map on G induces a Lipschitz map between metric spaces ( G, d Y ) → ( G, d X ). This is equivalent to the requirement that sup y ∈ Y | y | X < ∞ , where | · | X = d X (1 , · )denotes the word length with respect to X . It is easy to see that (cid:22) is a preorder on theset of generating sets of G and therefore it induces an equivalence relation in the standardway: X ∼ Y ⇔ X (cid:22) Y and Y (cid:22) X. This is equivalent to the condition that the Cayley graphs Γ(
G, X ) and Γ(
G, Y ) are G -equivariantly quasi-isometric. We denote by [ X ] the equivalence class of a generating set X , and by G ( G ) the set of all equivalence classes of generating sets of G . The preorder (cid:22) induces an order relation on G ( G ) by the rule[ X ] [ Y ] ⇔ X (cid:22) Y. Note that the above order relation is inclusion reversing . i.e. if Y ⊆ X , then [ X ] [ Y ].This definition is consistent with the following observation : if we take the generating set X = G , then the corresponding Cayley graph is a bounded space of diameter 1. This Cayleygraph retains practically no information about the structure of the group G , and so [ G ] isalways the smallest element in G ( G ). H ( G ) is the partially ordered set of hyperbolic structures on a group G . It contains equiv-alence classes of generating sets [ X ] such that Γ( G, X ) is a hyperbolic space. The partialorder on this poset is the one inherited from G ( G ). The subset AH ( G ) ⊂ H ( G ) is thepartially ordered set of acylindrically hyperbolic structures on a group G . It contains equiv-alence classes of generating sets [ X ] ∈ H ( G ) such that G y Γ( G, X ), is acylindrical.An element [ X ] ∈ H ( G ) (resp. AH ( G )) is called largest if for every [ Y ] ∈ H ( G ) (resp. every[ Y ] ∈ AH ( G )), we have that [ Y ] [ X ]. It is easy to see that if a largest element exists, itis unique. When H ( G ) (resp. AH ( G )) contains the largest element, then we say that thegroup G is H− accessible (resp. AH− accessible).3ote that while the question of H− accessibility makes sense for any group, the questionwhether a group is AH− accessible only makes sense in the case of acylindrically hyperbolicgroups. Indeed, if the group G is not acylindrically hyperbolic, then the cardinality of AH ( G ) is either 1 or 2; in either case the group is AH− accessible (see [1, Theorem 2.6]).Further note that for acylindrically hyperbolic groups, the property of being
AH− accessibleis strictly stronger than being acylindrically hyperbolic. Indeed, every (non-elementary)
AH− accessible group is acylindrically hyperbolic, but there exist acylindrically hyperbolicgroups that are not
AH− accessible (See [1, Theorem 2.17]).For further details of this poset, we refer the reader to [1, Sections 1,7]. In particular, wewould like to mention [1, Theorem 2.19], which proves that the following classes of groupsare
AH− accessible groups : mapping class groups of punctured closed surfaces, right-angledArtin groups (RAAGs) and fundamental groups of compact orientable 3-manifolds withempty or toroidal boundary. Main Results
We start by proving Proposition 1.4. The strategy used in the proof of Proposition 1.3in [8] is the following : Given the largest element [ X ] ∈ AH ( H ) and a finite set Y ofdistinct representatives of cosets of H in G , the authors show that [ X ∪ Y ] ∈ AH ( G ). Since H y Γ( H, X ) is non-elementary, it follows that so is G y Γ( G, X ∪ Y ). We will show thatthe structure [ X ∪ Y ] is the largest in AH ( G ). We will set this notation for the followingproof. Proof of Proposition 1.4.
Let [ Z ] be any element in AH ( G ). Then Γ( G, Z ) is hyperbolicand G y Γ( G, Z ) is an acylindrical action. We will show that [ Z ] [ X ∪ Y ], which willprove the result.Since H ≤ G , H y Γ( G, Z ) is also acylindrical. Since H has finite index in G , the action isalso cobounded. Indeed, the G -action is cobounded since we are considering an action ona Cayley graph and we have that [ h ∈ H hY = G . Since Y is finite, it is bounded in d Z . Byusing the Svarc-Milnor map from [1, Lemma 3.11], we get that there exists [ W ] ∈ AH ( H )such that H y Γ( H, W ) is equivalent to H y Γ( G, Z ) . (See [3] or [1, Section 3] for furtherdetails.)By definition, this means that there is a coarsely H − equivariant quasi-isometry φ : Γ( G, Z ) → Γ( H, W ). Thus there exists a constant C satisfying the following condi-tions for every g ∈ G and every h ∈ H . − C + 1 C d Z (1 , g ) ≤ d W ( φ (1) , φ ( g )) ≤ Cd Z (1 , g ) + C (1)sup h ∈ H d W ( hφ ( g ) , φ ( hg )) ≤ C (2)4ince [ X ] is the largest element in AH ( H ), we must have that [ W ] [ X ]. Thus there existsa constant K such that sup x ∈ X | x | W ≤ K. (3)Let x ∈ X . Then x ∈ H and by using (1) and (2) we get that d Z (1 , x ) ≤ Cd W ( φ (1) , φ ( x )) + C ≤ Cd W ( φ (1) , xφ (1)) + 2 C . By using the triangle inequality and (3), we get that d Z (1 , x ) ≤ Cd W ( φ (1) ,
1) + Cd W (1 , x ) + Cd W ( x, xφ (1) (cid:1) + 2 C ≤ Cd W (1 , φ (1)) + CK + 2 C . Since d W (1 , φ (1)) is a constant independent of the choice of x , we get thatsup x ∈ X | x | Z < ∞ . Since Y is finite, it follows that sup s ∈ X ∪ Y | s | Z < ∞ . Thus [ Z ] [ X ∪ Y ]. Corollary 3.1.
Right-angled coxeter groups (RACG) that contain a right-angled Artingroup (RAAG) as a finite index normal subgroup are
AH− accessible.Proof.
By [1, Theorem 2.19(c)], every RAAG is AH -accessible (where the largest actioncorresponds to the action on the extension graph). The result now follows from Proposition1.4.Note that by [4, Lemma 3], given any RAAG H , there exists a RACG K such that H ≤ K isfinite index and normal. Thus by Corollary 3.1, the associated RACG K is AH -accessible.Also note that the result that every RACG is AH -accessible is proved in [2, Theorem A(4)]. The proof contained therein uses techniques related to the hierarchically hyperbolicstructure of these groups, and shows that the largest action of a RACG corresponds to theaction of the group on an altered contact graph. However, in the cases when the RACGcontains a RAAG as a finite index normal subgroup, by Corollary 3.1, we may alternativelystudy the largest action of the RACG by considering the extension of the action of theRAAG on its extension graph (since the two actions are equivalent).We will now turn our attention to the proof of Proposition 1.5. The first part of the proofis practically identical to the first part of the proof of [8, Lemma 6], which we reproducehere for the sake of completion. In what follows, we set the notation that G is a group and H ≤ G is a finite index, normal subgroup such that H is H− accessible. Proof of Proposition 1.5.
Let [ X ] be the largest element in H ( H ), and let Y be a fixed setof distinct representatives of cosets of H in G , where the representative of H is 1. Obviously Y is finite and X ∪ Y is a generating set for G .5t follows from Section 2.3 of [1] that the action of Aut ( H ) on H ( H ) defined by α ([ W ]) =[ α ( W )] is a well-defined, order preserving action. Since [ X ] ∈ H ( H ) is the largest element,we must have α ([ X ]) = [ α ( X )] for every α ∈ Aut ( H ). In particular, since H is normal in G and Y is finite, there exists a constant L such that | y − xy | X = | y − x − y | X ≤ L (4)for all x ∈ X and for all y ∈ Y .We will show that the inclusion map from Γ( H, X ) → Γ( G, X ∪ Y ) is an H − equivariantquasi-isometry. Obviously this map is H − equivariant. To see that the map is coarselysurjective, observe that for every g ∈ G , there exists h ∈ H and y ∈ Y such that g = hy .Then d X ∪ Y ( g, h ) = d X ∪ Y ( hy, h ) = d X ∪ Y ( y, ≤ . Further, we have that for every h ∈ H , | h | X ∪ Y ≤ | h | X . So it remains to find a constant M such that for every h ∈ H , we have that | h | X ≤ M | h | X ∪ Y .To this end, let h ∈ H and let a a ....a n be the shortest word in X ± ∪ Y ± representing h .Let w = 1 and w i = a a ...a i for every i = 1 , , ...n . For every i , there exists y i ∈ Y suchthat w i y − i ∈ H .Then h = ( y a y − )( y a y − ) ... ( y n − a n − y − n − )( y n − a n y − n ) , where y = y n = 1. Observe that(1) y a y − = w y − ∈ H (2) y j a j +1 y − j +1 = ( y j w − j )( w j a j +1 y − j +1 ) = ( y j w − j )( w j +1 y − j +1 ) ∈ H (3) y n − a n y n = ( y n − w − n − )( w n − a n ) = ( y n − w − n − ) w n ∈ H since w n = h ∈ H This establishes that y i − a i y − i ∈ H for all i = 1 , , ...n . If a i ∈ X , then a i ∈ H . Bynormality, y i a i y − i ∈ H . Observe that y i − a i y − i = ( y i − y − i )( y i a i y − i ) . Since y i − a i y − i ∈ H , we must have that y i − y − i ∈ H , which implies that y i H = y i − H .Since Y was a collection of distinct coset representatives, this implies that y i = y i − . Itfollows from Equation 4, that | y i − a i y − i | X ≤ L .If a i ∈ Y , then | y i − a i y − i | Y ≤
3. Since Y is finite and y i − a i y − i ∈ H , we can find aconstant N (independent of i ) such that | y i − a i y − i | X ≤ N in this case as well. Take M = max { L, N } . Then | h | X ≤ M | h | X ∪ Y .Since Γ( H, X ) is hyperbolic, so is Γ(
G, X ∪ Y ). Hence [ X ∪ Y ] ∈ H ( G ). That this is thelargest element in H ( G ) follows from an argument almost identical to the one in the proofof Proposition 1.4, by considering [ Z ] ∈ H ( G ) instead (since we do not have to worry aboutacylindricity in this case). 6 eferenceseferences