Pulling back stability with applications to Out(Fn) and relatively hyperbolic groups

Aougab, Tarik; Durham, Matthew Gentry; Taylor, Selwyn

doi:10.1112/jlms.12071

Cited by 15 publications

(26 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There has been much interest in developing alternative characterizations [DMS10,CS15,ACGH17,ADT17] and understanding this phenomenon in various important contexts [Min96,Beh06,DMS10,DT15,ADT17]. This includes the theory of Morse boundaries, which encode all Morse geodesics of a group [CS15,Cor17,CH17,CD19,CM19].…”

Section: Introductionmentioning

confidence: 99%

Largest acylindrical actions and Stability in hierarchically hyperbolic groups

Abbott¹,

Behrstock²,

Durham³

2021

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, right-angled Coxeter groups, most 3-manifold groups, right-angled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a "best" one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and right-angled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. In the appendix, it is verified that any space satisfying the a priori weaker property of being an "almost hierarchically hyperbolic space" is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.

show abstract

Section: Introductionmentioning

confidence: 99%

Largest acylindrical actions and Stability in hierarchically hyperbolic groups

Abbott¹,

Behrstock²,

Durham³

2021

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

show abstract

“…The idea is that a short curve must pass near the 'middle third' of the subsegment of Z connecting its endpoints. The property, again only for quasi-geodesics, but for variable t, is called 'middle recurrence' in[3].…”

mentioning

confidence: 99%

A metrizable topology on the contracting boundary of a group

Cashen

Mackay

2019

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.2010 Mathematics Subject Classification. 20F65, 20F67.1 In even more recent developments, Behrstock [6] produces interesting examples of right-angled Coxeter groups whose Morse boundaries contain a circle, and Charney and Murray [13] give conditions that guarantee that a homeomorphism between Morse boundaries of CAT(0) spaces is induced by a quasi-isometry.

show abstract

“…If H is quasi-isometrically embedded in FF n under the orbit map, then H is stable in Out(F n ) by [9]. Hence by Theorem 5.1 there exists a fully irreducible element f such that no power of f is conjugate into H. We will show that such an element is transverse to H in FF n .…”

Section: Out(f N )mentioning

confidence: 91%

“…In many cases when G is a group with a non-elementary partially WPD action on a hyperbolic metric space X, the subgroups of G which are quasi-isometrically embedded in X are precisely the stable subgroups of G in the sense of Durham-Taylor [2,9,29,45]. We prove an analogue of [54,Proposition 1] in the setting of a stable subgroup H of an acylindrically hyperbolic group G (Theorem 5.1).…”

Section: Introductionmentioning

confidence: 98%

Random walks and quasi‐convexity in acylindrically hyperbolic groups

Abbott

Hull

2021

Journal of Topology

View full text Add to dashboard Cite

Arzhantseva proved that every infinite-index quasi-convex subgroup H of a torsion-free hyperbolic group G is a free factor in a larger quasi-convex subgroup of G. We give a probabilistic generalization of this result. That is, we show that when R is a subgroup generated by independent random walks in G, then H, R ∼ = H * R with probability going to one as the lengths of the random walks go to infinity and this subgroup is quasi-convex in G. Moreover, our results hold for a large class of groups acting on hyperbolic metric spaces and subgroups with quasi-convex orbits. In particular, when G is the mapping class group of a surface and H is a convex cocompact subgroup we show that H, R is convex cocompact and isomorphic to H * R.

show abstract

Pulling back stability with applications to Out(Fn) and relatively hyperbolic groups

Cited by 15 publications

References 33 publications

Largest acylindrical actions and Stability in hierarchically hyperbolic groups

Largest acylindrical actions and Stability in hierarchically hyperbolic groups

A metrizable topology on the contracting boundary of a group

Random walks and quasi‐convexity in acylindrically hyperbolic groups

Contact Info

Product

Resources

About