Largest acylindrical actions and Stability in hierarchically hyperbolic groups
By Carolyn R. Abbott, Jason A. Behrstock. Transactions of the American Mathematical Society. Series B, 2017
73 citations recorded by the source corpus.
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 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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5b87a901c26466a7946c62c1126c1a7abf880257
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- On strongly quasiconvex subgroups (2017, methodology): “ble in GΓ. Question 7.13. Can we characterize all strongly quasiconvex (stable) subgroups of right-angled Coxeter groups using defining graphs? We guess that the recent work of Abbott-Behrstock-Durham [ABD] can help us characterize all stable subgroups of right-angled Coxeter groups using defining gra”
- Hyperbolicities in CAT(0) cube complexes (2017, unknown): “) However, it was proved in [ABD17] that universal acylindrical actions exist for groups acting on hierarchically hyperbolic spaces, which include CAT(0) cube complexes, hence:”