Largest acylindrical actions and Stability in hierarchically hyperbolic groups
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- by Carolyn Abbott, Jason Behrstock and Matthew Gentry Durham; With an appendix by Daniel Berlyne and Jacob Russell
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 66-104
- DOI: https://doi.org/10.1090/btran/50
- Published electronically: February 16, 2021
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Abstract:
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.
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Bibliographic Information
- Carolyn Abbott
- Affiliation: Department of Mathematics, Columbia University, New York, New York
- MR Author ID: 1171294
- Email: abbott@math.columbia.edu
- Jason Behrstock
- Affiliation: Department of Mathematics, Lehman College and The Graduate Center, CUNY, New York, New York
- MR Author ID: 789183
- ORCID: 0000-0002-7652-0374
- Email: jason.behrstock@lehman.cuny.edu
- Matthew Gentry Durham
- Affiliation: Department of Mathematics, University of California, Riverside, Riverside, California
- MR Author ID: 1134962
- Email: matthew.durham@ucr.edu
- Daniel Berlyne
- ORCID: 0000-0002-3193-4848
- Jacob Russell
- MR Author ID: 1190611
- ORCID: 0000-0001-6480-9405
- Received by editor(s): May 31, 2019
- Received by editor(s) in revised form: January 17, 2020, May 11, 2020, and July 22, 2020
- Published electronically: February 16, 2021
- Additional Notes: The authors were supported in part by NSF grant DMS-1440140 while at the Mathematical Sciences Research Institute in Berkeley during Fall 2016 program in Geometric Group Theory. The first author was supported by the NSF RTG award DMS-1502553 and NSF award DMS-1803368. The second author was supported by NSF award DMS-1710890, and the third author was supported by NSF RTG award DMS-1045119 and NSF award DMS-1906487.
- © Copyright 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 66-104
- MSC (2020): Primary 20F55, 20F65, 20F67
- DOI: https://doi.org/10.1090/btran/50
- MathSciNet review: 4215647