Hierarchically hyperbolic groups are determined by their Morse boundaries
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We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of top-heavy hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are quasi-isometric if and only if there exists a homeomorphism between their Morse boundaries such that the map and its inverse are 2-stable, quasi-möbius. Our result extends a recent result of Charney–Murray, who prove such a classification for CAT(0) groups, and is new for mapping class groups and the fundamental groups of 3-manifolds without Nil or Sol components.
KeywordsMorse boundary Hierarchically hyperbolic group Quasi-möbius
Mathematics Subject Classification20F65 20F36 20F67
The authors would like to thank Devin Murray, Ruth Charney, and Matthew Cordes for helpful conversation about the Morse boundary and Mark Hagen for his assistance with hierarchically hyperbolic spaces. The authors are also very grateful for the support and guidance of their advisors, Chris Leininger and Jason Behrstock. The second author would like to thank Chris Leininger for his mentorship and hospitality during a visit funded by NSF Grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network), where the work on this project began.
- 1.Abbott, C., Behrstock, J., Durham, M.G.: Largest acylindrical action and stability in hierarchically hyperbolic groups (2017). arXiv:1509.00632v2
- 2.Behrstock, J., Hagen, M.F., Sisto, A.: Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015). arXiv:1509.00632v2
- 10.Charney, R., Cordes, M., Murray, D.: Quasi-möbius homeomorphisms of Morse boundaries. arXiv:1801.05315 (2018)
- 11.Charney, R., Murray, D.: A rank-one CAT(0) group is determined by its Morse boundary (2017). arXiv:1707.07028v1
- 14.Cordes, M.: A survey on Morse boundaries & stability (2017). arXiv:1704.07598v1
- 18.Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, vol. 182, pp. 1–295. Cambridge University Press, Cambridge (1993)Google Scholar
- 19.Hagen, M.F., Susse, T.: On hierarchical hyperbolicity of cubical groups (2016). arXiv:1609.01313v2
- 20.Kapovich, I., Benakli, N.: Boundaries of hyperbolic groups. In: Combinatorial and Geometric Group Theory (New York, 2000/Hoboken, NJ, 2001), Contemporary Mathematics, vol. 296, pp. 39–93. American Mathematical Society, Providence, RI (2002). https://doi.org/10.1090/conm/296/05068
- 26.Sisto, A.: What is a hierarchically hyperbolic space? (2017). arXiv:1707.00053
- 27.Vokes, K.M.: Hierarchical hyperbolicity of the separating curve graph (2017). arXiv:1711.03080