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Hierarchically hyperbolic groups are determined by their Morse boundaries

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Abstract

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.

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Notes

  1. After the release of a preprint of this paper, Charney et al. [10] showed that all finitely generated groups have the small cross-ratio property, thus extending our Main Theorems to all finitely generated groups.

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Acknowledgements

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.

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Authors and Affiliations

  1. University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, USA

    Sarah C. Mousley

  2. CUNY Graduate Center, 365 5th Ave, New York, NY, 10016, USA

    Jacob Russell

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  1. Sarah C. Mousley
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  2. Jacob Russell
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Correspondence to Sarah C. Mousley.

Additional information

Second author was supported by NSF Grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representation Varieties” (the GEAR Network).

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Mousley, S.C., Russell, J. Hierarchically hyperbolic groups are determined by their Morse boundaries. Geom Dedicata 202, 45–67 (2019). https://doi.org/10.1007/s10711-018-0402-x

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