Hierarchically hyperbolic groups are determined by their Morse boundaries

Mousley, Sarah C.; Russell, Jacob

doi:10.1007/s10711-018-0402-x

Cited by 3 publications

(2 citation statements)

References 25 publications

(41 reference statements)

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“…[17] for the definition of the Morse boundary). By [39,Lemma 9], there exists R > 0, depending on M, G, and S such that every bi-infinite geodesic in Cay(G, S) with endpoints U + and U − is contained in the R-neighborhood of p u . We can now follow the proof of [10, Theorem III.H.3.17] verbatim.…”

Section: Definition 38mentioning

confidence: 99%

The local-to-global property for Morse quasi-geodesics

2021

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We show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

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Section: Definition 38mentioning

confidence: 99%

The local-to-global property for Morse quasi-geodesics

2021

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“…The proofs in that setting are somewhat easier. We have also recently learned that Sarah Mousley and Jacob Russel have proved an analogous result for Hierarchically Hyperbolic Groups (HHGs) .…”

Section: Introductionmentioning

confidence: 90%

Quasi‐Mobius Homeomorphisms of Morse boundaries

Charney

Cordes

Murray

2019

Bull. London Math. Soc.

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The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic‐like behavior in the space. A key property of this boundary is that a quasi‐isometry between two such spaces induces a homeomorphism on their Morse boundaries. In this paper, we investigate when the converse holds. We prove that for X,Y proper, cocompact spaces, a homeomorphism between their Morse boundaries is induced by a quasi‐isometry if and only if the homeomorphism is quasi‐mobius and 2‐stable.

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Maps on the Morse boundary

Liu

2022

Conform. Geom. Dyn.

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For a proper geodesic metric space, the Morse boundary focuses on the hyperbolic-like directions in the space. The Morse boundary is a quasi-isometry invariant. That is, a quasi-isometry between two spaces induces a homeomorphism on their Morse boundaries. In this paper, we investigate additional structures on the Morse boundary which determine the space up to a quasi-isometry. We prove that a homeomorphism between the Morse boundaries of two proper, cocompact spaces is induced by a quasi-isometry if and only if both the homeomorphism and its inverse are bihölder, or quasi-symmetric, or strongly quasi-conformal.

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Hierarchically hyperbolic groups are determined by their Morse boundaries

Cited by 3 publications

References 25 publications

The local-to-global property for Morse quasi-geodesics

The local-to-global property for Morse quasi-geodesics

Quasi‐Mobius Homeomorphisms of Morse boundaries

Maps on the Morse boundary

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