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Coarse injectivity,
hierarchical hyperbolicity and semihyperbolicity
Thomas Haettel, Nima Hoda and Harry Petyt |
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Geometry & Topology 27 (2023) 1587–1633
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Abstract |
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We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is coarsely injective. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that every coarsely injective metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups — including mapping class groups of surfaces — are coarsely injective and coarsely injective groups are strongly shortcut. Using these results, we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, they have solvable conjugacy problem and finitely many conjugacy classes of finite subgroups, and their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing. |
Keywords
hierarchically hyperbolic, coarsely injective, strongly
shortcut, semihyperbolic, hierarchically quasiconvex,
bounded packing
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Mathematical Subject Classification
Primary: 20F65, 20F67, 51F30
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References |
Publication
Received: 13 January 2021
Revised: 24 June 2021
Accepted: 2 October 2021
Published: 15 June 2023
Proposed: Urs Lang
Seconded: Mladen Bestvina, Anna Wienhard
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Authors |
| Thomas Haettel | |
| Institut Montpelliérain Alexander
Grothendieck CNRS Université de Montpellier Montpellier France |
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| Nima Hoda | |
| DMA École normale supérieure Université PSL CNRS Paris France |
Department of Mathematics Cornell University Ithaca, NY United States |
| Harry Petyt | |
| School of Mathematics University of Bristol Bristol United Kingdom |
Mathematical Institute University of Oxford Oxford United Kingdom |
| © 2023 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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