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Coarse injectivity, hierarchical hyperbolicity and semihyperbolicity

Thomas Haettel, Nima Hoda and Harry Petyt

Geometry & Topology 27 (2023) 1587–1633

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.

hierarchically hyperbolic, coarsely injective, strongly shortcut, semihyperbolic, hierarchically quasiconvex, bounded packing
Mathematical Subject Classification
Primary: 20F65, 20F67, 51F30
Received: 13 January 2021
Revised: 24 June 2021
Accepted: 2 October 2021
Published: 15 June 2023
Proposed: Urs Lang
Seconded: Mladen Bestvina, Anna Wienhard
Thomas Haettel
Institut Montpelliérain Alexander Grothendieck
Université de Montpellier
Nima Hoda
École normale supérieure
Université PSL
Department of Mathematics
Cornell University
Ithaca, NY
United States
Harry Petyt
School of Mathematics
University of Bristol
United Kingdom
Mathematical Institute
University of Oxford
United Kingdom

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