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Convexity in hierarchically hyperbolic spaces

Jacob Russell, Davide Spriano and Hung Cong Tran

Algebraic & Geometric Topology 23 (2023) 1167–1248

Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3–manifold groups. We investigate strongly quasiconvex subsets in this class and characterize them in terms of their contracting properties, relative divergence, the coarse median structure, and the hierarchical structure itself. Along the way, we obtain new tools to study HHSs, including two new equivalent definitions of hierarchical quasiconvexity and a version of the bounded geodesic image property for strongly quasiconvex subsets. Utilizing our characterization, we prove that the hyperbolically embedded subgroups of hierarchically hyperbolic groups are precisely those that are almost malnormal and strongly quasiconvex, producing a new result in the case of the mapping class group. We also apply our characterization to study strongly quasiconvex subsets in several specific examples of HHSs. We show that while many commonly studied HHSs have the property that every strongly quasiconvex subset is either hyperbolic or coarsely covers the entire space, right-angled Coxeter groups exhibit a wide variety of strongly quasiconvex subsets.

hierarchically hyperbolic spaces, strong quasiconvexity, stability, contracting subsets
Mathematical Subject Classification
Primary: 20F65, 20F67
Received: 16 September 2020
Revised: 17 June 2021
Accepted: 26 September 2021
Published: 6 June 2023
Jacob Russell
Math Department
Rice University
Houston, TX
United States
Davide Spriano
Mathematical Institute
University of Oxford
United Kingdom
Hung Cong Tran
Department of Mathematics
The University of Oklahoma
Norman, OK
United States

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