Convexity in hierarchically hyperbolic spaces

Abstract

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 quasconvex subsets. Utilizing our characterization, we prove that the hyperbolically embedded subgroups of hierarchically hyperbolic groups are precisely those which 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 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.