Quasiconvexity in Hierarchically Hyperbolic Spaces

Jacob Russell (CUNY)

Abstract: Quasiconvex subsets of a hyperbolic spaces exhibit many strong geometric properties. In particular, every quasi-geodesic with endpoints on a quasiconvex subset lies within a uniformly bounded amount of that subset. This quasi-geodesic convexity is a quasi-isometry invariant even in non-hyperbolic spaces and has proved to be a key feature of many quasi-isometric rigidity results in hyperbolic spaces and beyond. We study quasi-geodesic convexity in the class of hierarchically hyperbolic spaces. Hierarchically hyperbolic spaces are a generalization of Gromov hyperbolic spaces that includes the mapping class group, right angled Artin groups and many 3-manifold groups. The hallmark of hierarchically hyperbolic spaces is that the coarse geometry can be recovered from a family of projections onto hyperbolic space and we classify quasi-geodesic convexity in these spaces in terms of these projections. As an application of this classification, we show that all almost malnormal quasi-geodesic convex subgroups of hierarchically hyperbolic groups are hyperbolically embedded.