From $\mathrm{CAT}(0)$ to relatively hyperbolic

Abstract

Hruska and Kleiner proved that a $\mathrm{CAT}(0)$ group with isolated flats is hyperbolic relative to the stabilizers of maximal flats using asymptotic cones. We give a different proof that a $\mathrm{CAT}(0)$ group with isolated flats is hyperbolic relative to the stabilizers of the maximal flats. The approach is entirely via the boundary and uses the classification of relatively hyperbolic groups due to Yaman. In the process, we construct the relatively hyperbolic boundary of the group as the “obvious” quotient of the $\mathrm{CAT}(0)$ boundary, which was first shown in Tran (2013). Along the way, we give reasonable conditions under which a $\mathrm{CAT}(0)$ group acts as a convergence group on a quotient of its boundary. We also define the domain of discontinuity (in the boundary) of a convex subgroup of a $\mathrm{CAT}(0)$ group (on which the convex subgroup acts properly discontinuously) and give conditions under which the convex subgroup acts cocompactly on its domain of discontinuity.