Quasi-convexity of hyperbolically embedded subgroups

Abstract

We show that any infinite order element g of a virtually cyclic hyperbolically embedded subgroup of a group G is Morse, that is to say any quasi-geodesic connecting points in the cyclic group C generated by g stays close to C. This answers a question of Dahmani–Guirardel–Osin. What is more, we show that hyperbolically embedded subgroups are quasi-convex. Finally, we give a definition of what it means for a collection of subspaces of a metric space to be hyperbolically embedded and we show that axes of pseudo-Anosovs are hyperbolically embedded in Teichmüller space endowed with the Weil-Petersson metric.