When dealing with geometric structures one natural question that arise is "when does a subset inherit the geometry of the ambient space"? In the case of hyperbolic space, the concept of quasi-convexity provides answer to this question. However, for a general metric space, being quasi-convex is not a quasi-isometric invariant. This motivates the notion of Morse subsets. In this talk we will motivate the definition and introduce some examples. Then we will introduce the class of hierarchically hyperbolic groups (HHG), and furnish a complete characterization of Morse subgroups of HHG. If time allows, we will discuss the relationship between Morse subgroups and hyperbolically-embedded subgroups. This is a joint work with Hung C. Tran and Jacob Russell.
