One of the most successful techniques for studying groups acting on metric spaceshas been to study actions on spaces which admit hyperbolic properties. We study a group G acting by isometries on a proper, geodesic metric space X by studying interactions between the group action on the space and hyperbolic-like boundaries for X. We present results regarding two different hyperbolic-like boundaries on X: the Morse boundary and the sublinearly Morse boundary. Both of these boundaries are quasi-isometry invariants for proper geodesic metric spaces.

Subgroup stability is a strong notion of quasiconvexity that generalizes convexcocompactness in a variety of settings. A characterization of convex cocompact Kleinian groups is that the limit set of the group is composed entirely of conical limit points in the boundary of the three dimensional hyperbolic space. We show that stable subgroups admit an identical conical limit point characterization in the Morse boundary. We also, additionally, show that stable subgroups are characterized by having an entirely horospherical limit set.

A group G is non-elementary if G is not virtually cyclic and if its boundary is notempty. We show that every non-elementary group acts minimally on its sublinearly Morse boundary, i.e., for any element in the sublinearly Morse boundary, the G orbit of that element is dense in the boundary. This result, which is joint with Yulan Qing and Elliott Vest, is an important step towards understanding the dynamics of groups acting on their own sublinearly Morse boundary.

We show that the sublinear Morse boundary of every CAT(0) space continuously injects into the Gromovboundary of a hyperbolic space, which was not previously known even for all CAT(0) cube complexes. Our work utilizes the curtain machinery introduced by Petyt-Spriano-Zalloum. Curtains are more general combinatorial analogues of hyperplanes in cube complexes, and we develop multiple curtain characterizations of the sublinear Morse property along the way. The hyperbolic space mentioned is the curtain model, and its role for a CAT(0) space has shown a striking comparison to the curve graph for a mapping class group of a finite type surface. We show that the curtain model is not a quasi-isometry invariant for all CAT(0) spaces and that quasi-flats are of bounded diameter in the curtain model. Our results answer multiple questions of Petyt-Spriano-Zalloum.