Density of filling annuli in the space of metric structures of a hyperbolic group

A major theme in geometric group theory is that finitely generated groups (and particularly hyperbolic groups) can be understood via their isometric actions on metric spaces. Hence, for a hyperbolic group G, it’s natural to consider the deformation space D(G) consisting of all “well-behaved” isometric actions of G and try to understand how such actions relate to one another. I will discuss ongoing work with Kerr exhibiting several dense subspaces of D(G). Time permitting; I will also give an example of a hyperbolic group admitting an exotic isometric action on a hyperbolic space ( a cobounded acylindrical action that fails the bounded backtracking property) answering a question by Cantrell-Reyes.