MS Thesis Defense: Ms. Monica Cabria

This thesis investigates the relationship between convergence group actions and hyperbolic spaces. Since the 1990s, this relationship has been a significant theme in geometric group theory, revealing deep connections between geometry and dynamics. The goal of this thesis is to provide a streamlined introduction to these ideas and to clarify the proofs of several foundational results.

In particular, we discuss a theorem of Tukia and Freden showing that properly discontinuous actions on hyperbolic spaces induce convergence group actions on their boundaries, and a theorem of Bowditch characterizing convergence group actions through the induced action on the space of distinct triples. Together, these results illustrate how geometric properties of hyperbolic spaces are reflected in the dynamics of group actions on their boundaries.