Quasi-isometric Hyperbolic and Cusped Relatively Hyperbolic Groups are Symmetric

Prof. Genevieve Walsh
Professor of Mathematics
Tufts University

This talk will first describe hyperbolic groups and relatively hyperbolic group pairs, and give some key examples. We then delve into understanding when the cusp space for a relatively hyperbolic group can be quasi-isometric to a hyperbolic group. For example, real hyperbolic spaces admit uniform and non-uniform lattices. We show that this is the exception. In particular, if a hyperbolic group is quasi-isometric to a cusped space for a relatively hyperbolic group, then both groups are lattices acting on a rank-1 symmetric space.

This is joint work with Daniel Groves, Emily Stark, and Kevin Whyte.