# The space of metric structures on hyperbolic groups

# The space of metric structures on hyperbolic groups

**In-Person and Online Talk **

Teichmuller and Outer spaces are classical constructions used in geometric group theory to understand mapping class groups and outer automorphisms of free groups. These spaces can be understood as moduli spaces of geometric actions of surface groups on the hyperbolic plane and free groups on metric trees, respectively. I will talk about a generalization of these spaces, where for an arbitrary hyperbolic group we consider a moduli space of its geometric actions on Gromov hyperbolic spaces. Equipped with a natural Thurston’s-like metric, this space is contractible and geodesic, and has a “thick” part that is cocompact for the isometric action of the outer automorphism group.

This is joint work with Stephen Cantrell.