Group actions on metric spaces: fixed points and free subgroups

TL;DR

This paper studies how groups act on metric spaces, especially hyperbolic spaces, classifies automorphisms, and explores the density of hyperbolic limit sets, with results extending to non-hyperbolic graphs.

Contribution

It provides a classification of automorphisms and new results on the density of hyperbolic limit sets in group actions on metric spaces.

Findings

Classified types of automorphisms on geodesic hyperbolic spaces

Proved density of hyperbolic limit sets in the whole limit set

Results extend to non-hyperbolic graphs

Abstract

We look at group actions on metric spaces, particularly at group actions on geodesic hyperbolic spaces. We classify the types of automorphisms on these spaces and prove several results about the density of the hyperbolic limit set of the group in the whole limit set of the group. In the case of graphs, our theorems hold also when the graphs are not hyperbolic.