Groups acting on hyperbolic spaces -- a survey

TL;DR

This survey reviews how groups like mapping class groups and Out(F_n) act on hyperbolic spaces, highlighting current understanding, differences, and open questions in their large-scale geometric properties.

Contribution

It provides a comprehensive overview of group actions on hyperbolic spaces, emphasizing the differences between mapping class groups and Out(F_n) and outlining open research directions.

Findings

Mapping class groups have well-understood actions on hyperbolic spaces.

Understanding of Out(F_n) actions on hyperbolic spaces is less developed.

The paper identifies key open questions in the field.

Abstract

This is a (very subjective) survey paper for nonspecialists covering group actions on Gromov hyperbolic spaces. The first section is about hyperbolic groups themselves, while the rest of the paper focuses on mapping class groups and $Out(F_n)$, and the way to understand their large scale geometry using their actions on various hyperbolic spaces constructed using projection complexes. This understanding for $Out(F_n)$ significantly lags behind that of mapping class groups and the paper ends with a few open questions.