The topology and geometry of automorphism groups of free groups

TL;DR

This paper explores the geometric and topological properties of automorphism groups of free groups, emphasizing the role of Outer space and its applications across mathematics and related fields.

Contribution

It reviews recent developments in understanding automorphism groups of free groups through geometric and topological methods, highlighting new connections and ongoing research directions.

Findings

Outer space provides a rich geometric framework for studying automorphism groups.

Connections between automorphism groups and diverse mathematical areas have been established.

Recent advances reveal new structural insights into $Out(F_n)$ and related groups.

Abstract

In the 1970s Stallings showed that one could learn a great deal about free groups and their automorphisms by viewing the free groups as fundamental groups of graphs and modeling their automorphisms as homotopy equivalences of graphs. Further impetus for using graphs to study automorphism groups of free groups came from the introduction of a space of graphs, now known as Outer space, on which the group $Out(F_n)$ acts nicely. The study of Outer space and its $Out(F_n)$ action continues to give new information about the structure of $Out(F_n)$, but has also found surprising connections to many other groups, spaces and seemingly unrelated topics, from phylogenetic trees to cyclic operads and modular forms. In this talk I will highlight various ways these ideas are currently evolving.