The topology of $out(F_n)$

TL;DR

This paper surveys the topology of Out(F_n), highlighting its complex structure, the construction of Outer Space, and the development of train track maps, with applications to growth rates, fixed subgroups, and the Tits alternative.

Contribution

It provides a comprehensive overview of the topological and combinatorial tools used to study Out(F_n), including the construction of Outer Space and train track maps, and discusses recent advances and applications.

Findings

Outer Space X_n is contractible and admits an Out(F_n) action.

Train track maps provide a normal form for automorphisms.

Applications include understanding growth rates and the Tits alternative.

Abstract

We will survey the work on the topology of $Out(F_n)$ in the last 20 years or so. Much of the development is driven by the tantalizing analogy with mapping class groups. Unfortunately, $Out(F_n)$ is more complicated and less well-behaved. Culler and Vogtmann constructed Outer Space $X_n$, the analog of Teichm\"uller space, a contractible complex on which $Out(F_n)$ acts with finite stabilizers. Paths in $X_n$ can be generated using ``foldings'' of graphs, an operation introduced by Stallings to give alternative solutions for many algorithmic questions about free groups. The most conceptual proof of the contractibility of $X_n$ involves folding. There is a normal form of an automorphism, analogous to Thurston's normal form for surface homeomorphisms. This normal form, called a ``(relative) train track map'', consists of a cellular map on a graph and has good properties with respect to iteration. One may think of building an automorphism in stages, adding to the previous stages a building block that either grows exponentially or polynomially. A complicating feature is that these blocks are not ``disjoint'' as in Thurston's theory, but interact as upper stages can map over the lower stages. Applications include the study of growth rates (a surprising feature of free group automorphisms is that the growth rate of $f$ is generally different from the growth rate of $f^{-1}$), of the fixed subgroup of a given automorphism, and the proof of the Tits alternative for $Out(F_n)$.