Automorphism groups of free groups, surface groups and free abelian groups

Abstract

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups ${\rm{GL}}(n,\Z)$, the groups of outer automorphisms of free groups of rank $n\geq 2$, and the mapping class groups of closed orientable surfaces of genus $g\geq 1$. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous. In this article we highlight a few of the most striking similarities and differences between these series of groups and present a list of open problems motivated by this philosophy.