The topology and geometry of automorphism groups of free groups

Free groups, free abelian groups and fundamental groups of

closed orientable surfaces are the most basic and well-understood examples

of infinite discrete groups. The automorphism groups of these groups, in

contrast, are some of the most complex and intriguing groups in all of

mathematics. I will give some general comments about geometric group

theory and then describe the basic geometric object, called Outer space,

associated to automorphism groups of free groups.

This Colloquium talk is the first of a series of three lectures given by

Professor Vogtmann, who is the European Mathematical Society Lecturer. In

this series of three lectures, she will discuss groups of automorphisms

of free groups, while drawing analogies with the general linear group over

the integers and surface mapping class groups. She will explain modern

techniques for studying automorphism groups of free groups, which include

a mixture of topological, algebraic and geometric methods.