The topology and geometry of automorphism groups of free groups

Abstract

In the 1970's 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(Fn) acts nicely. The study of Outer space and its Out(Fn) action continues to give new information about the structure of Out(Fn), 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.

Automorphism groups of cyclic products of groups

“…Only few automorphism groups have been studied from a geometric point of view. Indeed, while many groups come with interesting actions associated to them, there is no general recipe for constructing a 'nice' action of Aut(G) out of an action of G. Famous examples where automorphism groups have been studied from a geometric perspective include the (outer) automorphism groups of free groups, which act on their outer-spaces and various hyperbolic graphs (see [Vog02,Vog16]); and the (outer) automorphism groups of surface groups, which essentially coincide with the corresponding mapping class groups and thus act on their Teichmüller spaces and their curve complexes (see [Iva01]).…”

Moduli spaces of colored graphs

“…For a precise definition and further reading on these groups with applications in geometric group theory we refer to [CHKV16] and the survey in [Vog16].…”

Cube complexes and abelian subgroups of automorphism groups of RAAGs

“…In recent years these automorphism groups have been shown to share many properties, but also to differ in significant ways (see e.g. the survey articles [2,16]). In this paper we study automorphism groups of general RAAGs, concentrating on the aspects they share with automorphism groups of free groups.…”