"Geometry and Analysis on Groups" Research Seminar

Time: 2015.01.27, 15:00--17:00
Location: Seminarraum 8, Oskar-Morgenstern-Platz 1, 2.Stock
Title: "The Tits alternative for the automorphism group of a free product."
Speaker: Camille Horbez (Université Rennes 1)
Abstract: A group \(G\) is said to satisfy the Tits alternative if every subgroup of \(G\) either contains a nonabelian free subgroup, or is virtually solvable. The talk will aim at presenting a version of this alternative for the automorphism group of a free product of groups. A classical theorem of Grushko states that every finitely generated group \(G\) splits as a free product of the form \(G=G_1*...*G_k*F_N\), where \(F_N\) is a finitely generated free group, and all \(G_i\)'s are nontrivial, not isomorphic to \(Z\), and freely indecomposable. I prove that if all groups \(G_i\) and \(\mathrm{Out}(G_i)\) in this decomposition satisfy the Tits alternative, then so does the group \(\mathrm{Out}(G)\) of outer automorphisms of \(G\). I will present some applications of this theorem, especially to the case where \(G\) is a right-angled Artin group. I will then present a proof of this result, in parallel to a new proof of the Tits alternative for mapping class groups of surfaces. This relies on a study of the actions of some subgroups of \(\mathrm{Out}(G)\) on a version of the outer space for free products, and on a hyperbolic simplicial graph.