"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.