** "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.