Dynamics of Automorphism of Groups: Growth, Entropy, Random walks – DAGGER

Given a finitely generated group G we want to investigate its outer automorphism group Out(G). This natural object was the source of an intense work when G is one of the following examples: a free abelian group, a surface group or a free group. For these groups one observes a strong growth dichotomy: conjugacy classes grow under the iteration of an automorphism either polynomially or exponentially. This remark is the starting point of our project.
We want to understand in a global way what are the groups G satisfying an analogous dichotomy. To that end we will make use of geometric, dynamical and probabilistic tools. This question about growth naturally extends to more general problems regarding the dynamics of Out(G), especially random walks in Out(G).