M.Sc Thesis
M.Sc Student Hase Antonius A study of the Out(F2)-action on second bounded cohomology Hb2(Fn,R) Department of Mathematics Dr. Tobias Hartnick

Abstract

The Out(G)-action on the group cohomology H^k(G,R) of a group G is an important object of study in group theory. On the contrary, almost nothing is known about the corresponding Out(G)-action on the bounded group cohomology H^k_b(G,R). In this thesis we study the case where G = F_n is a free group and k = 2. Here we can identify the second bounded cohomology H^2_b(F_n,R) with the space of homogeneous quasimorphisms on the free group Q^~(F_n) up to homomorphisms. We concentrate in this thesis on the Brooks space B^~(F_n), which is a dense subspace of Q^~(F_n), and invariant under the action of Out(F_n). We introduce a norm on the Brooks space and on the corresponding subspace of H^2_b(F_n,R). This norm allows us to introduce a notion of speed of an element X Out(F_n) on an element [f] of the Brooks space: The speed of X on [f] measures how fast the norm of [f] grows asymptotically under repeated applications of the outer automorphism X. For a special element of Out(F_n) we find a way to compute the speed on any element of the Brooks space. In particular we show that no non trivial element of the Brooks space B^~(F_n) and of the corresponding subspace of H^2_b(F_n,R) is fixed by the Out(F_n)-action. This answers a question of Miklós Abért partially.