A refinement of bounded cohomology

Abstract

We introduce a refinement of bounded cohomology and prove that the suitable comparison homomorphisms vanish for an amenable group. We investigate in this context Thompson's group 𝐹 and provide further evidence towards its amenability. We show that the space of 1-bounded cocycles of degree two is essentially as big as the space of Lipschitz functions on the underlying group. We also explain that such classes define metrics on the group.