Mixing, malnormal subgroups and cohomology in degree one

Abstract

The aim of the current paper is to explore the implications on the group $G$ of the non-vanishing of the cohomology in degree one of one of its representation $\pi$, given some mixing conditions on $\pi$. For example, harmonic cocycles of weakly mixing unitary representations factorise by the FC-centre. In that case non-vanishing implies the FC-centre is trivial or fixes a vector. Next, for any subgroup $H<G$, $H$ will either be “small,” almost-malnormal or ${\pi }_{|H}$ also has non-trivial cohomology in degree one (in this statement, “small,” reduced vs unreduced cohomology and unitary vs generic depend on the mixing condition). The notion of q-normal subgroups is an important ingredient of the proof and results on the vanishing of the reduced ${\ell }^p$-cohomology in degree one are obtained as an intermediate step.