Mixing, malnormal subgroups and cohomology in degree one

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 $\pi$. In one direction, harmonic cocycles are used to show that the FC-centre should be finite (for mildly mixing unitary representations). 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.