Mixing, malnormal subgroups and cohomology in degree one

TL;DR

This paper investigates how mixing conditions on group representations influence the structure of the group, particularly regarding cohomology in degree one, malnormal subgroups, and the finiteness of the FC-centre.

Contribution

It introduces new connections between mixing properties, cohomology, and subgroup structure, including the concept of q-normal subgroups and results on reduced ll^p-cohomology.

Findings

The FC-centre is finite under mildly mixing conditions.

Subgroups are either small, almost-malnormal, or have non-trivial cohomology.

Results on vanishing of reduced ll^p-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 $\pi$. 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.