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 π, given some mixing conditions on π. 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 π |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 ℓ p -cohomology in degree one are obtained as an intermediate step.

Isoperimetric inequalities, shapes of Følner sets and groups with Shalom’s property H FD

“…As we have mentioned in the Introduction, a group G is said to have Shalom's property H FD if every orthogonal G-representation π with nonzero reduced cohomology H 1 (G, π) admits a finite-dimensional sub-representation. By a result of Gournay [17,Theorem 4.7], if the quotient of a group over its F C-centre has Shalom's property H FD , then the group also has this property. Recall that the conjugacy class of an element g ∈ G is the set {hgh −1 : h ∈ G}.…”

Shalom’s property HFD and extensions by ℤ of locally finite groups

“…The proof of Fact 3.4 applies. Again, the group ∆ is an extension of its FC-center by A × B ≀ Z, hence has Shalom's property H T by [Gou16].…”

“…Recall that given a unitary representation π : G → H, a cocycle b : G → H is called a virtual coboundary if b(g) = π(g)x − x for some x ∈ W \ H, where W is a vector space where the unitary representation π extends to a linear action on W . Finding virtual coboundaries is a useful tool to exhibit cocycles with certain properties, see for example [FV14], [Gou16].…”

“…In one direction, in Proposition 4.1 we show that the wreath product (Z/2Z) ≀ Z d with d ≥ 3 does not have property H FD , while it has finite Hirsch length. In the other direction, we construct a solvable group of infinite Hirsch length which has property H FD by applying a result of Gournay [Gou16], see Proposition 3.6.…”

“…In fact, examples of groups with property H FD not satisfying the assumption of Erschler-Ozawa's corollary mentioned above can be obtained without Theorem 1.1 using a result of Gournay [Gou16], who proved that harmonic cocycles of weakly mixing representations factorize by the FC-center Z FC , which consists of elements with finite conjugacy classes. Indeed, the construction of [BZ15, Section 2] can be modified by taking finite factor groups.…”