Reduced 1-cohomology and relative property (T)

Abstract

Shalom characterized property (T) in terms of the vanishing of all reduced first cohomology. We characterize group pairs having the property that the restriction map on all first reduced cohomology vanishes. We show that, in a strong sense, this is inequivalent to relative property (T).

Amenable invariant random subgroups

“…In case one considers non spanning IRSs, our methods lead to the following statement for G countable: The normal closure of an IRS with relative property (T) has relative property (FH). We refer to [FVM12] for the definition of property (FH) and its relative version. We emphasize that even if Shalom's theorem shows that property (T) and property (FH) coincide for compactly generated groups, the relative versions do not coincide.…”

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

“…The question asked if π is weakly mixing and there is an infinite N ⊳G with G/N cyclic and H 1 (N, π |N ) = 0, does H 1 (G, π) = 0? Note that when N is locally finite, it is a direct limit of finite groups, therefore H 1 (N, σ) = 0 for any unitary representation σ of N , see for example [FVM12,Lemma 5]. Therefore examples of locally-finite-by-Z groups without property H FD answer this question negatively.…”

The Mayer–Vietoris sequence for graphs of groups, property (T), and the first $\ell^2$-Betti number

“…(The relative analogue of Shalom's theorem fails in general [FVM12].) A locally compact group G has Serre's Property (FA) if every continuous, isometric action of G on a tree preserves a vertex or an edge.…”