Emilie Elkiær: Weak property (T Lp) for discrete groups

Abstract: Property \( (\mathrm{T}_{L^p}) \) of Bader, Furman, Gelander and Monod is a rigidity property concerning how a group may act on an \( L^p \)-space. A group \( \Gamma\) has property \( (\mathrm{T}_{L^p}) \) if any net of almost \( \Gamma \)-invariant unit vectors in a \( L^p \)-space must approach the subspace of \( \Gamma \)-invariant vectors. We show that, for a countable discrete group \( \Gamma \), property \( (\mathrm{T}_{L^p}) \) is equivalent to the property that, whenever an \( L^p \)-representation of \( \Gamma \) admits a net of almost \( \Gamma \)-invariant unit vectors, it has a non-zero \( \Gamma \)-invariant vector. Central in the proof is to show that the closure of the group of \( \mathbb{T} \)-valued 1-coboundaries is a sufficient criteria for strong ergodicity of ergodic p.m.p. actions. This talk is based on the recent preprint arXiv:2403.05312.