Weak property $({\mathrm{T}}_{L^p})$ for discrete groups

We show that, for a countable discrete group $\mathrm{\Gamma }$, property $({\mathrm{T}}_{L^p})$ of Bader, Furman, Gelander and Monod is equivalent to the property that, whenever an $L^p$-representation of $\mathrm{\Gamma }$ admits a net of almost invariant unit vectors, it has a non-zero 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.