On fixed point property for $L_p$-representations of Kazhdan groups

Let $G$ be a topological group with finite Kazhdan set, let $Ω$ be a standard Borel space and $μ$ a finite measure on $Ω$. We prove that for any $p\in [1, \infty)$, any affine isometric action $G \curvearrowright L_p(Ω, μ)$ whose linear part arises from an ergodic measure-preserving action $G \curvearrowright (Ω, μ)$, has a fixed point.

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