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

TL;DR

This paper proves that for Kazhdan groups, any affine isometric action on Lp spaces with ergodic measure-preserving linear parts has a fixed point, extending fixed point properties to a broad class of Banach spaces.

Contribution

It establishes a fixed point property for Kazhdan groups acting on Lp spaces with ergodic linear parts, generalizing previous results in the area.

Findings

Affine isometric actions on Lp spaces have fixed points for Kazhdan groups.

The fixed point property holds for actions with ergodic measure-preserving linear parts.

The result applies to all p in [1, ∞).

Abstract

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