Kazhdan and Haagerup properties from the median viewpoint

TL;DR

This paper explores the relationship between Kazhdan and Haagerup properties and median spaces, establishing new connections and generalizations involving actions on median spaces, measured walls, and $L^p$-spaces.

Contribution

It introduces a novel viewpoint linking measured walls and median spaces, and generalizes the dynamical characterization of property (T).

Findings

Connected measured walls and median spaces.

Relates properties (T) and Haagerup to actions on median spaces.

Answers an open problem on dynamical characterization of property (T).

Abstract

We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affine isometric actions on $L^p$-spaces. It also allows us to answer an open problem on a dynamical characterization of property (T), generalizing results of Robertson-Steger.