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We prove the existence of a close connection between spaces with measuredwalls 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 ofproperty (T), generalizing results of Robertson-Steger.

Kazhdan and Haagerup properties from the median viewpoint