Spaces with measured walls, the Haagerup property and property (T)

We introduce the notion of a space with measured walls, generalizing the concept of a space with walls due to Haglund and Paulin (Simplicite de groupes d'automorphismes d'espaces courbure negative. Geom. Topol. Monograph 1 (1998), 181-248). We observe that if a locally compact group G acts properly on a space with measured walls, then G has the Haagerup property. We conjecture that the converse holds and we prove this conjecture for the following classes of groups: discrete groups with the Haagerup property, closed subgroups of SO(n, 1), groups acting properly on real trees, SL2(K) where K is a global field and amenable groups.