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A la mémoire de Martine BabillotAbstract. We introduce the notion of a space with measured walls, generalizing the concept of a space with walls due to Haglund and Paulin (Simplicité de groupes d'automorphismes d'espacesà courbure négative. 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, SL 2 (K) where K is a global field and amenable groups.

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We introduce and systematically study a profile function whose asymptotic behavior quantifies the dimension or the size of a metric approximation of a finitely generated group G by a family of groups F = {(Gα, dα, kα, εα)}α∈I, where each group Gα is equipped with a bi-invariant metric dα and a dimension kα, for strictly positive real numbers εα such that infα εα > 0. Through the notion of a residually amenable profile that we introduce, our approach generalizes classical isoperimetric (aka Følner) profiles of amenable groups and recently introduced functions quantifying residually finite groups. Our viewpoint is much more general and covers hyperlinear and sofic approximations as well as many other metric approximations such as weakly sofic, weakly hyperlinear, and linear sofic approximations.

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