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 and Steger.
Abstract. We explain how to adapt a construction due to M. Sageev in order to construct a proper action of a group on a CAT(0) cube complex starting from a proper action of the group on a wall space.
Abstract. We define a bounded cohomology class, called the median class, in the second bounded cohomology -with appropriate coefficients -of the automorphism group of a finite dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the median class of an action by automorphisms of X.We show that the median class of a non-elementary action by automorphisms does not vanish and we show to which extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show for example that any irreducible lattice in the product of at least two locally compact connected groups acts on a finite dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X.In the course of the proof we construct a Γ-equivariant measurable map from a Poisson boundary of Γ with values in the non-terminating ultrafilters on the Roller boundary of X.
We explain simple methods to establish the property of Rapid Decay for a number of groups arising geometrically. Those lead to new examples of groups with the property of Rapid Decay, notably including non-cocompact lattices in rank one Lie groups.
International audienceFor a locally compact group, property RD gives a control on the convolution norm of any compactly supported measure in terms of the $L^2$-norm of its density and the diameter of its support. We give a complete classification of those Lie groups with property RD
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