Boundary representations of non-positively curved groups

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The proposed research generally fits into the framework of Noncommutative Geometry, in particular research related to the Baum--Connes conjecture, analysis on groups, and representation theory. The Baum--Connes conjecture connects geometry, topology and algebra. From one point of view, it proposes a way to understand the algebraic topology (K-theory) of (a part of) the representation space of a group. While it is possible to effectively describe all the representations of (semisimple) Lie groups, this task is impossible for discrete groups in general. Here we propose to construct explicit families of representations for large classes of discrete groups, using geometry (non-positive curvature) and boundaries. They directly address important questions (Shalom's conjecture), relate to existing approaches to the Baum--Connes conjecture, and harmonic analysis on discrete groups. The proposed pathway combines ideas from analytic and geometric group theory, representation theory of Lie groups and random walks.The philosophy of this project is to capitalise on, and further develop, connections between Geometric Group Theory and Analysis/Noncommutative Geometry. We propose to construct a "compact picture" for (uniformly bounded) representations of prominent classes of non-positively curved groups.First, we deal with the case where one can do ``combinatorial harmonic analysis'', i.e. the case of groups acting properly on (finite dimensional) CAT(0) cube complexes.Second, we distill the main features of the construction and perform it with hyperbolic groups, thus establishing Shalom's conjecture.