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I will discuss geometric properties of the regular representation of hyperbolic groups. Given a torsion free hyperbolic group G, one can build a natural quotient Q of a Hilbert sphere from the regular representation of G. I am interested in the geometry of this infinite dimensional Riemannian space Q, and also of its ultralimit Q_omega. This ultralimit encodes the asymptotic geometry of Q, for instance possible limits of the regular representation. I will mention an application: the spherical volume (a topological invariant defined by Besson-Courttois-Gallot) of a negatively curved manifold is realized by a minimal surface inside the corresponding Q_omega. This result is motivated by the problem of constructing a "minimal surface geometry" on closed manifolds.