Given a torsion free hyperbolic group G, one can build a natural quotient Q of a Hilbert sphere from the regular representation of G. We will talk about the geometry of this infinite dimensional Riemannian space Q, and also of its ultralimit. This discussion involves studying properties of limits of the regular representation. It has applications to the construction of minimal surfaces related to a topological invariant for manifolds, called the spherical volume. I will also mention some possible connections between this setup and Cannon's conjecture.

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