Hyperbolic groups and spherical minimal surfaces

Let $M$ be a closed, oriented, negatively curved, $n$-dimensional manifold with fundamental group $\mathrm{\Gamma }$. Let $S^{\mathrm{\infty }}$ be the unit sphere in ${\mathrm{\ell }}^2(\mathrm{\Gamma })$, on which $\mathrm{\Gamma }$ acts by the regular representation. The spherical volume of $M$ is a topological invariant introduced by Besson-Courtois-Gallot. We show that it is equal to the area of an $n$-dimensional area-minimizing minimal surface inside the ultralimit of $S^{\mathrm{\infty }}\mathrm{/}\mathrm{\Gamma }$, in the sense of Ambrosio-Kirchheim. Our proof combines the theory of metric currents with a study of limits of the regular representation of torsion-free hyperbolic groups.