Hyperbolic groups and spherical minimal surfaces

Abstract

Let $M$ be a closed, oriented, negatively curved, $n$-dimensional manifold with fundamental group $\Gamma$. Let $S^{\infty }$ be the unit sphere in ${\ell }^2(\Gamma )$, on which $\Gamma$ 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 $S^{\infty }/\Gamma$, 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.