Hyperbolic groups and spherical minimal surfaces

Understand

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.