  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.
