Geometry of the Regular Representation of Hyperbolic Groups

Explore geometric properties of the regular representation of hyperbolic groups in this 55-minute lecture from the Hausdorff Center for Mathematics. Antoine Song examines how a natural quotient Q of a Hilbert sphere can be constructed from the regular representation of a torsion-free hyperbolic group G. Discover the geometry of this infinite-dimensional Riemannian space Q and its ultralimit Q_ω, which encodes Q's asymptotic geometry and possible limits of the regular representation. Learn about properties of both spaces and an important application: how the spherical volume (a topological invariant defined by Besson-Courtois-Gallot) of a negatively curved manifold is realized by a minimal surface inside the corresponding Q_ω. This research contributes to the broader challenge of constructing a "minimal surface geometry" on closed manifolds.