Kähler groups, real hyperbolic spaces and the Cremona group. With an appendix by Serge Cantat

Abstract

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a Kähler group on the real hyperbolic space of dimension at least three factors through a homomorphism onto a cocompact discrete subgroup of PSL 2 (R). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL 2 (R) on these spaces, and give an application to the study of the Cremona group.

Random walks on weakly hyperbolic groups

“…Manin [Man74] showed that the Cremona group acts faithfully by isometries on an infinite-dimensional hyperbolic space, known as the Picard-Manin space, which is not separable. However, any finite-generated subgroup preserves a totally geodesic closed subspace, which is separable, see for example Delzant and Py [DP12].…”

Geometric Perspectives on Reproducing Kernels

“…5.1]) for Banach vector bundles (with infinite-dimensional base). Results in this direction were recently obtained in [15] and [35].…”

Superrigidity in infinite dimension and finite rank via harmonic maps

“…It is a standard fact that the most difficult part to obtain smoothness of weak harmonic maps is the first regularity step, which is the continuity of the harmonic map (see for example [Jos11,8.4]). In our situation, we already know that the harmonic map is Lipschitz and we can easily adapt the argument given in [DP12], where the target is the infinite dimensional hyperbolic space. Proof.…”