K\"ahler groups, real hyperbolic spaces and the Cremona group

TL;DR

This paper extends classical results on K"ahler groups acting on hyperbolic spaces, showing such actions factor through specific subgroups, and explores their actions on infinite-dimensional hyperbolic spaces with applications to the Cremona group.

Contribution

It generalizes Carlson and Toledo's theorem to higher dimensions and infinite-dimensional spaces, providing new insights into K"ahler group actions and their relation to the Cremona group.

Findings

Any Zariski dense isometric action of a K"ahler group on real hyperbolic space of dimension ≥3 factors through a cocompact subgroup of PSL(2,R)

Descriptions of exotic actions of PSL(2,R) on infinite-dimensional hyperbolic spaces

Applications to the structure and actions of the Cremona group

Abstract

Generalizing a classical theorem of Carlson and Toledo, we prove that any Zariski dense isometric action of a K\"{a}hler group on the real hyperbolic space of dimension at least 3 factors through a homomorphism onto a cocompact discrete subgroup of PSL(2,R). We also study actions of K\"{a}hler 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.