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

Part of: Birational geometry Global differential geometry Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  30 November 2011

Thomas Delzant  and
Pierre Py
Thomas Delzant
Affiliation:
IRMA, Université de Strasbourg & CNRS, 7 rue René Descartes, 67084 Strasbourg, France (email: delzant@math.unistra.fr)
Pierre Py
Affiliation:
University of Chicago, Chicago, IL 60637, USA (email: pierre.py@math.uchicago.edu)
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Abstract

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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 PSL2(ℝ). We also study actions of Kähler groups on infinite-dimensional real hyperbolic spaces, describe some exotic actions of PSL2(ℝ) on these spaces, and give an application to the study of the Cremona group.

Keywords

Kähler manifoldsharmonic mapsreal hyperbolic space

MSC classification

Secondary: 14E07: Birational automorphisms, Cremona group and generalizations 20E99: None of the above, but in this section 53C55: Hermitian and Kählerian manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 153 - 184
Copyright
Copyright © Foundation Compositio Mathematica 2011

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