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Local quaternionic rigidity for complex hyperbolic lattices

Part of: Lie groups

Published online by Cambridge University Press:  01 September 2010

Inkang Kim ,
Bruno Klingler  and
P. Pansu
Inkang Kim
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, Hoegiro 87, Dongdaemun-gu, Seoul, 130-722, Korea (inkang@kias.re.kr)
Bruno Klingler
Affiliation:
Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, 75013 Paris, France (klingler@math.jussieu.fr) and Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA
P. Pansu
Affiliation:
Laboratoire de Mathématiques d'Orsay, UMR 8628 du CNRS, Université Paris-Sud, 91405 Orsay Cédex, France (pierre.pansu@math.u-psud.fr)
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Abstract

Let be a lattice in the real simple Lie group L. If L is of rank at least 2 (respectively locally isomorphic to Sp(n, 1)) any unbounded morphism ρ : Γ → G into a simple real Lie group G essentially extends to a Lie morphism ρL : L → G (Margulis's superrigidity theorem, respectively Corlette's theorem). In particular any such morphism is infinitesimally, thus locally, rigid. On the other hand, for L = SU(n, 1) even morphisms of the form are not infinitesimally rigid in general. Almost nothing is known about their local rigidity. In this paper we prove that any cocompact lattice Γ in SU(n, 1) is essentially locally rigid (while in general not infinitesimally rigid) in the quaternionic groups Sp(n, 1), SU(2n, 2) or SO(4n, 4) (for the natural sequence of embeddings SU(n, 1) ⊂ Sp(n, 1) ⊂ SU(2n, 2) ⊂ SO(4n, 4)).

Keywords

rigiditylatticescomplex hyperbolicdeformation

MSC classification

Secondary: 22E40: Discrete subgroups of Lie groups 57S25: Groups acting on specific manifolds
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 11 , Issue 1 , January 2012 , pp. 133 - 159
Copyright
Copyright © Cambridge University Press 2010

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