First cohomology and local rigidity of group actions

Details First cohomology and local rigidity of group actions First cohomology and local rigidity of group actions

2005-05-24

arxiv.org/abs/math/0505520v5

Mathematics

Differential Geometry

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Scientific paper

Given a topological group $G$, and a finitely generated group $\G$, a homomorphism $\pi:\G{\to}G$ is {\em locally rigid} if any nearby by homomorphism $\pi'$ is conjugate to $\pi$ by a small element of $G$. In 1964, Weil gave a criterion for local rigidity of a homomorphism from a finitely generated group $\G$ to a finite dimensional Lie group $G$ in terms of cohomology of $\G$ with coefficients in the Lie algebra of $G$. Here we generalize Weil's result to a class of homomorphisms into certain infinite dimensional Lie groups, namely groups of diffeomorphism compact manifolds. This gives a criterion for local rigidity of group actions which implies local rigidity of: $(1)$ all isometric actions of groups with property $(T)$, $(2)$ all isometric actions of irreducible lattices in products of simple Lie groups and certain more general locally compact groups and $(3)$ a certain class of isometric actions of a certain class of cocompact lattices in SU(1,n).

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