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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Rigidity of ergodic volume-preserving actions of semisimple groups of higher rank on compact manifolds


Author: Guillaume Seydoux
Journal: Trans. Amer. Math. Soc. 345 (1994), 753-776
MSC: Primary 58F11; Secondary 57S20
DOI: https://doi.org/10.1090/S0002-9947-1994-1250826-1
MathSciNet review: 1250826
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Abstract: Let M be a compact manifold, H a semisimple Lie group of higher rank (e.g., $ H = SL(n,{\mathbf{R}})$ with $ n \geq 3$) and $ a \in \mathcal{A}(H,M)$ an ergodic H-action on M which preserves a volume v. Such an H-action is conjectured to be "locally rigid": if $ a \prime$ is a sufficiently $ {C^1}$-small perturbation of a, then there should exist a diffeomorphism $ \Phi $ of the manifold M which conjugates $ a \prime$ to a. This conjecture would imply that if $ \omega $ is an a-invariant geometrical structure on M, then there should exist an. $ a \prime$-invariant geometrical structure $ \omega \prime$ on M of the same type. Using Kazhdan property, superrigidity for cocycles, and Sobolev spaces techniques we prove, under suitable conditions, two such results with $ \omega = v $ and with $ \omega $ a Riemannian metric along the leaves of a foliation of M.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1250826-1
Article copyright: © Copyright 1994 American Mathematical Society