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., with ) and an ergodic *H*-action on *M* which preserves a volume *v*. Such an *H*-action is conjectured to be "locally rigid": if is a sufficiently -small perturbation of *a*, then there should exist a diffeomorphism of the manifold *M* which conjugates to *a*. This conjecture would imply that if is an *a*-invariant geometrical structure on *M*, then there should exist an. -invariant geometrical structure 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 and with 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