Some computations of 1-cohomology groups and construction of non orbit equivalent actions

Details Some computations of 1-cohomology groups and construction of non orbit equivalent actions Some computations of 1-cohomology groups and construction of non orbit equivalent actions

2004-07-12

arxiv.org/abs/math/0407199v6

Mathematics

Operator Algebras

24 pages (final version, with slight additions and change of title on 09/20/04)

Scientific paper

For each group $G$ having an infinite normal subgroup with the relative property (T) (for instance $G = H \times K$ where $H$ is infinite with property (T) and $K$ is arbitrary), and any countable abelian group $\Lambda$ we construct free ergodic measure preserving actions $\sigma_\Lambda$ of $G$ on the probability space such that the 1'st cohomology group of $\sigma_\Lambda$, $H^1(\sigma_\Lambda)$, is equal to Char$(G) \times \Lambda$. We deduce that $G$ has uncountably many non stably orbit equivalent actions. We also calculate 1-cohomology groups and show existence of ``many'' non stably orbit equivalent actions for free products of groups as above.

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