Entropy of Stationary Measures and Bounded Tangential de-Rham Cohomology of Semisimple Lie Group Actions |
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Authors: | Email author" target="_blank">Amos?NevoEmail author Robert?J?Zimmer |
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Institution: | (1) Department of Mathematics, Israel Institue of Technology, Technion, 32000 Haifa, Israel;(2) Brown University, Providence, RI, U.S.A. |
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Abstract: | Let G be a connected noncompact semisimple Lie group with finite center, K a maximal compact subgroup, and X a compact manifold (or more generally, a Borel space) on which G acts. Assume that ν is a μ -stationary measure on X, where μ is an admissible measure on G, and that the G-action is essentially free. We consider the foliation of K\ X with Riemmanian leaves isometric to the symmetric space K\ G, and the associated tangential bounded de-Rham cohomology, which we show is an invariant of the action. We prove both vanishing
and nonvanishing results for bounded tangential cohomology, whose range is dictated by the size of the maximal projective
factor G/Q of (X, ν). We give examples showing that the results are often best possible. For the proofs we formulate a bounded tangential version
of Stokes’ theorem, and establish a bounded tangential version of Poincaré’s Lemma. These results are made possible by the
structure theory of semisimple Lie groups actions with stationary measure developed in Nevo and Zimmer Ann of Math. 156, 565--594]. The structure theory assert, in particular, that the G-action is orbit equivalent to an action of a uniquely determined parabolic subgroup Q. The existence of Q allows us to establish Stokes’ and Poincaré’s Lemmas, and we show that it is the size of Q (determined by the entropy) which controls the bounded tangential cohomology.
Supported by BSF and ISF.
Supported by BSF and NSF. |
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Keywords: | semisimple Lie groups parabolic subgroups stationary measures entropy Riemannian foliations de-Rham cohomology tangential bounded cohomology |
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