Journal of Modern Dynamics (JMD)

The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory

Pages: 61 - 92, Issue 1, January 2007      doi:10.3934/jmd.2007.1.61

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David Mieczkowski - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)

Abstract: We make use of representation theory to study the first smooth almost-cohomology of some higher-rank abelian actions by parabolic operators. First, let $N$ be the upper-triangular group of $SL(2,\mathbb{C})$, $\Gamma$ any lattice and $\pi = L^2(SL(2,\mathbb{C})$/$\Gamma)$ the usual left-regular representation. We show that the first smooth almost-cohomology group $H_a^1(N, \pi)$ $H_a^1(SL(2,\mathbb{C}) , \pi)$. In addition, we show that the first smooth almost-cohomology of actions of certain higher-rank abelian groups $A$ acting by left translation on $(SL(2,\mathbb{R}) \times G)$/$\Gamma$ trivialize, where $G = SL(2,\mathbb{R})$ or $SL(2,\mathbb{C})$ and $\Gamma$ is any irreducible lattice. The abelian groups $A$ are generated by various mixtures of the diagonal and/or unipotent generators on each factor. As a consequence, for these examples we prove that the only smooth time changes for these actions are the trivial ones (up to an automorphism).

Keywords:  first cohomology, parabolic actions,higher-rank abelian actions, representation theory.
Mathematics Subject Classification:  28Dxx, 43A85, 22E27, 22E40, 58J42.

Received: May 2006;      Available Online: October 2006.