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January  2007, 1(1): 61-92. doi: 10.3934/jmd.2007.1.61

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

David Mieczkowski 1,

1. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received  May 2006 Published  October 2006

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, higher-rank abelian actions, representation theory., parabolic actions.
Mathematics Subject Classification: 28Dxx, 43A85, 22E27, 22E40, 58J42. .
Citation: David Mieczkowski. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. Journal of Modern Dynamics, 2007, 1 (1) : 61-92. doi: 10.3934/jmd.2007.1.61
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