July  2009, 3(3): 335-357. doi: 10.3934/jmd.2009.3.335

Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups


Department of Mathematics, University of Michigan, Ann Arbor, MI 48104, United States

Received  October 2008 Revised  March 2009 Published  August 2009

We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra - one nilpotent and the other semisimple. Second, we consider actions arising from two commuting unipotent flows that come from an embedded copy of $\overline{\SL(2,\RR)}^{k} \times \overline{\SL(2,\RR)}^{l}$. In both cases we show that any smooth $\RR$-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function. These results build on theorems of D. Mieczkowski, where the same is shown for actions on $(\SL(2,\RR) \times \SL(2,\RR))$/Γ.
Citation: Felipe A. Ramírez. Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups. Journal of Modern Dynamics, 2009, 3 (3) : 335-357. doi: 10.3934/jmd.2009.3.335

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