American Institute of Mathematical Sciences

December  2014, 34(12): 5211-5227. doi: 10.3934/dcds.2014.34.5211

Cocycle rigidity and splitting for some discrete parabolic actions

 1 Department of Mathematics, Rice University, 6100 Main st, Houston, TX 77005, United States, United States

Received  September 2013 Revised  May 2014 Published  June 2014

We prove trivialization of the first cohomology with coefficients in smooth vector fields, for a class of $\mathbb{Z}^2$ parabolic actions on $(SL(2, \mathbb R)\times SL(2, \mathbb R))/\Gamma$, where the lattice $\Gamma$ is irreducible and co-compact. We also obtain a splitting construction involving first and second coboundary operators in the cohomology with coefficients in smooth vector fields.
Citation: Danijela Damjanović, James Tanis. Cocycle rigidity and splitting for some discrete parabolic actions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5211-5227. doi: 10.3934/dcds.2014.34.5211
References:
 [1] D. Damjanović, Perturbations of smooth actions with non-trivial cohomology,, Preprint., ().   Google Scholar [2] D. Damjanović and A. Katok, Local rigidity of homogeneous parabolic actions: I. A model case,, Journal of Modern Dynamics, 5 (2011), 203.  doi: 10.3934/jmd.2011.5.203.  Google Scholar [3] R. Feres and A. Katok, Ergodic theory and dynamics of G-spaces (with special em- phasis on rigidity phenomena),, Handbook of dynamical systems, 1A (2002), 665.  doi: 10.1016/S1874-575X(02)80011-X.  Google Scholar [4] R. Godemont, Sur la théori des représentations unitaires,, Ann. of Math., 53 (1951), 68.  doi: 10.2307/1969343.  Google Scholar [5] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke J. of Math, 119 (2003), 465.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar [6] F. Mautner, Unitary representations of locally compact groups. I,, Ann. of Math. (2), 51 (1950), 1.  doi: 10.2307/1969494.  Google Scholar [7] F. Mautner, Unitary representations of locally compact groups. II,, Ann. of Math. (2), 52 (1950), 528.  doi: 10.2307/1969431.  Google Scholar [8] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory,, Journal of Modern Dynamics, 1 (2007), 61.  doi: 10.3934/jmd.2007.1.61.  Google Scholar [9] D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces,, Invent. Math., 138 (1999), 451.  doi: 10.1007/s002220050350.  Google Scholar [10] D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $PSL(2, \mathbbR)$,, J. Eur. Math. Soc., 11 (2009), 283.  doi: 10.4171/JEMS/151.  Google Scholar [11] F. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups,, J. Mod. Dyn., 3 (2009), 335.  doi: 10.3934/jmd.2009.3.335.  Google Scholar [12] J. Tanis, The cohomological equation and invariant distributions for horocycle maps,, Ergodic Theory and Dynamical systems, 34 (2014), 299.  doi: 10.1017/etds.2012.125.  Google Scholar

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References:
 [1] D. Damjanović, Perturbations of smooth actions with non-trivial cohomology,, Preprint., ().   Google Scholar [2] D. Damjanović and A. Katok, Local rigidity of homogeneous parabolic actions: I. A model case,, Journal of Modern Dynamics, 5 (2011), 203.  doi: 10.3934/jmd.2011.5.203.  Google Scholar [3] R. Feres and A. Katok, Ergodic theory and dynamics of G-spaces (with special em- phasis on rigidity phenomena),, Handbook of dynamical systems, 1A (2002), 665.  doi: 10.1016/S1874-575X(02)80011-X.  Google Scholar [4] R. Godemont, Sur la théori des représentations unitaires,, Ann. of Math., 53 (1951), 68.  doi: 10.2307/1969343.  Google Scholar [5] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke J. of Math, 119 (2003), 465.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar [6] F. Mautner, Unitary representations of locally compact groups. I,, Ann. of Math. (2), 51 (1950), 1.  doi: 10.2307/1969494.  Google Scholar [7] F. Mautner, Unitary representations of locally compact groups. II,, Ann. of Math. (2), 52 (1950), 528.  doi: 10.2307/1969431.  Google Scholar [8] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory,, Journal of Modern Dynamics, 1 (2007), 61.  doi: 10.3934/jmd.2007.1.61.  Google Scholar [9] D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces,, Invent. Math., 138 (1999), 451.  doi: 10.1007/s002220050350.  Google Scholar [10] D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $PSL(2, \mathbbR)$,, J. Eur. Math. Soc., 11 (2009), 283.  doi: 10.4171/JEMS/151.  Google Scholar [11] F. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups,, J. Mod. Dyn., 3 (2009), 335.  doi: 10.3934/jmd.2009.3.335.  Google Scholar [12] J. Tanis, The cohomological equation and invariant distributions for horocycle maps,, Ergodic Theory and Dynamical systems, 34 (2014), 299.  doi: 10.1017/etds.2012.125.  Google Scholar
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