Cocycle rigidity and splitting for some discrete parabolic actions

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

Cocycle rigidity and splitting for some discrete parabolic actions

Danijela Damjanović ^1, and James Tanis ^1,

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

Abstract
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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.

Keywords: splitting, Cocycle rigidity, discrete parabolic actions..

Mathematics Subject Classification: Primary: 37A20; Secondary: 37A1.

Citation: Danijela Damjanović, James Tanis. Cocycle rigidity and splitting for some discrete parabolic actions. Discrete and Continuous Dynamical Systems, 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., ().
[2]	D. Damjanović and A. Katok, Local rigidity of homogeneous parabolic actions: I. A model case, Journal of Modern Dynamics, 5 (2011), 203-235. doi: 10.3934/jmd.2011.5.203.
[3]	R. Feres and A. Katok, Ergodic theory and dynamics of G-spaces (with special em- phasis on rigidity phenomena), Handbook of dynamical systems, North-Holland, Amsterdam, 1A (2002), 665-763. doi: 10.1016/S1874-575X(02)80011-X.
[4]	R. Godemont, Sur la théori des représentations unitaires, Ann. of Math., 53 (1951), 68-124. doi: 10.2307/1969343.
[5]	L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke J. of Math, 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.
[6]	F. Mautner, Unitary representations of locally compact groups. I, Ann. of Math. (2), 51 (1950), 1-25. doi: 10.2307/1969494.
[7]	F. Mautner, Unitary representations of locally compact groups. II, Ann. of Math. (2), 52 (1950), 528-556. doi: 10.2307/1969431.
[8]	D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, Journal of Modern Dynamics, 1 (2007), 61-92. doi: 10.3934/jmd.2007.1.61.
[9]	D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. doi: 10.1007/s002220050350.
[10]	D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $PSL(2, \mathbbR)$, J. Eur. Math. Soc., 11 (2009), 283-313. doi: 10.4171/JEMS/151.
[11]	F. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups, J. Mod. Dyn., 3 (2009), 335-357. doi: 10.3934/jmd.2009.3.335.
[12]	J. Tanis, The cohomological equation and invariant distributions for horocycle maps, Ergodic Theory and Dynamical systems, 34 (2014), 299-340. doi: 10.1017/etds.2012.125.

show all references

References:

[1]	D. Damjanović, Perturbations of smooth actions with non-trivial cohomology,, Preprint., ().
[2]	D. Damjanović and A. Katok, Local rigidity of homogeneous parabolic actions: I. A model case, Journal of Modern Dynamics, 5 (2011), 203-235. doi: 10.3934/jmd.2011.5.203.
[3]	R. Feres and A. Katok, Ergodic theory and dynamics of G-spaces (with special em- phasis on rigidity phenomena), Handbook of dynamical systems, North-Holland, Amsterdam, 1A (2002), 665-763. doi: 10.1016/S1874-575X(02)80011-X.
[4]	R. Godemont, Sur la théori des représentations unitaires, Ann. of Math., 53 (1951), 68-124. doi: 10.2307/1969343.
[5]	L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke J. of Math, 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.
[6]	F. Mautner, Unitary representations of locally compact groups. I, Ann. of Math. (2), 51 (1950), 1-25. doi: 10.2307/1969494.
[7]	F. Mautner, Unitary representations of locally compact groups. II, Ann. of Math. (2), 52 (1950), 528-556. doi: 10.2307/1969431.
[8]	D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, Journal of Modern Dynamics, 1 (2007), 61-92. doi: 10.3934/jmd.2007.1.61.
[9]	D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. doi: 10.1007/s002220050350.
[10]	D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $PSL(2, \mathbbR)$, J. Eur. Math. Soc., 11 (2009), 283-313. doi: 10.4171/JEMS/151.
[11]	F. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups, J. Mod. Dyn., 3 (2009), 335-357. doi: 10.3934/jmd.2009.3.335.
[12]	J. Tanis, The cohomological equation and invariant distributions for horocycle maps, Ergodic Theory and Dynamical systems, 34 (2014), 299-340. doi: 10.1017/etds.2012.125.

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