Article Contents
Article Contents

# Cocycle rigidity and splitting for some discrete parabolic actions

• 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.
Mathematics Subject Classification: Primary: 37A20; Secondary: 37A15.

 Citation:

•  [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.