Cohomological equation and cocycle rigidity of discrete parabolic actions

Abstract

We study the cohomological equation for discrete horocycle maps on SL(2, R) and SL(2, R) × SL(2, R) via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of SL(2, R). Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a codimension one subspace of sl(2, R), and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for SL(2, R) in which all cases of irreducible, unitary representations of SL(2, R) can be studied simultaneously.Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.

Transversal local rigidity of discrete abelian actions on Heisenberg nilmanifolds

“…Namely, in [8], we carried out analysis of the first cohomology for the discrete parabolic homogeneous action on . However, the inverse of the second coboundary operator turned out not to be tame; in fact, Tanis and Wang [14] proved that there can be no tame inverse (see also [15, Theorem 2.2]). No local classification results have been obtained for this example.…”