Cohomological equation and cocycle rigidity of discrete parabolic actions

TL;DR

This paper advances the understanding of the cohomological equation for discrete horocycle maps on SL(2,R), providing sharp Sobolev estimates, tame cocycle rigidity results, and implications for local rigidity of parabolic actions in Lie groups.

Contribution

It introduces improved Sobolev estimates for solutions of the cohomological equation, establishes tame cocycle rigidity for certain discrete actions, and extends results to general simple Lie groups.

Findings

Sharp Sobolev non-tame estimates for horocycle map solutions

Tame cocycle rigidity for two-parameter discrete actions

Extension of cohomology results to simple Lie groups

Abstract

We study the cohomological equation for discrete horocycle maps on $SL(2, \mathbb{R})$ and $SL(2,\mathbb R)\times SL(2, \mathbb{R})$ via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of $SL(2,\mathbb R)$. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of $sl(2, \mathbb R)$, 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 $SL(2, \mathbb R)$ in which all cases of irreducible, unitary representations of $SL(2, \mathbb R)$ 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.