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- W2890624132 abstract "We study the cohomological equation for discrete horocycle maps on begin{document}$ {rm SL}(2, mathbb{R}) $end{document} and begin{document}$ {rm SL}(2, mathbb{R})times {rm SL}(2, mathbb{R}) $end{document} via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of begin{document}$ {rm SL}(2, mathbb{R}) $end{document} . Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of begin{document}$ operatorname{mathfrak smathfrak l}(2, mathbb{R}) $end{document} , and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for begin{document}$ {rm SL}(2, mathbb{R}) $end{document} in which all cases of irreducible, unitary representations of begin{document}$ {rm SL}(2, mathbb{R}) $end{document} 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." @default.
- W2890624132 created "2018-09-27" @default.
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- W2890624132 date "2019-01-01" @default.
- W2890624132 modified "2023-09-23" @default.
- W2890624132 title "Cohomological equation and cocycle rigidity of discrete parabolic actions" @default.
- W2890624132 doi "https://doi.org/10.3934/dcds.2019160" @default.
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