Abstract
We show that for a parabolic Rd-action on PSL(2,R)d/Γ, the cohomologies in degrees 1 through d − 1 trivialize, and we give the obstructions to solving the degree-d coboundary equation, along with bounds on Sobolev norms of primitives. In previous papers, we have established these results for certain Anosov systems. This work extends the methods of those papers to systems that are not Anosov. The main new idea is defining special elements of representation spaces that allow us to modify the arguments from the previous papers. We discuss how to generalize this strategy to Rd-systems coming from a product of Lie groups, as in the systems analyzed here.
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This work was completed while the author was employed at the University of Bristol and supported by ERC.
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Ramírez, F.A. Higher cohomology of parabolic actions on certain homogeneous spaces. JAMA 131, 255–275 (2017). https://doi.org/10.1007/s11854-017-0008-5
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DOI: https://doi.org/10.1007/s11854-017-0008-5