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26 April 2024
 
  » arxiv » math.DG/0009038

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Hodge theory on hyperbolic manifolds of infinite volume
Martin Olbrich ;
Date 4 Sep 2000
Subject Differential Geometry; Representation Theory MSC-class: 58J, 22E40 | math.DG math.RT
AbstractLet $Y=Gammaackslash H^n$ be a quotient of the hyperbolic space by the action of a discrete convex-cocompact group of isometries. We describe certain spaces of $Gamma$-invariant currents on the sphere at infinity of $H^n$ with support on the limit set of $Gamma$. These spaces are finite-dimensional. The main result identifies the cohomology of $Y$ with a quotient of such spaces. We explain in which sense this result generalizes the classical Hodge theorem for compact quotients. We obtain analogous results for the cohomology groups $H^p(Gamma,F)$, where $F$ is a finite-dimensional representation of the full group of orientation preserving isometries of $H^n$.
Source arXiv, math.DG/0009038
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