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Courants invariants et formes automorphes d'un groupe kleinéen élémentaire

Abstract : The cohomology of a discrete group with real (or complex) coefficients can be seen as the de Rham cohomology of the quotient of a contractile manifold by a free and proper action of this group. It is then natural to consider the existence of subcomplexes of the complex of invariant differentia! forms that induce the same cohomology, i.e. for which the inclusion is a quasi-isomorphism. This question is studied in the case of the quotient of the real hyperbolic space by a Kleinian group, the considered subcomplex being that of automorphic forms. This problem is then known as the Borel conjecture, and allows a variant, sometimes called the Borel-Harder conjecture. This conjecture is solved by the classical Hodge theory when the quotient is compact and was proved by Franke when the quotient has finite volume. In this work, we examine the simplest case where the quotient has infinite volume : elementary groups. We use the Poisson transformation to move the problem to the sphere at infinity of the hyperbolic space. Then we compute explicitely the cohomology of invariant currents on this sphere thanks to a decomposition in a regular part on the discontinuity domain and an irregular part on the limit set. This calculus is first made in the cases of infinite cyclic groups of hyperbolic type (generated by a loxodromy), then of parabolic type (generated by a translation in dimension 2). By an average process we can then extend the computations to bigger classes of elementary groups. We obtain explicitely the cohomology of coclosed harmonie automorphic forms via the Poisson transformation, and, by comparing the results with the de Rham cohomology of the quotient when the dimension is even, we can positively answer to the Borel-Harder conjecture for the groups above.
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https://hal-uphf.archives-ouvertes.fr/tel-03002101
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Submitted on : Thursday, November 12, 2020 - 4:33:51 PM
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Frédéric Delacroix. Courants invariants et formes automorphes d'un groupe kleinéen élémentaire. Mathématiques [math]. Université de Valenciennes et du Hainaut-Cambrésis, UVHC, (France), 2001. Français. ⟨NNT : 2001VALE0016⟩. ⟨tel-03002101⟩

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