On the non-vanishing of the first Betti number of hyperbolic three manifolds
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We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in SL(1,D), where D is a quaternion division algebra defined over a number field E contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbolic three manifolds, with non-torsion first homology group, confirming a conjecture of Waldhausen. The proof uses the characterisation of the image of solvable base change by the author, and the construction of cusp forms with non-zero cusp cohomology by Labesse and Schwermer.
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- 1.Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups and representations of reductive groups. Ann. Math. Stud. 94, Princeton Univ. Press, 1980Google Scholar
- 3.Jacquet, H., Langlands, R.: Automorphic forms on GL(2). Lect. Notes in Math. 114, Berlin, Springer 1970Google Scholar
- 4.Labesse, J.-P., Langlands, R.: L-indistinguishability for SL(2). Can. J. Math. 31 726–785 (1979)Google Scholar
- 6.Langlands, R.: Base change for GL(2). Ann. Math. Stud. 96 (1980), Princeton Univ. PressGoogle Scholar