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On the non-vanishing of the first Betti number of hyperbolic three manifolds

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Abstract.

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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Authors and Affiliations

  1. Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay, 400 005, India

    C. S. Rajan

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  1. C. S. Rajan
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Correspondence to C. S. Rajan.

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Mathematics Subject Classification (2000): 11F75, 22E40, 57M50

Revised version: 18 February 2004

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Rajan, C. On the non-vanishing of the first Betti number of hyperbolic three manifolds. Math. Ann. 330, 323–329 (2004). https://doi.org/10.1007/s00208-004-0552-z

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  • DOI: https://doi.org/10.1007/s00208-004-0552-z

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