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.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups and representations of reductive groups. Ann. Math. Stud. 94, Princeton Univ. Press, 1980
Clozel, L.: On the cuspidal cohomology of arithmetic subgroups of SL(2n) and the first Betti number of arithmetic 3-manifolds. Duke Math. J. 55(2), 475–486 (1987)
Jacquet, H., Langlands, R.: Automorphic forms on GL(2). Lect. Notes in Math. 114, Berlin, Springer 1970
Labesse, J.-P., Langlands, R.: L-indistinguishability for SL(2). Can. J. Math. 31 726–785 (1979)
Labesse, J.-P., Schwermer, J.: On liftings and cusp cohomology of arithmetic groups. Invent. math. 83, 383–401 (1986)
Langlands, R.: Base change for GL(2). Ann. Math. Stud. 96 (1980), Princeton Univ. Press
Li, J. S., Millson, J. J.: On the first Betti number of a hyperbolic manifold with an arithmetic fundamental group. Duke Math. J. 71(2), 365–401 (1993)
Millson, J. J.: On the first Betti number of a constant negatively curved manifold. Ann. Math. 104, 235–247 (1976)
Rajan, C. S.: On the image and fibres of solvable base change. Math. Res. Letters 9(4), 499–508 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 11F75, 22E40, 57M50
Revised version: 18 February 2004
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-004-0552-z