Mathematische Annalen

, Volume 330, Issue 2, pp 323–329

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


DOI: 10.1007/s00208-004-0552-z

Cite this article as:
Rajan, C. Math. Ann. (2004) 330: 323. doi:10.1007/s00208-004-0552-z


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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Tata Institute of Fundamental ResearchBombayIndia