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

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.