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

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

2003-02-19

arxiv.org/abs/math/0302226v1

Mathematics

Representation Theory

7 pages

Scientific paper

We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in $SL(1,D)$, where $D$ is a quaternion division algebras 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 Thurston. 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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