On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds

Details On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds

2012-04-17

arxiv.org/abs/1204.3750v1

Mathematics

Number Theory

23 pages

Scientific paper

We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce the following theorem: Given an arithmetically defined cocompact subgroup of SL(2,C), provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence of finite index subgroups of such that the first Betti number grows at least as fast as the square root of the index.

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