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

Abstract

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 $\Gamma \subset \mathrm {SL}\_2(\mathbb C)$, provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence ${\Gamma\_i}\_i$ of finite index congruence subgroups of $\Gamma$ such that the first Betti number satisfies $$ b\_1(\Gamma\_i) \gg \[\Gamma:\Gamma\_i]^{1/2} $$ as $i$ goes to infinity.