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

as $i$ goes to infinity.