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

### Joachim Schwermer

Universität Wien, Austria### Steffen Kionke

Heinrich Heine Universität Düsseldorf, Düsseldorf, Germany

## 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.

## Cite this article

Joachim Schwermer, Steffen Kionke, On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds. Groups Geom. Dyn. 9 (2015), no. 2, pp. 531–565

DOI 10.4171/GGD/320