Groups, Geometry, and Dynamics


Full-Text PDF (350 KB) | Metadata | Table of Contents | GGD summary
Volume 9, Issue 2, 2015, pp. 531–565
DOI: 10.4171/GGD/320

Published online: 2015-06-08

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

Steffen Kionke[1] and Joachim Schwermer[2]

(1) Heinrich Heine Universität Düsseldorf, Düsseldorf, Germany
(2) Universität Wien, Austria

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.

Keywords: Arithmetic groups, cohomology, hyperbolic manifolds, Betti numbers

Kionke Steffen, Schwermer Joachim: On the growth of the first Betti number of arithmetic hyperbolic 3-manifolds. Groups Geom. Dyn. 9 (2015), 531-565. doi: 10.4171/GGD/320