JournalsggdVol. 9, No. 2pp. 531–565

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

Abstract

We give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 33-manifolds and we deduce the following theorem. Given an arithmetically defined cocompact subgroup ΓSL2(C)\Gamma \subset \mathrm {SL}_2(\mathbb C), provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence {Γi}i\{\Gamma_i\}_i of finite index congruence subgroups of Γ\Gamma such that the first Betti number satisfies

b1(Γi)[Γ:Γi]1/2b_1(\Gamma_i) \gg [\Gamma:\Gamma_i]^{1/2}

as ii 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