Betti numbers of Shimura curves and arithmetic three-orbifolds

Frączyk, Mikołaj; Raimbault, Jean

doi:10.2140/ant.2019.13.2359

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Betti numbers of Shimura curves and arithmetic three-orbifolds

Mikołaj Frączyk and Jean Raimbault

Vol. 13 (2019), No. 10, 2359–2382

DOI: 10.2140/ant.2019.13.2359

Abstract

We show that asymptotically the first Betti number $b_{1}$ of a Shimura curve satisfies the Gauss–Bonnet equality $2 π (b_{1} - 2) = vol$ where $vol$ is hyperbolic volume; equivalently $2 g - 2 = (1 + o (1)) vol$ where $g$ is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3-orbifold asymptotically vanishes relatively to hyperbolic volume, that is $b_{1} ∕ vol \to 0$ . This generalizes previous results obtained by Frączyk, on which we rely, and uses the same main tool, namely Benjamini–Schramm convergence.

Keywords

Shimura curves, arithmetic hyperbolic manifolds, Betti numbers

Mathematical Subject Classification 2010

Primary: 11F06

Secondary: 20H10

Milestones

Received: 30 November 2018

Revised: 19 April 2019

Accepted: 15 August 2019

Published: 6 January 2020

Authors

Mikołaj Frączyk
Alfréd Rényi Institute of Mathematics Budapest Hungary

Jean Raimbault
Institut de Mathématiques de Toulouse Université de Toulouse CNRS, UPS IMT France

Vol. 13, No. 10, 2019