Vol. 13, No. 10, 2019

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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
Abstract

We show that asymptotically the first Betti number b1 of a Shimura curve satisfies the Gauss–Bonnet equality 2π(b1 2) = vol where vol is hyperbolic volume; equivalently 2g 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 b1vol 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