#### 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 ${b}_{1}$ of a Shimura curve satisfies the Gauss–Bonnet equality $2\pi \left({b}_{1}-2\right)=vol$ where $vol$ is hyperbolic volume; equivalently $2g-2=\left(1+o\left(1\right)\right)vol$ 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
Primary: 11F06
Secondary: 20H10