Betti numbers of Shimura curves and arithmetic three-orbifolds

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.