Betti numbers of Shimura curves and arithmetic three-orbifolds

We show that asymptotically the first Betti number b(1) of a Shimura curve satisfies the Gauss-Bonnet equality 2 pi(b(1) - 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 b(1)/vol -> 0. This generalizes previous results obtained by Fraczyk, on which we rely, and uses the same main tool, namely Benjamini-Schramm convergence.