Betti numbers of Shimura curves and arithmetic three--orbifolds

TL;DR

This paper investigates the asymptotic behavior of Betti numbers in Shimura curves and hyperbolic 3-orbifolds, revealing their relation to geometric invariants like volume and genus, and establishing new asymptotic equalities and vanishing results.

Contribution

It establishes the asymptotic Gauss--Bonnet equality for the first Betti number of Shimura curves and shows the vanishing of the first Betti number in congruence hyperbolic 3-orbifolds relative to volume.

Findings

First Betti number of Shimura curves satisfies Gauss--Bonnet equality asymptotically.

First Betti number of hyperbolic 3-orbifolds asymptotically vanishes compared to volume.

Provides new insights into the topological invariants of arithmetic hyperbolic manifolds.

Abstract

We show that asymptotically the first Betti number, or the arithmetic genus, of a Shimura curve satisfies the Gauss--Bonnet equality. We also show that the first Betti number of a congruence hyperbolic 3--orbifold asymptotically vanishes relatively to hyperbolic volume.