On the convergence of arithmetic orbifolds

Abstract

We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric ("Benjamini--Schramm") sense for hyperbolic three--space and a product of hyperbolic planes. We also deal with arbitrary sequences of maximal arithmetic three--dimensional hyperbolic lattices defined over a quadratic or cubic field. A motivating application is the study of Betti numbers of Bianchi groups.