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.

Final version, 51 pages (journal layout). Minor correction to the main theorem from the first version

References in corpus (2)

• New non-arithmetic complex hyperbolic lattices
• Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

Cited by in corpus (8)

• Invariant random subgroups of linear groups
• Strong Limit Multiplicity for arithmetic hyperbolic surfaces and $3$-manifolds
• Bounds for twisted symmetric square $L$-functions via half-integral weight periods
• Generic IRS in free groups, after Bowen
• Betti numbers of Shimura curves and arithmetic three--orbifolds
• Limit multiplicities for ${\mathrm{SL}}_2({\mathcal{O}}_F)$ in ${\mathrm{SL}}_2({\mathbb{R}}^{r_1}\oplus {\mathbb{C}}^{r_2})$
• New variants of arithmetic quantum ergodicity
• Géométrie et topologie des variétés hyperboliques de grand volume