Commensurability of hyperbolic manifolds with geodesic boundary

Suppose n > 3, let M1,M2 be n-dimensional connected complete finitevolume hyperbolic manifolds with non-empty geodesic boundary, and suppose that π1(M1) is quasi-isometric to π1(M2) (with respect to the word metric). Also suppose that if n = 3, then ∂M1 and ∂M2 are compact. We show that M1 is commensurable with M2. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M1 above and G is a finitely generated group which is quasi-isometric to π1(M), then there exists a hyperbolic manifold with geodesic boundaryM ′ with the following properties: M ′ is commensurable with M , and G is a finite extension of a group which contains π1(M ) as a finite-index subgroup. MSC (2000): 20F65 (primary), 30C65, 57N16 (secondary).