Commensurability of hyperbolic manifolds with geodesic boundary

TL;DR

This paper proves that hyperbolic manifolds with geodesic boundary are determined up to commensurability by their fundamental group's quasi-isometry, with specific results for 3-dimensional cases and group extensions.

Contribution

It establishes the commensurability of hyperbolic manifolds with geodesic boundary from quasi-isometric fundamental groups and explores related non-commensurable examples and group extension properties.

Findings

Hyperbolic manifolds with geodesic boundary are commensurable if their fundamental groups are quasi-isometric.

Existence of non-commensurable, homotopically equivalent hyperbolic 3-manifolds with non-compact boundary.

Quasi-isometric groups relate to finite extensions of fundamental groups of certain hyperbolic manifolds.

Abstract

Suppose n>2, let M,M' be n-dimensional connected complete finite-volume hyperbolic manifolds with non-empty geodesic boundary, and suppose that the fundamental group of M is quasi-isometric to the fundamental group of M' (with respect to the word metric). Also suppose that if n=3, then the boundaries of M and of M' are compact. We show that M is commensurable with M'. 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 above and G is a finitely generated group which is quasi-isometric to the fundamental group of M, then there exists a hyperbolic manifold with geodesic boundary M'' with the following properties: M'' is commensurable with M, and G is a finite extension of a group which contains the fundamental group of M'' as a finite-index subgroup.