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Commensurability of hyperbolic manifolds with geodesic boundary

Geometriae Dedicata, ISSN: 0046-5755, Vol: 118, Issue: 1, Page: 105-131
2006
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Suppose n ≥ 3, let M , M be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that π (M ) is quasi-isometric to π (M ) (with respect to the word metric). Also suppose that if n=3, then ∂M and ∂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 M above and G is a finitely generated group which is quasi-isometric to π (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 π (M′) as a finite-index subgroup. © Springer 2006.

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