Commensurability of Hyperbolic Manifolds with Geodesic Boundary |
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Authors: | Roberto Frigerio |
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Affiliation: | (1) Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, Pisa, 56127, Italy |
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Abstract: | Suppose, let M 1, M 2 be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that π1 (M 1) is quasi-isometric to π1 (M 2) (with respect to the word metric). Also suppose that if n=3, then ∂M 1 and ∂M 2 are compact. We show that M 1 is commensurable with M 2. 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 1 above and G is a finitely generated group which is quasi-isometric to π1 (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 π1 (M′) as a finite-index subgroupMathematics Subject Classification (2000). Primary: 20F65; secondary: 30C65, 57N16 |
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Keywords: | fundamental group Cayley graph quasi-isometry quasi-conformal homeomorphism hyperbolic manifold |
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