Erratum to: Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams |
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Authors: | Emily Stark |
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Institution: | 1.Section de Mathématiques, University of Geneva,Geneva,Switzerland;2.Dipartimento di Matematica,Università di Pisa,Pisa,Italy;3.Department Mathematik,ETH Zentrum,Zürich,Switzerland |
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Abstract: | A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume \(\Vert M\Vert \) of M is equal to \(\mathrm{Vol}(M)/v_n\), where \(v_n\) is a constant depending only on the dimension of M. The same result also holds for complete finite-volume hyperbolic manifolds without boundary, while Jungreis proved that the ratio \(\mathrm{Vol}(M)/\Vert M\Vert \) is strictly smaller than \(v_n\) if M is compact with nonempty geodesic boundary. We prove here a quantitative version of Jungreis’ result for \(n\ge 4\), which bounds from below the ratio \(\Vert M\Vert /\mathrm{Vol}(M)\) in terms of the ratio \(\mathrm{Vol}(\partial M)/\mathrm{Vol}(M)\). As a consequence, we show that, for \(n\ge 4\), a sequence \(\{M_i\}\) of compact hyperbolic n-manifolds with geodesic boundary satisfies \(\lim _i \mathrm{Vol}(M_i)/\Vert M_i\Vert =v_n\) if and only if \(\lim _i \mathrm{Vol}(\partial M_i)/\mathrm{Vol}(M_i)=0\). We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension 3. |
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