Equidecomposability, volume formulae and orthospectra
| Autor: | Greg McShane, Hidetoshi Masai |
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| Rok vydání: | 2013 |
| Předmět: |
volume
Pure mathematics Dimension (graph theory) Mathematical analysis hyperbolic manifold Boundary (topology) Isometry (Riemannian geometry) Mathematics::Geometric Topology Linear subspace 57M50 Manifold orthospectrum Unit tangent bundle Totally geodesic Geometry and Topology 32Q45 Mathematics Volume (compression) |
| Zdroj: | Algebr. Geom. Topol. 13, no. 6 (2013), 3135-3152 |
| ISSN: | 1472-2739 1472-2747 |
| DOI: | 10.2140/agt.2013.13.3135 |
| Popis: | Bridgeman‐Kahn and Calegari derived formulae for the volumes of compact hyperbolic n‐manifolds with totally geodesic boundary in terms of the orthospectrum. Their methods are apparently different from each other, and involve computing the volume of different subspaces of unit tangent bundle of hyperbolic n‐space. In this paper, we show that the two volume formulae coincide. We also derive a closed form of the formula in dimension 3. 57M50; 32Q45 Bridgeman‐Kahn and Calegari derived formulae for the volumes of compact hyperbolic n‐manifolds with totally geodesic boundary in terms of the orthospectrum of the manifold. Both methods for producing the formulae are based on decomposing the unit tangent bundle into countably many pieces, each of which is naturally associated to a unique orthogeodesic. In fact, each of these pieces is congruent to a model piece, respectively B.l/ for the Bridgeman‐Kahn decomposition and C.l/ for Calegari’s, determined up to isometry by the length l of the corresponding orthogeodesic . So the volume of the unit tangent bundle can be expressed as a sum of the volumes of these pieces and each volume only depends on the length of an orthogeodesic. The formulae obtained are valid for all compact hyperbolic n‐manifolds with totally geodesic boundary, however, the decompositions used by Bridgeman‐Kahn and Calegari are quite different. It is natural to ask how the terms in the two formula are related. We show that the two formulae coincide, that is, for each orthogeodesic the associated Bridgeman‐Kahn model piece and the Calegari model piece have the same volume regardless of the dimension. Throughout voln will denote the volume of an n‐dimensional orientable manifold. |
| Databáze: | OpenAIRE |
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