Geodesic
Organizing volumes of right-angled hyperbolic polyhedra
Inoue, Taiyo1
1Department of Mathematics, University of California, Berkeley, CA 94720, USA
Algebraic and Geometric Topology, Tome 8 (2008) no. 3, pp. 1523-1565
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This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angled objects, until it is a disjoint union of Löbell polyhedra, a class of polyhedra which generalizes the dodecahedron. Furthermore, these combinatorial operations are shown to have geometric realizations which are volume decreasing. This allows for an organization of the volumes of right-angled hyperbolic polyhedra and allows, in particular, the determination of the polyhedra with smallest and second smallest volumes.

DOI : 10.2140/agt.2008.8.1523
Keywords: hyperbolic, geometry, right-angled, polyhedra

Inoue, Taiyo  1

1 Department of Mathematics, University of California, Berkeley, CA 94720, USA 
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Inoue, Taiyo. Organizing volumes of right-angled hyperbolic polyhedra. Algebraic and Geometric Topology, Tome 8 (2008) no. 3, pp. 1523-1565. doi: 10.2140/agt.2008.8.1523

[1] I Agol, P A Storm, W P Thurston, Lower bounds on volumes of hyperbolic Haken $3$–manifolds, J. Amer. Math. Soc. 20 (2007) 1053

[2] D V Alekseevskij, È B Vinberg, A S Solodovnikov, Geometry of spaces of constant curvature, from: "Geometry, II", Encyclopaedia Math. Sci. 29, Springer (1993) 1

[3] E M Andreev, Convex polyhedra in Lobačevskiĭspaces, Mat. Sb. $($N.S.$)$ 81 (123) (1970) 445

[4] K Appel, W Haken, Every planar map is four colorable, Contemporary Math. 98, Amer. Math. Soc. (1989)

[5] R Charney, M Davis, Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds, Amer. J. Math. 115 (1993) 929

[6] A F Costa, E Martínez, On hyperbolic right-angled polygons, Geom. Dedicata 58 (1995) 313

[7] C D Hodgson, Deduction of Andreev's theorem from Rivin's characterization of convex hyperbolic polyhedra, from: "Topology '90 (Columbus, OH, 1990)", Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 185

[8] C D Hodgson, I Rivin, A characterization of compact convex polyhedra in hyperbolic $3$–space, Invent. Math. 111 (1993) 77

[9] A Mednykh, A Vesnin, Colourings of polyhedra and hyperelliptic $3$–manifolds, from: "Recent advances in group theory and low-dimensional topology (Pusan, 2000)", Res. Exp. Math. 27, Heldermann (2003) 123

[10] A V Pogorelov, Regular decomposition of the Lobačevskiĭspace, Mat. Zametki 1 (1967) 3

[11] R K W Roeder, J H Hubbard, W D Dunbar, Andreev's theorem on hyperbolic polyhedra, Ann. Inst. Fourier (Grenoble) 57 (2007) 825

[12] A Y Vesnin, Volumes of Löbell $3$–manifolds, Mat. Zametki 64 (1998) 17

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