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Uspekhi Mat. Nauk, 2017, Volume 72, Issue 2(434), Pages 147–190 (Mi umn9762)  

This article is cited in 5 scientific papers (total in 5 papers)

Right-angled polyhedra and hyperbolic 3-manifolds

A. Yu. Vesnin

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Hyperbolic 3-manifolds whose fundamental groups are subgroups of finite index in right-angled Coxeter groups are under consideration. The construction of such manifolds is associated with of the faces of polyhedra and, in particular, with 4-colourings. The following questions are discussed: the structure of the set of right-angled polytopes in Lobachevskii space; examples of orientable and non-orientable manifolds, including the classical Löbell manifold constructed in 1931; connections between the Hamiltonian property of a polyhedron and the existence of hyperelliptic involutions of manifolds; the volumes and complexity of manifolds; isometry between hyperbolic manifolds constructed from 4-colourings.
Bibliography: 89 titles.

Keywords: right-angled reflection groups, hyperbolic 3-manifolds, volumes of manifolds, colourings of polyhedra, Hamiltonian graphs, small covers.

Funding Agency Grant Number
Russian Science Foundation 16-41-02006
This research was supported by the Russian Science Foundation (grant no. 16-41-02006).


DOI: https://doi.org/10.4213/rm9762

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English version:
Russian Mathematical Surveys, 2017, 72:2, 335–374

Bibliographic databases:

UDC: 514.132+515.162
MSC: Primary 52B10, 52B11, 57N10; Secondary 22E40, 51M10
Received: 31.01.2017
Revised: 16.02.2017

Citation: A. Yu. Vesnin, “Right-angled polyhedra and hyperbolic 3-manifolds”, Uspekhi Mat. Nauk, 72:2(434) (2017), 147–190; Russian Math. Surveys, 72:2 (2017), 335–374

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Inoue T., “Exploring the List of Smallest Right-Angled Hyperbolic Polyhedra”, Exp. Math.  crossref  isi  scopus
    2. V. M. Buchstaber, N. Yu. Erokhovets, “Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes”, Izv. Math., 81:5 (2017), 901–972  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. V. M. Buchstaber, N. Yu. Erokhovets, “Fullerenes, polytopes and toric topology”, Combinatorial and toric homotopy, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 35, World Sci. Publ., Hackensack, NJ, 2018, 67–178  crossref  mathscinet  zmath  isi
    4. N. Erokhovets, “Construction of fullerenes and Pogorelov polytopes with 5-, 6-and one 7-gonal face”, Symmetry, 10:3 (2018), 67, 28 pp.  crossref  zmath  isi
    5. A. Yu. Vesnin, S. V. Matveev, E. A. Fominykh, “New aspects of complexity theory for 3-manifolds”, Russian Math. Surveys, 73:4 (2018), 615–660  mathnet  crossref  crossref  adsnasa  isi  elib
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