Spherical Coxeter groups and hyperelliptic 3-manifolds
type: Item
instance of: scholarly article
title: Spherical Coxeter groups and hyperelliptic 3-manifolds
zbMATH Open document ID: 0948.57010
DOI: 10.1007/BF02674867
author: Andrei Vesnin
author: Alexander Mednykh
published in: Mathematical Notes
publication date: 2000-08-02T00:00:00Z
review text: A 3-manifold \(M\) is called hyperelliptic if there exists an isometric involution \(\tau\) such that the quotient space \(M/\langle\tau\rangle\) is homeomorphic to the 3-sphere \(S^3\). Let \(X\) be one of the 3-dimensional geometries \(H^3\), \(E^3\), \(S^3\), \(H^2 \times E^1\), or \(S^2 \times E^1\), and let \(P\) be a Coxeter polyhedron (that is, each dihedral angle of \(P\) is of the form \(\pi/n\) for some integer \(n\)) in \(X\). Let \(\Delta(P)\) denote the group of isometries of \(X\) generated by reflections in the faces of \(P\). Under a condition relating \(P\) to some Coxeter polyhedron in \(S^3\), the authors show that \(\Delta(P)\) contains a torsion free, finite index subgroup \(G\) such that \(X^3/G\) is a hyperelliptic manifold. This result generalizes earlier work of the second author [Ann. Global Anal. Geom. 8, No. 1, 13-19 (1990; Zbl 0712.57007)].
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 20F55
zbMATH DE Number: 1429622
zbMATH Keywords: spherical Coxeter polyhedron
zbMATH Keywords: reflection group
zbMATH Keywords: hyperelliptic manifold
zbMATH Keywords: Coxeter group
zbMATH Keywords: skeleton of a Coxeter polyhedron
zbMATH Keywords: Thurston geometries
reviewed by: James Hebda
MaRDI profile type: Publication
cites work: The Geometries of 3-Manifolds
cites work: Three-dimensional hyperelliptic manifolds
cites work: Discrete groups generated by reflections
cites work: Q3219491
cites work: Q3243614
full work available at URL: https://doi.org/10.1007/bf02674867
OpenAlex ID: W2069791075
description: scientific article; zbMATH DE number 1429622