Geometric orbifolds with torsion free derived subgroup
type: Item
instance of: scholarly article
title: Geometric orbifolds with torsion free derived subgroup
zbMATH Open document ID: 1211.57017
published in: Siberian Mathematical Journal
publication date: 2010-11-15T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/141534
review text: The paper under review deals with orientable geometric three-orbifolds with underlying space the three-sphere. It gives a criterion to decide whether the maximal abelian covering is a manifold. Being a manifold is equivalent to saying that its branching locus is nonempty. In addition, when the geometry is not spherical, this is also equivalent to saying that the orbifold fundamental group of the covering has no torsion. By Selberg's lemma, every geometric 3-orbifold has always a finite sheeted covering which is a manifold, however the proof is not constructive and it can be quite difficult to determine the precise covering. This paper gives a precise statement that is straightforward to check. Notice that since we assume that the underlying space of the orbifold is the three-sphere, the orbifold fundamental group is generated by torsion elements, in particular the maximal abelian covering is finite. The criterion is stated in terms of the branching locus (which is a link or trivalent graph in the sphere, as we assume that the orbifold is orientable) and the isotropy groups. The criterion proved in this paper says the following: the maximal abelian covering is a manifold if, and only if, the following two conditions hold: (1) the isotropy group of each singular vertex is the abelian dihedral group of four elements, and (2) the branching locus has no separating edge. An edge of the branching locus is called separating if the number of components of the singular graph increases when we remove this edge. In the paper, to have a separating edge is called 1-connected, which should no be confused with simply connected. In particular, when the branching locus of the orbifold has no vertices, then the maximal abelian covering is always a manifold.
reviewed by: Joan Porti
Mathematics Subject Classification ID: 57R18
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 5816218
zbMATH Keywords: manifold
zbMATH Keywords: three-orbifold
zbMATH Keywords: geometry
zbMATH Keywords: isometry
zbMATH Keywords: covering
author: Rubén A. Hidalgo
author: Alexander Mednykh
describes a project that uses: GAP
describes a project that uses: SnapPea
MaRDI profile type: Publication
OpenAlex ID: W2065291958
cites work: On Dehn's algorithms for the conjugacy and word problems, with applications
cites work: On Greendlinger's lemma
cites work: Q3972207
cites work: Q4226362
cites work: On free subgroups of generalized triangle groups
cites work: Three dimensional manifolds, Kleinian groups and hyperbolic geometry
cites work: The Geometries of 3-Manifolds
cites work: Q3748518
cites work: Abelian Groups of Automorphisms of Compact Riemann Surfaces
cites work: Torsion free commutator subgroups of generalized Coxeter groups.
cites work: Q4789131
cites work: Generalized Humbert curves
cites work: Generalized Fermat curves
cites work: Q3219491
cites work: Three-dimensional hyperelliptic manifolds
cites work: Three-dimensional hyperbolic manifolds of small volume with three hyperelliptic involutions
cites work: Q3802262
cites work: Q4827469
cites work: Some honey-combs in hyperbolic 3-space
cites work: COVERING PROPERTIES OF SMALL VOLUME HYPERBOLIC 3-MANIFOLDS
cites work: Q4381886
cites work: Q4786566
cites work: Q3533424
DOI: 10.1007/S11202-010-0005-8
description: scientific article; zbMATH DE number 5816218