Some generalized Coxeter groups and their orbifolds
type: Item
instance of: scholarly article
title: Some generalized Coxeter groups and their orbifolds
zbMATH Open document ID: 0891.57015
DOI: 10.4171/RMI/230
author: Marcel Hagelberg
author: Rubén A. Hidalgo
published in: Revista Matemática Iberoamericana
publication date: 1998-07-19T00:00:00Z
full work available at URL: https://eudml.org/doc/39534
review text: Classes of geometric 3-orbifolds are constructed whose (orbifold-) fundamental groups are generalized Coxeter groups (of the form \(\langle a_1, \dots, a_n\mid a_i^{u_i}= (a_{i+1} a_i^{-1})^{v_i} =1\rangle\); for \(u_i=v_i=2\) these are Coxeter groups), certain \(\mathbb{Z}\)-extensions of such generalized Coxeter groups and also some generalized triangle groups. The singular sets of these orbifolds are obtained by connecting two concentric circles by segments (rungs) resp. by cyclic quotients of such orbifolds. It is shown that these orbifolds belong to hyperbolic geometry, and also to some others of Thurston's eight 3-dimensional geometries (depending on the branching orders associated to the singular sets), by constructing explicitly fundamental polyhedra for the universal covering groups and applying Poincaré's theorem.
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 1136592
zbMATH Keywords: geometric 3-orbifold
zbMATH Keywords: Coxeter group
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
OpenAlex ID: W2043166433
description: scientific article; zbMATH DE number 1136592