Quotients of Coxeter complexes, fundamental groupoids and regular graphs
type: Item
instance of: scholarly article
title: Quotients of Coxeter complexes, fundamental groupoids and regular graphs
zbMATH Open document ID: 0807.57002
DOI: 10.1007/BF02571944
author: Stephen P. Humphries
published in: Mathematische Zeitschrift
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174690
review text: Let \(W = \langle a_ 1, \dots, a_ n | (a_ ia_ j)^{m_ ij} \rangle\) be a Coxeter group with standard generating set \(s = \{a_ 1, \dots, a_ n\}\), and let \(\Sigma = \Sigma (W,S)\) be the corresponding dual Coxeter complex. Assume that if \(W_ i = \langle S \backslash a_ i \rangle\), then each \(W_ i\) is spherical; such \(W\)'s we call compact. We study quotients of such Coxeter complexes by torsion-free, normal subgroups \(N\) of finite index in compact \(W\); these are \((n-1)\)-manifolds \(M^{n-1} (W/N) = \Sigma/N\). We give a presentation for a fundamental groupoid of \(M^{n-1} (W/N)\) and use this presentation to give lower bounds for the ranks of \(H_ 1 (M^{n-1} (W/N); \mathbb{Z}_ 2)\) and \(H_ 1 (M^{n-1} (W/N); \mathbb{Z})\) in various circumstances. We also find edge- transitive, bipartite, regular graphs that are embedded in \(M^{n-1} (W/N)\) and which are invariant under the action of \(W/N\) on \(M^{n-1} (W/N)\). We make various calculations using our techniques for some interesting (sporadic simple) groups \(W/N\).
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57Q99
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 704948
zbMATH Keywords: quotients of Coxeter complexes
zbMATH Keywords: manifolds
zbMATH Keywords: presentation for fundamental groupoid
zbMATH Keywords: Coxeter group
zbMATH Keywords: dual Coxeter complex
zbMATH Keywords: graphs
describes a project that uses: Cayley
MaRDI profile type: Publication
cites work: Q3708174
cites work: Q4040708
cites work: Q4000288
cites work: Q4039784
cites work: Q4114528
cites work: Q3998721
cites work: Immersions and Embeddings of Totally Geodesic Surfaces
cites work: A lower bound for the volume of hyperbolic 3-orbifolds
cites work: Totally geodesic surfaces in hyperbolic 3-manifolds
OpenAlex ID: W2002308410
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description: scientific article; zbMATH DE number 704948