Quotients of Coxeter complexes, fundamental groupoids and regular graphs
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 \(W/N\).
scientific article; zbMATH DE number 704948
scientific article