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Semi-edges, reflections and Coxeter groups

Authors:	Ralf Gramlich Georg W Hofmann Karl-Hermann Neeb

Institution:	TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany ; TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany ; TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany

Abstract:	We combine the theory of Coxeter groups, the covering theory of graphs introduced by Malnic, Nedela and Skoviera and the theory of reflections of graphs in order to obtain the following characterization of a Coxeter group: Let $\pi : \Gamma \rightarrow (v,D,\iota,-1)$ be a -covering of a monopole admitting semi-edges only. The graph $\Gamma$ is the Cayley graph of a Coxeter group if and only if $\pi$ is regular and any deck transformation in $\Delta(\pi)$ that interchanges two neighboring vertices of $\Gamma$ acts as a reflection on $\Gamma$ .

Keywords:	Coxeter groups coverings of graphs reflections on graphs voltage spaces

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