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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Semi-edges, reflections and Coxeter groups


Authors: Ralf Gramlich, Georg W. Hofmann and Karl-Hermann Neeb
Journal: Trans. Amer. Math. Soc. 359 (2007), 3647-3668
MSC (2000): Primary 05C25, 20F55, 55U10
Published electronically: March 7, 2007
MathSciNet review: 2302510
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Abstract | References | Similar Articles | Additional Information

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 $ 1$-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$.


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Additional Information

Ralf Gramlich
Affiliation: TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany
Email: gramlich@mathematik.tu-darmstadt.de

Georg W. Hofmann
Affiliation: TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany
Address at time of publication: 5669 Merkel Street, Halifax, Nova Scotia B3K 2J1, Canada
Email: ghofmann@mathematik.tu-darmstadt.de, hofmann@mathstat.dal.ca

Karl-Hermann Neeb
Affiliation: TU Darmstadt, FB Mathematik / AG 5, Schloßgartenstraße 7, 64289 Darmstadt, Germany
Email: neeb@mathematik.tu-darmstadt.de

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04040-8
PII: S 0002-9947(07)04040-8
Keywords: Coxeter groups, coverings of graphs, reflections on graphs, voltage spaces
Received by editor(s): October 27, 2004
Received by editor(s) in revised form: March 22, 2005
Published electronically: March 7, 2007
Article copyright: © Copyright 2007 American Mathematical Society