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Euler Characteristics and imbeddings of hyperbolic Coxeter groups

Part of: Special aspects of infinite or finite groups General convexity

Published online by Cambridge University Press:  09 April 2009

George Maxwell
George Maxwell
Affiliation:
Mathematics Department University of British Columbia#121-1984 Mathematics Road Vancouver, B. C. V6T 1Y4Canada
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Abstract

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This paper has a twofold purpose. The first is to compute the Euler characteristics of hyperbolic Coxeter groups Ws of level 1 or 2 by a mixture of theoretical and computer aided methods. For groups of level 1 and odd values of |S|, the Euler characteristic is related to the volume of the fundamental region of Ws in hyperbolic space. Secondly we note two methods of imbedding such groups in each other. This reduces the amount of computation needed to determine the Euler characteristics and also reduces the number of essentially different hyperbolic groups that need to be considered.

MSC classification

Secondary: 20F55: Reflection and Coxeter groups 52A38: Length, area, volume
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 2 , April 1998 , pp. 149 - 161
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Böhm, J. and Hertel, E., Polyedergeometrie in n-dimensionalen Räumen konstanter Krümmung (Birkhäuser, Berlin, 1981).Google Scholar
[2]Bott, R., ‘An application of the Morse theory to the topology of Lie groups’, Bull. Soc. Math. France 84 (1956) 251–281.CrossRefGoogle Scholar
[3]Bourbaki, N., Groupes et algèbres Lie, Chapters IV–VI (Hermann, Paris, 1968).Google Scholar
[4]Chiswell, I. M., ‘The Euler characteristic of graph products and of Coxeter groups’, in: Discrete groups and geometry, London Math. Soc. Lecture Note Ser. 173 (Cambridge Univ. Press, Cambridge, 1992), pp. 36–46.CrossRefGoogle Scholar
[5]Coxeter, H. S. M., Regular polytopes, 3rd edition (Dover, New York).Google Scholar
[6]Heckman, G. J., ‘The volume of hyperbolic Coxeter polytopes of even dimension’, Indag. Math. 6 (1995), 189–196.CrossRefGoogle Scholar
[7]Humphreys, J. E., Reflection groups and Coxeter groups (Cambridge Univ. Press, Cambridge, 1990).Google Scholar
[8]Hof, H.-C. Im, ‘A class of hyperbolic Coxeter groups’, Exposition Math. 3 (1985), 179–186.Google Scholar
[9]Kellerhals, R., ‘On Schläfli's reduction formula’, Math. Z. 206 (1991), 193–210.Google Scholar
[10]Kellerhals, R., ‘On the volumes of hyperbolic 5-orthoschemes and the trilogarithm’, Comment. Math. Helv. 67 (1992), 648–663.Google Scholar
[11]Maxwell, G., ‘Sphere packings and hyperbolic reflection groups’, J. Algebra 79 (1982), 78–97.Google Scholar
[12]Meyerhoff, R., ‘A lower bound for the volume of hyperbolic 3-orbifolds’, Duke Math. J. 57 (1988), 185–203.Google Scholar
[13]Serre, J. P., Cohomologie des groupes discrets, Ann. of Math. Stud. 70 (Princeton University Press, Princeton, 1971), pp. 77–169.Google Scholar