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The spherical growth series of Dyer groups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  21 December 2023

Luis Paris  and
Olga Varghese
Luis Paris
Affiliation:
IMB, UMR 5584, CNRS, Université de Bourgogne, Dijon, France (lparis@u-bourgogne.fr)
Olga Varghese
Affiliation:
Institute of Mathematics, Heinrich-Heine-University Düsseldorf, Universitätsstraße 1, Düsseldorf, Germany (olga.varghese@hhu.de)
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Abstract

Graph products of cyclic groups and Coxeter groups are two families of groups that are defined by labelled graphs. The family of Dyer groups contains these both families and gives us a framework to study these groups in a unified way. This paper focuses on the spherical growth series of a Dyer group D with respect to the standard generating set. We give a recursive formula for the spherical growth series of D in terms of the spherical growth series of standard parabolic subgroups. As an application we obtain the rationality of the spherical growth series of a Dyer group. Furthermore, we show that the spherical growth series of D is closely related to the Euler characteristic of D.

Keywords

Dyer groupsCoxeter groupsright-angled Artin groupsthe spherical growth series of groupsthe rational Euler characteristic

MSC classification

Primary: 20F36: Braid groups; Artin groups 20F55: Reflection and Coxeter groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 1 , February 2024 , pp. 168 - 187
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Benson, M., Growth series of finite extensions of Zn are rational, Invent. Math. 73(2): (1983), 251–269.CrossRefGoogle Scholar
Bourbaki, N., Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337 Elements de mathématique. Fasc. XXXIV. Groupes et al gèbres de Lie (Hermann, Paris, 1968).Google Scholar
Brown, K. S., Cohomology of groups, (Springer-Verlag, 1982).CrossRefGoogle Scholar
Cannon, J. W., The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16(2): (1984), 123–148.CrossRefGoogle Scholar
Cannon, J. W. and Wagreich, P., Growth functions of surface groups, Math. Ann. 293(2): (1992), 239–257.CrossRefGoogle Scholar
Chiswell, I. M., The growth series of a graph product, Bull. Lond. Math. Soc. 26(3): (1994), 268–272.CrossRefGoogle Scholar
Dyer, M., Reflection subgroups of Coxeter systems, J. Algebra 135(1): (1990), 57–73.CrossRefGoogle Scholar
Ghys, E. and de la Harpe, P., La propriété de Markov pour les groupes hyperboliques. Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988). Progr. Math., Volume 83, (Birkhäuser Boston, Boston, MA, 1990), 165–187.Google Scholar
Gromov, M., Hyperbolic groups. Essays in group theory. Math. Sci. Res. Inst. Publ., Volume 8, (Springer, New York, 1987), 75–263.Google Scholar
Lewin, J., The growth function of some free products of groups, Comm. Alg. 19(9): (1991), 2405–2418.CrossRefGoogle Scholar
Mann, A., How groups grow. (London Mathematical Society Lecture Note Series), (Cambridge University Press, Cambridge, 2011).Google Scholar
Marjanski, L., Solon, E., Zheng, F. and Zopff, K., Geodesic Growth of Numbered Graph Products, J. Groups Complex. Cryptol. 14(2): (2023), doi.org/10.46298/jgcc.2023.14.2.10019.Google Scholar
Paris, L. and Soergel, M., Word problem and parabolic subgroups in Dyer groups, Bull. Lond. Math. Soc. 55(6): 2928–2947.CrossRefGoogle Scholar
Rota, G. -C., On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368.CrossRefGoogle Scholar
Serre, J. -P., Cohomologie des groupes discrete, in Prospects in Mathematics. Ann. Math. Stud. No. 70, Princeton, 1971.Google Scholar
Soergel, M., A generalization of the Davis-Moussong complex for Dyer groups (2023), ArXiv:2212.03017.Google Scholar
Solomon, L., The orders of the finite Chevalley groups, J. Algebra 3(3): (1966), 376–393.CrossRefGoogle Scholar
Stanley, R. P., Enumerative combinatorics, Second edition, Volume 1, (Cambridge University Press, Cambridge, 2012), Cambridge Studies in Advanced Mathematics, 49.Google Scholar
Stoll, M., Rational and transcendental growth series for the higher Heisenberg group, Invent. Math. 126(1): (1996), 85–109.CrossRefGoogle Scholar