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The growth function of Coxeter garlands in \({\mathbb{H}^{4}}\)

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Abstract

The growth function W(t) of a Coxeter group W relative to a Coxeter generating set is always a rational function. We prove by an explicit construction that there are infinitely many cocompact Coxeter groups W in hyperbolic 4-space with the following property. All the roots of the denominator of W(t) are on the unit circle except exactly two pairs of real roots.

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References

  • Cannon J., Wagreich P.: Growth functions of surface groups. Math. Ann. 293, 239–257 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Charney , R., Davis, M.: Reciprocity of growth functions of Coxeter groups. Geometriae Dedicata 39, 373–378 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Esselmann F.: The classification of compact hyperbolic Coxeter d-polytopes with d + 2 facets. Comment. Math. Helvetici 71, 229–242 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Humphreys, J.E.: Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29 (1990)

  • Kaplinskaja I.M.: Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces. Math. Notes 15, 88–91 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  • Kempner A.J.: On the complex roots of algebraic equations. Bull. Am. Math. Soc. 41, 809–843 (1935)

    Article  MathSciNet  Google Scholar 

  • Makarov V. S.: On Fedorov’s groups in four- and five-dimensional Lobachevskij spaces. Issled. po obshch. algebre Kishinev 1, 120–129 (1968)

    Google Scholar 

  • Parry W.: Growth series of Coxeter groups and Salem numbers. J. Algebra 154, 406–415 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Perren, G.: Growth of cocompact hyperbolic Coxeter groups and their rate, Ph.D. thesis, Universitè de Fribourg (2009)

  • Serre, J.-P.: Cohomologie des groupes discrets. In: Prospects in Mathematics. Ann. Math. Studies, vol. 70, pp. 77–169. Princeton University Press, Princeton (1971)

  • Steinberg, R.: Endomorphisms of linear algebraic groups. Mem. Am. Math. Soc. 80 (1968)

  • Vinberg, E.B. (ed.): Geometry II, Encyclopedia of Mathematical Sciences, vol. 29. Springer, Berlin (1993)

    Google Scholar 

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Authors and Affiliations

  1. Département de mathématiques, Université de Fribourg, Chemin du Museé 23, 1700, Fribourg, Switzerland

    Thomas Zehrt

  2. Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Christine Zehrt-Liebendörfer

Authors
  1. Thomas Zehrt
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  2. Christine Zehrt-Liebendörfer
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Correspondence to Thomas Zehrt.

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partially supported by SNF No. 200020-113199.

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Zehrt, T., Zehrt-Liebendörfer, C. The growth function of Coxeter garlands in \({\mathbb{H}^{4}}\) . Beitr Algebra Geom 53, 451–460 (2012). https://doi.org/10.1007/s13366-011-0073-3

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  • DOI: https://doi.org/10.1007/s13366-011-0073-3

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