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June 2019 An infinite sequence of ideal hyperbolic Coxeter 4-polytopes and Perron numbers
Tomoshige Yukita
Kodai Math. J. 42(2): 332-357 (June 2019). DOI: 10.2996/kmj/1562032833

Abstract

In [7], Kellerhals and Perren conjectured that the growth rates of cocompact hyperbolic Coxeter groups are Perron numbers. By results of Floyd, Parry, Kolpakov, Nonaka-Kellerhals, Komori and the author [1], [3], [8], [10], [12], [13], [21], [22], the growth rates of 2- and 3-dimensional hyperbolic Coxeter groups are always Perron numbers. Kolpakov and Talambutsa showed that the growth rates of right-angled Coxeter groups are Perron numbers [9]. For certain families of 4-dimensional cocompact hyperbolic Coxeter groups, the conjecture holds as well (see [7], [19] and also [23]). In this paper, we construct an infinite sequence of ideal non-simple hyperbolic Coxeter 4-polytopes giving rise to growth rates which are distinct Perron numbers. This is the first explicit example of an infinite family of non-compact finite volume Coxeter polytopes in hyperbolic 4-space whose growth rates are of the conjectured arithmetic nature as well.

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Tomoshige Yukita. "An infinite sequence of ideal hyperbolic Coxeter 4-polytopes and Perron numbers." Kodai Math. J. 42 (2) 332 - 357, June 2019. https://doi.org/10.2996/kmj/1562032833

Information

Published: June 2019
First available in Project Euclid: 2 July 2019

zbMATH: 07108015
MathSciNet: MR3981308
Digital Object Identifier: 10.2996/kmj/1562032833

Rights: Copyright © 2019 Tokyo Institute of Technology, Department of Mathematics

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Vol.42 • No. 2 • June 2019
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