An infinite sequence of ideal hyperbolic coxeter 4-polytopes and perron numbers

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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.

Original languageEnglish
Pages (from-to)332-357
Number of pages26
JournalKodai Mathematical Journal
Volume42
Issue number2
DOIs
Publication statusPublished - 2019

Keywords

  • Coxeter group
  • Perron number
  • growth function
  • growth rate

ASJC Scopus subject areas

  • Mathematics(all)

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