### 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 language | English |
---|---|

Pages (from-to) | 332-357 |

Number of pages | 26 |

Journal | Kodai Mathematical Journal |

Volume | 42 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2019 Jan 1 |

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### Keywords

- Coxeter group
- growth function
- growth rate
- Perron number

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**An infinite sequence of ideal hyperbolic coxeter 4-polytopes and perron numbers.** / Yukita, Tomoshige.

Research output: Contribution to journal › Article

*Kodai Mathematical Journal*, vol. 42, no. 2, pp. 332-357. https://doi.org/10.2996/kmj/1562032833

}

TY - JOUR

T1 - An infinite sequence of ideal hyperbolic coxeter 4-polytopes and perron numbers

AU - Yukita, Tomoshige

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Coxeter group

KW - growth function

KW - growth rate

KW - Perron number

UR - http://www.scopus.com/inward/record.url?scp=85071141455&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071141455&partnerID=8YFLogxK

U2 - 10.2996/kmj/1562032833

DO - 10.2996/kmj/1562032833

M3 - Article

VL - 42

SP - 332

EP - 357

JO - Kodai Mathematical Journal

JF - Kodai Mathematical Journal

SN - 0386-5991

IS - 2

ER -