TY - JOUR

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

AU - Yukita, Tomoshige

N1 - Funding Information:
The author wishes to express his gratitude to Professor Ruth Kellerhals and Professor Yohei Komori for fruitful discussions and helpful comments around Sturm’s theorem. This work was partially supported by Grant-in-Aid for JSPS Fellows number 17J05206.

PY - 2019

Y1 - 2019

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 - Perron number

KW - growth function

KW - growth rate

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U2 - 10.2996/kmj/1562032833

DO - 10.2996/kmj/1562032833

M3 - Article

AN - SCOPUS:85071141455

VL - 42

SP - 332

EP - 357

JO - Kodai Mathematical Journal

JF - Kodai Mathematical Journal

SN - 0386-5991

IS - 2

ER -