On the growth rate of ideal Coxeter groups in hyperbolic 3-space

Abstract

We study the set G of growth rates of of ideal Coxeter groups in hyperbolic 3-space which consists of real algebraic integers greater than 1. We show that (1) G is unbounded above while it has the minimum, (2) any element of G is a Perron number, and (3) growth rates of of ideal Coxeter groups with n generators are located in the closed interval [n − 3, n − 1].2010 Mathematics Subject Classification. Primary 20F55, Secondary 20F65.

Growth Rates of 3-dimensional Hyperbolic Coxeter Groups are Perron Numbers

“…Theorem B settles the 3-dimensional case of a conjecture by Kellerhals and Perren [6]. Results in the closer vein have recently been published by Kellerhals and Nonaka [11] and Komori and Yukita [10]. Also, since the set of growth rates of 3-dimensional hyperbolic Coxeter groups is now shown to comprise only Perron numbers by Theorem B, the minimal growth rate becomes a necessary point to be mentioned; Kellerhals determined the minimal growth rate among all 3-dimensional cofinite hyperbolic Coxeter groups [4], while Kellerhals and Kolpakov found the minimal growth rate in the case of 3-dimensional cocompact hyperbolic Coxeter groups [5].…”

Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

“…The actual conjecture describes a detailed distribution of the poles of the associated growth series, and it implies that the growth rate is a Perron number. Several results confirming the latter fact have appeared recently in [19,20,25,30,31,32].…”

“…The original conjecture by Kellerhals and Perren has been confirmed in several cases [19,20,25,30] by applying Steinberg's formula [29] and with extensive use of hyperbolic geometry, notably Andreev's theorem [1,2]. Recently in [31,32] it was established that the growth rates of all 3-dimensional hyperbolic Coxeter groups are Perron numbers.…”

On the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form $\fracπ{m}$ for $m=2,3,4,5,6$

“…Kellerhals and Nonaka showed that the growth rates of three-dimensional ideal hyperbolic Coxeter groups are Perron numbers ( [5]); a hyperbolic Coxeter group is called ideal if all of the vertices of its fundamental polyhedron are located on the ideal boundary of hyperbolic space. Komori and Yukita also showed the same result independently ( [8]). In this paper, we consider the growth rates of cofinite Coxeter groups in hyperbolic 3-space and we shall prove the following theorem.…”