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

Abstract

Abstract. In this paper we consider the growth rates of -dimensional hyperbolic Coxeter polyhedra with at least one dihedral angle of the form π k for an integer k ≥ . Combining a classical result by parry with a previous result of ours, we prove that the growth rates of -dimensional hyperbolic Coxeter groups are Perron numbers.

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. In the present paper, we prove that the spherical and geodesic growth rates of RACGs and RAAGs are also Perron numbers, even when there is no cocompact or finite covolume action.…”

Growth rates of Coxeter groups and Perron numbers

“…It follows from the results of [10,19,22,23] that the growth rates of Coxeter groups acting on H 2 and H 3 with finite co-volume are Perron numbers. Moreover, a conjecture by Kellerhals and Perren in [14] suggests a very particular distribution of the poles of the growth function ω G (z) = ∞ k=0 w k z k , which implies that the word growth rate ω(G) is a Perron number.…”

“…The word growth rates of their associated reflection groups are Perron numbers by [22,23], and their geodesic growth rates are Perron numbers by Theorem 4.1. Indeed, any Coxeter polyhedron P polyhedron combinatorially isomorphic to a Löbell polyhedron L n has the following property: each of its faces has at most n neighbours, while L n has 2n + 2 faces in total.…”

An infinite sequence of ideal hyperbolic Coxeter 4-polytopes and Perron numbers

“…Recall that a real algebraic number τ > 1 is a Perron number, if and only if all of its other algebraic conjugates are less than τ in absolute value. It is known that the growth rates of 2 and 3-dimensional hyperbolic Coxeter polytopes are always Perron numbers ( [1], [4], [8], [16], [17]). From now on, we consider the growth rates of hyperbolic Coxeter 4-polytopes.…”