Growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers

TL;DR

This paper proves that the growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers, extending understanding of their algebraic properties by combining classical and recent results.

Contribution

It establishes that all growth rates of 3D hyperbolic Coxeter groups are Perron numbers, a significant algebraic property, using a novel combination of existing theorems.

Findings

Growth rates are Perron numbers.

Applicable to groups with certain dihedral angles.

Extends previous algebraic classifications.

Abstract

In this paper we consider the growth rates of 3-dimensional hyperbolic Coxeter polyhedra some of its dihedral angles are $\frac{\pi}{m}$ for $m\geq{7}$. By combining with the classical result by Parry \cite{Pa} and the main result of \cite{Y}, we prove that the growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers.