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

TL;DR

This paper investigates the growth rates of certain 3D hyperbolic Coxeter groups with specific dihedral angles, demonstrating that these rates are always Perron numbers, which has implications for their algebraic and geometric properties.

Contribution

It establishes that the growth rates of these particular hyperbolic Coxeter groups are always Perron numbers, providing new insights into their algebraic structure.

Findings

Growth rates are always Perron numbers.

The study focuses on groups with dihedral angles $rac{ extpi}{m}$ for m=2,3,4,5,6.

Arithmetic properties of these growth rates are characterized.

Abstract

We study arithmetic properties of the growth rates of cofinite 3-dimensional hyperbolic Coxeter groups whose dihedral angles are of the form $\frac{\pi}{m}$ for $m=2,3,4,5,6$ and show that the growth rates are always Perron numbers.