On 3-dimensional hyperbolic Coxeter pyramids

TL;DR

This paper classifies 3D hyperbolic Coxeter pyramids, computes their growth functions and volumes, and explores the relationship between their geometric properties and growth rates, revealing that growth rates are always Perron numbers.

Contribution

It provides a complete classification of 3D hyperbolic Coxeter pyramids and links their geometric volumes with algebraic growth rates, a novel connection in the field.

Findings

Growth rates are always Perron numbers.

Volumes are explicitly calculated and compared with growth rates.

A geometric ordering of pyramids based on growth rates is proposed.

Abstract

After classifying 3-dimensional hyperbolic Coxeter pyramids by means of elementary plane geometry, we calculate growth functions of corresponding Coxeter groups by using Steinberg formula and conclude that growth rates of them are always Perron numbers. We also calculate hyperbolic volumes of them and compare volumes with their growth rates. Finally we consider a geometric ordering of Coxeter pyramids comparable with their growth rates.