Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers

TL;DR

This paper explores the relationship between the growth functions of hyperbolic Coxeter groups, their geometric limits, and algebraic integers like Salem and Pisot numbers, revealing new connections in hyperbolic geometry.

Contribution

It establishes a link between the poles of growth functions and algebraic integers, demonstrating geometric convergence and relating Salem and Pisot numbers in hyperbolic Coxeter groups.

Findings

Connection between growth function poles and algebraic integers

Geometric convergence of fundamental domains

Relation between Salem and Pisot numbers

Abstract

A connection between real poles of the growth functions for Coxeter groups acting on hyperbolic space of dimensions three and greater and algebraic integers is investigated. In particular, a geometric convergence of fundamental domains for cocompact hyperbolic Coxeter groups with finite-volume limiting polyhedron provides a relation between Salem numbers and Pisot numbers. Several examples conclude this work.