Salem numbers, spectral radii and growth rates of hyperbolic Coxeter groups

TL;DR

This paper investigates the relationship between Salem numbers and growth rates of hyperbolic Coxeter groups, revealing limitations and providing new proofs regarding spectral radii and growth rate properties.

Contribution

It demonstrates that not all Salem numbers are growth rates of cocompact hyperbolic Coxeter groups and clarifies the spectral radius connection for planar and tetrahedral cases.

Findings

Not all Salem numbers are growth rates of cocompact hyperbolic Coxeter groups

A new proof links planar hyperbolic Coxeter group growth rates to spectral radii

Growth rates of hyperbolic tetrahedral Coxeter groups may not correspond to spectral radii

Abstract

We show that not every Salem number appears as the growth rate of a cocompact hyperbolic Coxeter group. We also give a new proof of the fact that the growth rates of planar hyperbolic Coxeter groups are spectral radii of Coxeter transformations, and show that this need not be the case for growth rates of hyperbolic tetrahedral Coxeter groups.