The geometry of bi-Perron numbers with real or unimodular Galois conjugates

TL;DR

This paper characterizes bi-Perron numbers with specific Galois conjugates as those related to pseudo-Anosov homeomorphisms and bipartite Coxeter transformations, revealing deep connections between algebraic numbers and geometric structures.

Contribution

It provides a complete characterization of certain bi-Perron numbers in terms of their Galois conjugates and their relation to pseudo-Anosov homeomorphisms and Coxeter transformations.

Findings

Bi-Perron numbers with real or unimodular Galois conjugates are characterized.

Such numbers admit a power that is a stretch factor of a pseudo-Anosov homeomorphism.

Equivalent to admitting a power as the spectral radius of a bipartite Coxeter transformation.

Abstract

Among all bi-Perron numbers, we characterise those all of whose Galois conjugates are real or unimodular as the ones that admit a power which is the stretch factor of a pseudo-Anosov homeomorphism arising from Thurston's construction. This is in turn equivalent to admitting a power which is the spectral radius of a bipartite Coxeter transformation.