Pseudo-Anosov mappings and toral automorphisms

TL;DR

This paper constructs pseudo-Anosov mappings associated with certain toral automorphisms, demonstrating that specific algebraic numbers can serve as stretch factors, and extends these results to higher-dimensional tori.

Contribution

It provides explicit constructions linking irreducible toral automorphisms with pseudo-Anosov mappings, confirming Fried's conjecture for degree 3 and extending to higher dimensions.

Findings

Any norm-1 cubic Pisot number is a stretch factor of a pseudo-Anosov mapping.

Constructs pseudo-Anosov mappings for 3-torus automorphisms with positive eigenvalue product.

Extends the construction to higher-dimensional tori under specific eigenvalue conditions.

Abstract

For every irreducible automorphism $\phi\in\text{SL}_3({\mathbb Z})$ of the $3$-torus, for which the product of the expanding eigenvalues is positive, we construct a pseudo-Anosov mapping $f$ of an associated surface, semi-conjugate and almost-isomorphic to $\phi$, whose stretch factor is the product of the expanding eigenvalues of $\phi$. This shows that any norm-$1$ cubic Pisot number occurs as the stretch factor of a pseudo-Anosov mapping, proving a conjecture of Fried in degree $3$. A similar construction works for the $4$-torus on condition that $\phi$ has exactly two eigenvalues outside the unit circle (and whose product is positive). Furthermore for any irreducible hyperbolic automorphism $\phi\in\text{SL}_n({\mathbb Z})$ of the $n$-torus, $n\ge 4$, we construct a pseudo-Anosov mapping semiconjugate and almost-isomorphic to any sufficiently large power of $\phi$.