New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

TL;DR

This paper characterizes when pseudo-Anosov maps have a vanishing Sah-Arnoux-Fathi invariant, linking it to the minimal polynomial of their dilatation, and constructs new infinite families of such maps, especially in genus 3.

Contribution

It provides a complete characterization of vanishing Sah-Arnoux-Fathi invariant in pseudo-Anosov maps and constructs explicit new infinite families using Veech's method.

Findings

Vanishing Sah-Arnoux-Fathi invariant occurs iff the dilatation's minimal polynomial is not reciprocal.

Explicit constructions of pseudo-Anosov maps with vanishing invariant in genus 3.

New infinite families of such maps are identified and described.

Abstract

We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction of pseudo-Anosov maps to give explicit pseudo-Anosov maps of vanishing Sah-Arnoux-Fathi invariant. In particular, we give new infinite families of such maps in genus 3.