Distinctness of two pseudo-Anosov maps

TL;DR

This paper proves that two pseudo-Anosov maps with the same genus and stretch factor, originally constructed by Arnoux-Yoccoz and Fried, are distinct by analyzing their mapping tori and cross sections.

Contribution

It demonstrates the difference between two pseudo-Anosov maps with identical stretch factors by examining their mapping tori and cross sections, clarifying their distinctness.

Findings

The Arnoux-Yoccoz pseudo-Anosov map's mapping torus has no torus cross section with two points blown-up.

The two pseudo-Anosov maps, despite sharing the same stretch factor, are proven to be different.

The method involves reversing Fried's construction to analyze the structure of the mapping tori.

Abstract

In 1981, Arnoux and Yoccoz gave the first examples of pseudo-Anosov maps with odd degree stretch factors. In 1985, D.~Fried deduced the existence of a pseudo-Anosov map in genus three with the same stretch factor as the Arnoux-Yoccoz example in that genus, and asked if these were the same. We show that they are distinct. We do this by, in a sense, reversing Fried's construction; we show that the mapping torus of the pseudo-Anosov map induced by the Arnoux-Yoccoz map on the surface obtained by blowing-up its two singularities has no cross section which is a torus with two points blown-up.