Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant

TL;DR

This paper proves that lifts of pseudo-Anosov homeomorphisms from nonorientable surfaces always have a zero SAF invariant, and offers a criterion to identify when a pseudo-Anosov map is not such a lift.

Contribution

It establishes a universal property of lifts of pseudo-Anosov maps from nonorientable surfaces and introduces a new criterion for distinguishing these lifts.

Findings

Lifts of pseudo-Anosov maps from nonorientable surfaces have vanishing SAF invariant.

A criterion is provided to determine when a pseudo-Anosov map is not a lift.

The result connects surface topology with dynamical invariants.

Abstract

We show that any pseudo-Anosov map that is a lift of pseudo-Anosov homeomorphism of a nonorientable surface has vanishing SAF invariant. We also provide a criterion to certify that a pseudo-Anosov map is not such a lift.