Minimal pseudo-Anosov stretch factors on nonoriented surfaces

TL;DR

This paper identifies the minimal stretch factors for specific classes of pseudo-Anosov maps on nonorientable and orientable surfaces, revealing new algebraic properties and limitations of existing techniques.

Contribution

It determines the smallest stretch factors for pseudo-Anosov maps on various nonorientable and orientable surfaces, and shows these factors lack Galois conjugates on the unit circle.

Findings

Smallest stretch factors identified for multiple surface genera.

Stretch factors lack Galois conjugates on the unit circle.

Techniques used to disprove Penner's conjecture are ineffective in nonorientable cases.

Abstract

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliations on the closed orientable surfaces of genus 1, 3, 5, 7, 9 and 11. As a byproduct, we obtain that the stretch factor of a pseudo-Anosov map on a nonorientable surface or an orientation-reversing pseudo-Anosov map on an orientable surface does not have Galois conjugates on the unit circle. This shows that the techniques that were used to disprove Penner's conjecture on orientable surfaces are ineffective in the nonorientable cases.