Small dilatation pseudo-Anosov mapping classes coming from the simplest hyperbolic braid

TL;DR

This paper investigates the smallest dilatation pseudo-Anosov mapping classes derived from a specific hyperbolic 3-manifold, revealing new minimal dilatations and their relation to orientability across various genera.

Contribution

It introduces new examples of minimal dilatation pseudo-Anosov classes from a hyperbolic braid, connecting these to known and conjectured minima across multiple genera.

Findings

Minimum dilatations include those for genus 2,3,4,5,8.

Minimum dilatation for orientable classes exceeds that for non-orientable classes in certain genera.

Examples support conjectures about minimal dilatations in specific genera.

Abstract

In this paper we study the minimum dilatation pseudo-Anosov mapping classes coming from fibrations over the circle of a single 3-manifold, the mapping torus for the "simplest pseudo-Anosov braid". The dilatations that arise include the minimum dilatations for orientable mapping classes for genus g=2,3,4,5,8 as well as Lanneau and Thiffeault's conjectural minima for orientable mapping classes, when g = 2,4 (mod 6). Our examples also show that the minimum dilatation for orientable mapping classes is strictly greater than the minimum dilatation for non-orientable ones when g = 4,6,8.