Braid Forcing, Hyperbolic Geometry, and Pseudo-Anosov Sequences of Low Entropy

Abstract

We view braids as automorphisms of punctured disks and define a partial order on pseudo-Anosov braids called the “forcing order”. The order measures whether one automorphism induces another given automorphism on the surface. Pseudo-Anosov growth rate decreases relative to the order and appears to give a good measure of braid complexity. Unfortunately it appears difficult computationally to determine explicitly the partial order structure by hand. We use several computer algorithms to study the bottom part of the partial order when the number of braid strands is fixed. From the algorithms, we build sequences of low entropy pseudo-Anosov n-strand braids that are minimal in the sense that they do not force any other pseudo-Anosov braids on the same number of strands. The sequences are an extension of work done by Hironaka and Kin, and we conjecture the sequences to achieve minimal entropy among certain non-trivial classes of braids. In general, the lowest entropy pseudo-Anosov braids appear to have mapping tori that come from Dehn surgery on very low volume hyperbolic 3-manifolds, and we begin to analyze the relation between entropy and hyperbolic volume. Moreover, the low-growth families contain non-trivial low-growth families of horseshoe braids and we proceed to study dynamics of the horseshoe map as well.