Least dilatation of pure surface braids

TL;DR

This paper investigates the minimal dilatation of pseudo-Anosov pure surface braids, establishing bounds that grow with genus and punctures, and providing insights into their asymptotic behavior.

Contribution

It offers new bounds on dilatation for pure surface braids, revealing how these bounds depend on genus and puncture count, and extends understanding of their asymptotic properties.

Findings

Bounds on dilatation grow with genus for fixed punctures

Dilatation is bounded away from zero for pure surface braids

Constant upper bound on dilatation for sufficiently many punctures

Abstract

We study the minimal dilatation of pseudo-Anosov pure surface braids and provide upper and lower bounds as a function of genus and the number of punctures. For a fixed number of punctures, these bounds tend to infinity as the genus does. We also bound the dilatation of pseudo-Anosov pure surface braids away from zero and give a constant upper bound in the case of a sufficient number of punctures.