Thurston construction mapping classes with minimal dilatation

TL;DR

This paper explicitly computes the minimal dilatation for Thurston construction pseudo-Anosov maps on surfaces, using spectral radius minimization in a congruence subgroup, advancing understanding of minimal dilatations in surface dynamics.

Contribution

It provides an explicit method to determine the minimal dilatation for Thurston construction maps based on filling pairs, connecting spectral radius minimization with geometric intersection bounds.

Findings

Explicit minimal dilatation values for given filling pairs

Method to compute minimal spectral radius in a congruence subgroup

Lower bounds on intersection numbers for minimal dilatation maps

Abstract

Given a pair of filling curves $\alpha, \beta$ on a surface of genus $g$ with $n$ punctures, we explicitly compute the mapping classes realizing the minimal dilatation over all the pseudo-Anosov maps given by the Thurston construction on $\alpha,\beta$. We do so by solving for the minimal spectral radius in a congruence subgroup of $\text{PSL}_2(\mathbb{Z})$. We apply this result to realized lower bounds on intersection number between $\alpha$ and $\beta$ to give the minimal dilatation over any Thurston construction pA map on $\Sigma_{g,n}$ given by a filling pair $\alpha \cup \beta$.