Meanders, hyperelliptic pillowcase covers, and the Johnson filtration

TL;DR

This paper constructs specific meanders and hyperelliptic pillowcase covers to demonstrate the existence of deep pseudo-Anosov elements in the Johnson filtration within certain moduli spaces of quadratic differentials.

Contribution

It introduces minimal constructions of meanders and pillowcase covers to produce pseudo-Anosov maps with deep Johnson filtration properties.

Findings

Existence of ratio-optimising pseudo-Anosovs in hyperelliptic components.

Construction of pillowcase covers with single horizontal and vertical cylinders.

Deep elements in the Johnson filtration stabilizing Teichmüller disks.

Abstract

We provide minimal constructions of meanders with particular combinatorics. Using these meanders, we give minimal constructions of hyperelliptic pillowcase covers with a single horizontal cylinder and simultaneously a single vertical cylinder so that one or both of the core curves are separating curves on the underlying surface. In the case where both of the core curves are separating, we use these surfaces in a construction of Aougab-Taylor in order to prove that for any hyperelliptic connected component of the moduli space of quadratic differentials with no poles there exist ratio-optimising pseudo-Anosovs lying arbitrarily deep in the Johnson filtration and stabilising the Teichm\"uller disk of a quadratic differential lying in this connected component.