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Meanders, hyperelliptic pillowcase covers, and the Johnson filtration

Luke Jeffreys

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 Teichmüller disk of a quadratic differential lying in this connected component.

Meanders, hyperelliptic pillowcase covers, and the Johnson filtration

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 Teichmüller disk of a quadratic differential lying in this connected component.
Paper Structure (19 sections, 16 theorems, 13 equations, 30 figures, 1 table)

This paper contains 19 sections, 16 theorems, 13 equations, 30 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quadratic differentials
  3. Pillowcase covers
  4. Hyperelliptic connected components
  5. Meanders
  6. Anchored meanders
  7. The stratum of a meander
  8. Meanders from hyperelliptic $[1,1]$-pillowcase covers
  9. Minimal constructions
  10. Minimal closed meanders
  11. Minimal singly-anchored open meanders
  12. Minimal doubly-anchored open meanders and semi-meanders
  13. Lifting to hyperelliptic connected components
  14. Ratio-optimisers in the Johnson filtration
  15. Ratio-optimising pseudo-Anosovs
  16. ...and 4 more sections

Key Result

Theorem 1.2

The minimum number of squares required to produce $[1,1]$-pillowcase covers in the hyperelliptic components of the strata of the moduli space of quadratic differentials are as in Table tab:theorem.

Figures (30)

  • Figure 1: Two half-translation surfaces. Sides with the same label are identified by translation, or by half-translation when arrows are indicated. The surface on the left is also a translation surface.
  • Figure 2: Two pillowcase covers. The surface at the top has a single-horizontal cylinder but two vertical cylinders. The surface at the bottom is a $[1,1]$-pillowcase cover.
  • Figure 3: An open meander on the left and an open semi-meander on the right.
  • Figure 4: A closed meander on the left and a closed semi-meander on the right.
  • Figure 5: A doubly-anchored open meander on the left and a doubly-anchored open semi-meander on the right.
  • ...and 25 more figures

Theorems & Definitions (19)

  • Theorem 1.2
  • Theorem 1.4
  • Proposition 4.1
  • Claim 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Proposition 4.5
  • Proposition 4.6
  • Lemma 4.7
  • Remark 4.8
  • ...and 9 more