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Cylinder decompositions on geometric armadillo tails

Dami Lee, Josh Southerland

Abstract

We study a class of finite-area, infinite-type translation surfaces, and find an explicit cylinder decomposition on these surfaces which do not manifest on finite-type translation surfaces. Each cylinder decomposition contains a special curve which we show is an obstruction to the existence of certain affine diffeomorphisms.

Cylinder decompositions on geometric armadillo tails

Abstract

We study a class of finite-area, infinite-type translation surfaces, and find an explicit cylinder decomposition on these surfaces which do not manifest on finite-type translation surfaces. Each cylinder decomposition contains a special curve which we show is an obstruction to the existence of certain affine diffeomorphisms.

Paper Structure

This paper contains 11 sections, 19 theorems, 85 equations, 10 figures.

Table of Contents

  1. Introduction and Definitions
  2. Main results
  3. Related work
  4. Acknowledgements
  5. The spine and the first cylinder (or a particular direction on geometric armadillo tails)
  6. Framing (or finding the bottom saddle connection of each cylinder by induction)
  7. Constructing $\mathrm{bsc}_k^{\prime}$
  8. Constructing $\mathrm{bsc}_k$
  9. Infinitely many cylinders (or finding successive cylinders in the $\frac{1}{2-r}$-direction)
  10. Measurements of cylinders (or computations reinforcing work in previous sections)
  11. No parabolic element preserves the cylinder decomposition

Key Result

Theorem 1.4

There exists a cylinder decomposition with a rigid spine on any geometric armadillo tail surface of parameter $\frac{1}{q}$, $q \in \mathbb{N}\setminus \{1\}$. Moreover, there is no parabolic affine diffeomorphism of the surface that fixes this cylinder decomposition.

Figures (10)

  • Figure 1: The harmonic armadillo tail surface
  • Figure 2: A cylinder decomposition of an armadillo tail surface in the horizontal direction
  • Figure 3: A cylinder decomposition of an armadillo tail surface in the vertical direction
  • Figure 4: A geometric armadillo tail ($r=4/5$) with a trajectory of slope $5/6$
  • Figure 5: The cylinder $\mathrm{cyl}_1$ on a geometric armadillo tail with $r=1/2$ (left) and $1/3$ (right)
  • ...and 5 more figures

Theorems & Definitions (45)

  • Example 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • ...and 35 more