Cylinder decompositions on geometric armadillo tails

TL;DR

This paper explores a class of infinite-type translation surfaces, providing explicit cylinder decompositions that reveal obstructions to affine symmetries, advancing understanding of their geometric and dynamical properties.

Contribution

It introduces explicit cylinder decompositions for infinite-type surfaces, highlighting obstructions to affine diffeomorphisms not present in finite-type cases.

Findings

Explicit cylinder decompositions on infinite-type surfaces

Identification of obstructions to affine diffeomorphisms

Differences between finite and infinite-type translation surfaces

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.