# Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs

## TL;DR

This paper constructs minimal square-tiled surfaces with specific cylinder structures across all strata of Abelian differentials and shows that ratio-optimising pseudo-Anosov homeomorphisms are widespread in these strata, with applications to filling pairs on punctured surfaces.

## Contribution

It provides explicit constructions of minimal square-tiled surfaces with one vertical and one horizontal cylinder in all non-hyperelliptic strata, and demonstrates the ubiquity of ratio-optimising pseudo-Anosov homeomorphisms.

## Key findings

- Constructed minimal square-tiled surfaces in all non-hyperelliptic strata.
- Proved that ratio-optimising pseudo-Anosov homeomorphisms are widespread.
- Applied results to construct filling pairs with equal algebraic and geometric intersection numbers.

## Abstract

In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds. Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichm\"uller to curve graph translation length are, in a reasonable sense, ubiquitous in the connected components of strata of Abelian differentials. Finally, we present a further application to filling pairs on punctured surfaces by constructing filling pairs whose algebraic and geometric intersection numbers are equal.