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

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 Teichmüller 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.

Origami edge-paths in the curve graph

“…An origami for a closed surface S g≥2 is obtained from a finite number of Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. Our first result, which has also been independently shown by Jeffreys [14], establishes a correspondence between a coherent filling pair and an origami on S g . Theorem 1.1.…”

“…Origamis are of significant interest in Teichmüller theory. (See [6,14,13,20,25].) Specifically, an origami determines a Riemann surface along with a translation structure.…”

“…The combinatorial structure of origamis specific to a coherent filling pair of curvesa [1, 1]-origami-is a current area of investigation and we recommend the excellent discussion in the introduction of [14].…”