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

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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.
Original languageEnglish
Pages (from-to)5739-5781
JournalTransactions of the American Mathematical Society
Volume374
Early online date20 May 2021
DOIs
Publication statusE-pub ahead of print - 20 May 2021

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