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

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