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Given $\phi$ a pseudo-Anosov map, let $\ell_\mathcal{T}(\phi)$ denote thetranslation length of $\phi$ in the Teichm\"uller space, and let$\ell_\mathcal{C}(\phi)$ denote the stable translation length of $\phi$ in thecurve graph. Gadre--Hironaka--Kent--Leininger showed that, as a function ofEuler characteristic $\chi(S)$, the minimal possible ratio $\tau(\phi) =\frac{\ell_\mathcal{T}(\phi)}{\ell_\mathcal{C}(\phi)}$ is $\log(|\chi(S)|)$, upto uniform additive and multiplicative constants. In this short note, weintroduce a new construction of such ratio optimizers and demonstrate theirabundance in the mapping class group. Further, we show that ratio optimizerscan be found arbitrarily deep into the Johnson filtration as well as in thepoint pushing subgroup.

Pseudo-Anosovs optimizing the ratio of Teichm\"uller to curve graph translation length