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

TL;DR

This paper introduces new constructions of pseudo-Anosov maps that optimize the ratio of translation lengths in Teichmüller space and the curve graph, showing their abundance and presence in deep subgroups.

Contribution

The paper presents a novel method for constructing ratio-optimizing pseudo-Anosov maps and proves their widespread existence within the mapping class group.

Findings

Constructed new ratio-optimizing pseudo-Anosov maps

Demonstrated abundance of these maps in the mapping class group

Found such maps deep in the Johnson filtration and in the point pushing subgroup

Abstract

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