Comparing Teichm\"uller and curve graph translation lengths

Abstract

A pseudo-Anosov mapping class acts on Teichm\"uller space $\mathcal{T}$ as well as on the curve graph $\mathcal{C}$ with so called north-south dynamics. We can measure a stable translation length $l_\mathcal{T}$ and $l_\mathcal{C}$ of the respective actions. Boissy and Lanneau compute the minimal Teichm\"uller translation length over all pseudo Anosovs in a fixed genus that lie in a hyperelliptic component of translation surfaces. In particular, this minimum is always greater than $\log(\sqrt{2}),$ independently of the genus. Here, we show that the minimal stable curve graph translation length over the same family of pseudo-Anosovs behaves differently: Namely, for a genus $g$ surface this minimal translation length is of order $\frac{1}{g}.$ To prove this result, we combine techniques that are used to find upper and lower bounds for the stable curve graph translation length with the Rauzy-Veech induction machinery. We proceed with showing that for a fixed genus $g$ there is a sequence of pseudo-Anosovs $f_n$ with $$\lim\limits_{n \to \infty} l_\mathcal{T}(f_n) = \infty \text{ and } l_\mathcal{C}(f_n) \le \frac{1}{g-1}$$ for all $n \in \mathbb{N}.$ As a corollary, we obtain that there are stable curve graph translation lengths with infinite multiplicity, i.e. there exists $q \in \mathbb{Q}$ and infinitely many, non-conjugate pseudo-Anosovs $f_n$ with $l_\mathcal{C}(f_n) = q$ for all $n.$