Comparing Teichm\"uller and curve graph translation lengths

TL;DR

This paper compares translation lengths of pseudo-Anosov mapping classes on Teichmüller space and the curve graph, revealing different asymptotic behaviors and constructing sequences with specific translation length properties.

Contribution

It demonstrates that minimal stable curve graph translation lengths can decrease with genus, contrasting with Teichmüller translation lengths, and constructs sequences with infinite multiplicity of certain lengths.

Findings

Teichmüller translation length is bounded below by log(√2).

Curve graph translation length can be of order 1/g.

Existence of infinitely many pseudo-Anosovs with the same curve graph translation length.

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 $n.$