Minimal asymptotic translation lengths on curve complexes and homology of mapping tori

TL;DR

This paper investigates the minimal asymptotic translation lengths of pseudo-Anosov mapping classes on curve complexes, providing bounds that interpolate known extreme cases and constructing examples with rapidly decaying lengths as genus increases.

Contribution

It extends the understanding of translation lengths from Teichmüller spaces to curve complexes, offering bounds and explicit examples that reveal new decay phenomena.

Findings

Bounds on $L_{\mathcal{C}}(k, g)$ interpolate known cases.

Constructed pseudo-Anosov mapping classes with rapidly decaying translation lengths.

Restrictions on minimal translation lengths based on genus and phenomena similar to Teichmüller spaces.

Abstract

Let $S_g$ be a closed orientable surface of genus $g > 1$. Consider the minimal asymptotic translation length $L_{\mathcal{T}}(k, g)$ on the Teichm\"uller space of $S_g$, among pseudo-Anosov mapping classes of $S_g$ acting trivially on a $k$-dimensional subspace of $H_1(S_g)$, $0 \le k \le 2g$. The asymptotics of $L_{\mathcal{T}}(k, g)$ for extreme cases $k = 0, 2g$ have been shown by several authors. Jordan Ellenberg asked whether there is a lower bound for $L_{\mathcal{T}}(k, g)$ interpolating the known results on $L_{\mathcal{T}}(0, g)$ and $L_{\mathcal{T}}(2g, g)$, which was affirmatively answered by Agol, Leininger, and Margalit. In this paper, we study an analogue of Ellenberg's question, replacing Teichm\"uller spaces with curve complexes. We provide lower and upper bound on the minimal asymptotic translation length $L_{\mathcal{C}}(k, g)$ on the curve complex, whose lower bound interpolates the known results on $L_{\mathcal{C}}(0, g)$ and $L_{\mathcal{C}}(2g, g)$. Finally, for each $g$, we construct a non-Torelli pseudo-Anosov $f_g \in \operatorname{Mod}(S_g)$ which does not normally generates $\operatorname{Mod}(S_g)$ and so that the asymptotic translation length of $f_g$ on curve complexes decays more quickly than a constant multiple of $1/g$ as $g \to \infty$. From this, we provide a restriction on how small the asymptotic translation lengths on curve complexes should be if the similar phenomenon as in the work of Lanier and Margalit on Teichm\"uller spaces holds for curve complexes.