Minimal Asymptotic Translation Lengths on Curve Complexes and Homology of Mapping Tori

Abstract

Let ${\mathit{S}}_{\mathit{g}}$ be a closed orientable surface of genus $\mathit{g}>1$. Consider the minimal asymptotic translation length ${\mathit{L}}_{\mathcal{T}}(\mathit{k},\mathit{g})$ on the Teichmüller space of ${\mathit{S}}_{\mathit{g}}$, among pseudo-Anosov mapping classes of ${\mathit{S}}_{\mathit{g}}$ acting trivially on k-dimensional subspaces of ${\mathit{H}}_1({\mathit{S}}_{\mathit{g}})$, $0\le \mathit{k}\le 2\mathit{g}$. The asymptote of ${\mathit{L}}_{\mathcal{T}}(\mathit{k},\mathit{g})$ for extreme cases $\mathit{k}=0,2\mathit{g}$ have been shown by several authors. Jordan Ellenberg asked whether there is a lower bound for ${\mathit{L}}_{\mathcal{T}}(\mathit{k},\mathit{g})$ interpolating the known results on ${\mathit{L}}_{\mathcal{T}}(0,\mathit{g})$ and ${\mathit{L}}_{\mathcal{T}}(2\mathit{g},\mathit{g})$, which was affirmatively answered by Agol, Leininger, and Margalit.

In this paper, we study an analogue of Ellenberg’s question, replacing Teichmüller spaces with curve complexes. We provide lower and upper bound on the minimal asymptotic translation length ${\mathit{L}}_{\mathcal{C}}(\mathit{k},\mathit{g})$ on the curve complex, whose lower bound interpolates the known results on ${\mathit{L}}_{\mathcal{C}}(0,\mathit{g})$ and ${\mathit{L}}_{\mathcal{C}}(2\mathit{g},\mathit{g})$.

Finally, for each g, we construct a non-Torelli pseudo-Anosov ${\mathit{f}}_{\mathit{g}}\in Mod({\mathit{S}}_{\mathit{g}})$ which does not normally generate $Mod({\mathit{S}}_{\mathit{g}})$, so that the asymptotic translation length of ${\mathit{f}}_{\mathit{g}}$ on the curve complex decays faster than a constant multiple of $1/\mathit{g}$ as $\mathit{g}\to \infty$. 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 Teichmüller spaces holds for curve complexes.