Short curves of end-periodic mapping tori

Abstract

Let $S$ be a boundaryless infinite-type surface with finitely many ends and consider an end-periodic homeomorphism $f$ of S. The end-periodicity of $f$ ensures that its associated mapping torus $M_f$ has a compactification as a $3$-manifold with boundary; further, if $f$ is atoroidal then $M_f$ admits a hyperbolic metric. Such maps admit invariant positive and negative Handel--Miller laminations, ${\Lambda }^{+}$, ${\Lambda }^{-}$, whose leaves naturally project to the arc and curve complex of a compact subsurface $Y\subset S$. As an end-periodic analogy to work of Minsky in the finite-type setting, we show that there exists a hyperbolic structure $s$ of $M_f$ such that for every $\epsilon >0$ there exists $K>0$ (depending only on $\epsilon$ and the capacity of $f$) for which $d_Y({\Lambda }^{+},{\Lambda }^{-})\ge K$ implies ${\ell }_s(\partial Y)\le \epsilon$. Here ${\ell }_s(\partial Y)$ denotes the total length of the geodesic representative of $\partial Y$ in $(M_f,s)$. This work additionally produces the following: for every $\epsilon >0$ and closed connected surface $\Sigma$ of genus $g\ge 2$, we provide a closed fibered hyperbolic $3$-manifold in which $\Sigma$ embeds as a totally geodesic surface whose systole length is at most $\epsilon$.

45 pages, 7 figures. Significant edits to clarify exposition in Sections 2 and 6, per referee comments. To appear in Adv. Math