Short curves of end-periodic mapping tori
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, $Λ^+$, $Λ^-$, 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 $\varepsilon>0$ there exists $K> 0$ (depending only on $\varepsilon$ and the capacity of $f$) for which $d_Y (Λ^+, Λ^-)\geq K$ implies $\ell_s(\partial Y) \leq \varepsilon$. 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 $\varepsilon>0$ and closed connected surface $Σ$ of genus $g\geq 2$, we provide a closed fibered hyperbolic $3$-manifold in which $Σ$ embeds as a totally geodesic surface whose systole length is at most $\varepsilon$.