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 $M_f$, its associated mapping torus, 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 \emph{positive and negative Handel-Miller laminations}, $\Lambda^+$, $\Lambda^-$, whose leaves naturally project to the arc and curve complex of a given compact subsurface $Y\subset S$. As an end-periodic analogy to work of Minsky in the finite-type setting, we show that for every $\epsilon>0$ there exists $K> 0$ (depending only on $\epsilon$ and the \emph{capacity} of $f$) for which $d_Y (\Lambda^+, \Lambda^-)\geq K$ implies $\inf_{\sigma\in \text{AH}(M_f)}\{\ell_\sigma(\partial Y)\} \leq \epsilon$. Here $\ell_\sigma (\partial Y)$ denotes the total geodesic length of $\partial Y$ in $(M_f, \sigma)$, and the infimum is taken over all hyperbolic structures on $M_f$. This work produces the following: given a closed surface $\Sigma$, we provide a family of closed, fibered hyperbolic manifolds in which $\Sigma$ is totally geodesically embedded, (almost) transverse to the pseudo-Anosov flow, with arbitrarily small systole.