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 Mf, its associated mapping torus, has a compactification as a $3$-manifold with boundary; further, if f is atoroidal, then Mf 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 given compact subsurface Y⊂ S. As an end-periodic analogy to work of Minsky in the finite-type setting, we show that for every ε>0 there exists $K> 0$ (depending only on ε and the capacity of f) for which dY (Λ⁺, Λ⁻)≥ K implies infσ∈ AH(Mf)_σ(∂ Y)\ ≤ ε. Here _σ (∂ Y) denotes the total geodesic length of ∂ Y in (Mf, σ), and the infimum is taken over all hyperbolic structures on Mf. This work produces the following: given a closed surface Σ, we provide a family of closed, fibered hyperbolic manifolds in which Σ is totally geodesically embedded, (almost) transverse to the pseudo-Anosov flow, with arbitrarily small systole.