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 , Λ + , Λ − , whose leaves naturally project to the arc and curve complex of a compact subsurface 𝑌 ⊂ 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 𝜀 > 0 there exists 𝐾 > 0 (depending only on ε and the capacity of f ) for which 𝑑 𝑌 ⁡ ( Λ + , Λ − ) ≥ 𝐾 implies ℓ 𝑠 ⁡ ( ∂ 𝑌 ) ≤ 𝜀 . Here ℓ 𝑠 ⁡ ( ∂ 𝑌 ) denotes the total length of the geodesic representative of ∂ Y in ( 𝑀 𝑓 , 𝑠 ) . This work additionally produces the following: for every 𝜀 > 0 and closed connected surface Σ of genus 𝑔 ≥ 2 , we provide a closed fibered hyperbolic 3-manifold in which Σ embeds as a totally geodesic surface whose systole length is at most ε .