Short curves of end-periodic mapping tori

TL;DR

This paper studies the geometry of end-periodic mapping tori, establishing a relationship between the distance of certain laminations and the length of boundary curves, and constructs hyperbolic 3-manifolds with small systoles containing a given surface.

Contribution

It extends finite-type surface theory to infinite-type surfaces, relating lamination distances to boundary lengths, and constructs hyperbolic manifolds with prescribed embedded surfaces and small systoles.

Findings

Distance between laminations controls boundary geodesic lengths

Existence of hyperbolic structures with small systoles containing a given surface

Construction of fibered hyperbolic manifolds with totally geodesic embedded surfaces

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.