The geometry 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 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 ε>0 there exists $K> 0$ (depending only on ε and the \emph{capacity} of $f$) for which d_Y (\Lambda⁺, \Lambda^-)\geq K implies \inf_{σ\in \text{AH}(Mf)}\{\ell_\sigma(\partial Y)\} \leq ε. Here \ell_\sigma (\partial Y) denotes the total geodesic length of $\partial Y$ in (M_f, σ), 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.

Grantor dc:publisher

Temple University. Libraries

Year dc:date.issued

2025

Author dc:creator

• Whitfield, Brandis

Advisor dc:contributor.advisor

• Taylor, Samuel J.

Committee members dc:contributor.committeemember

• Futer, David
• Stover, Matthew
• Aougab, Tarik

• Mathematics

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Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

eng

Repository record dc:identifier.uri

https://scholarshare.temple.edu/handle/20.500.12613/11336

OAI identifier oai:identifier

oai:scholarshare.temple.edu:20.500.12613/11336