Short curves of end-periodic mapping tori
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917747858866176 |
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| author | Whitfield, Brandis |
| author_facet | Whitfield, Brandis |
| contents | 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}, $Λ^+$, $Λ^-$, 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 $d_Y (Λ^+, Λ^-)\geq K$ implies $\inf_{σ\in \text{AH}(M_f)}\{\ell_σ(\partial Y)\} \leq ε$. Here $\ell_σ(\partial Y)$ 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 $Σ$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07044 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Short curves of end-periodic mapping tori Whitfield, Brandis Geometric Topology 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}, $Λ^+$, $Λ^-$, 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 $d_Y (Λ^+, Λ^-)\geq K$ implies $\inf_{σ\in \text{AH}(M_f)}\{\ell_σ(\partial Y)\} \leq ε$. Here $\ell_σ(\partial Y)$ 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 $Σ$, 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. |
| title | Short curves of end-periodic mapping tori |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2408.07044 |