{"id":{"repo_id":"temple","oai_identifier":"oai:scholarshare.temple.edu:20.500.12613/11336"},"canonical_url":"https://search.dev.ndltd.org/etd/temple/oai:scholarshare.temple.edu:20.500.12613/11336","repository":{"repo_id":"temple","name":"Temple University","base_url":"https://scholarshare.temple.edu/server/oai/request"},"display":{"title":"The geometry 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 $M_f$, 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.","abstract_html":"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 <span class=\"etd-inline-math\">M<sub>f</sub></span>, its associated mapping torus, has a compactification as a $3$-manifold with boundary; further, if $f$ is atoroidal, then <span class=\"etd-inline-math\">M<sub>f</sub></span> admits a hyperbolic metric. Such maps admit invariant \\emph{positive and negative Handel-Miller laminations}, <span class=\"etd-inline-math\">\\Lambda<sup>+</sup></span>, <span class=\"etd-inline-math\">\\Lambda<sup>-</sup></span>, 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 <span class=\"etd-inline-math\">&epsilon;&gt;0</span> there exists $K&gt; 0$ (depending only on <span class=\"etd-inline-math\">&epsilon;</span> and the \\emph{capacity} of $f$) for which <span class=\"etd-inline-math\">d<sub>Y</sub> (\\Lambda<sup>+</sup>, \\Lambda<sup>-</sup>)\\geq K</span> implies <span class=\"etd-inline-math\">\\inf<sub>&sigma;\\in \\text{AH}(M<sub>f</sub>)</sub>\\{\\ell<sub>\\</sub>sigma(\\partial Y)\\} \\leq &epsilon;</span>. Here <span class=\"etd-inline-math\">\\ell<sub>\\</sub>sigma (\\partial Y)</span> denotes the total geodesic length of $\\partial Y$ in <span class=\"etd-inline-math\">(M<sub>f</sub>, &sigma;)</span>, and the infimum is taken over all hyperbolic structures on <span class=\"etd-inline-math\">M<sub>f</sub></span>. 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.","abstract_has_math":true,"creators":["Whitfield, Brandis"],"institution":"Temple University. Libraries","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Taylor, Samuel J."],"committee_chairs":[],"committee_members":["Futer, David","Stover, Matthew","Aougab, Tarik"],"year":2025,"date_issued":"2025-05","date_published":"2025-05","updated_at":"2026-07-27T21:20:03Z","subjects":["Mathematics"],"languages":["eng"],"rights":["IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available."],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://scholarshare.temple.edu/handle/20.500.12613/11336","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Taylor, Samuel J."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Futer, David","Stover, Matthew","Aougab, Tarik"]},{"key":"dc:creator","label":"Author","values":["Whitfield, Brandis"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-10-06T18:13:59Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-10-06T18:13:59Z"]},{"key":"dc:date.issued","label":"Date","values":["2025-05"]},{"key":"dc:publisher","label":"Institution","values":["Temple University. 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Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://scholarshare.temple.edu/handle/20.500.12613/11336"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["The geometry of end-periodic mapping tori"]}]}],"canonical_facts":{"dc:contributor.advisor":["Taylor, Samuel J."],"dc:contributor.committeemember":["Futer, David","Stover, Matthew","Aougab, Tarik"],"dc:creator":["Whitfield, Brandis"],"dc:date.accessioned":["2025-10-06T18:13:59Z"],"dc:date.available":["2025-10-06T18:13:59Z"],"dc:date.issued":["2025-05"],"dc:description.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."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://scholarshare.temple.edu/handle/20.500.12613/11336"],"dc:language.iso":["eng"],"dc:publisher":["Temple University. Libraries"],"dc:rights":["IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available."],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Mathematics"],"dc:title":["The geometry of end-periodic mapping tori"],"dc:type":["Text"]},"updated_at":"2026-07-27T21:20:03Z"}