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The Algebra, Geometry, and Topology of Cusped Mapping Tori
Oakley, John
Oakley, John
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Thesis/Dissertation
Date
2025-08
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Mathematics
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https://doi.org/10.34944/ny4m-jp07
Abstract
In this thesis, we study the relationship between the algebra, geometry, and topology of cusped mapping tori. In particular, we prove that for hyperbolic fibered knots in any closed, connected, oriented 3-manifold the volume and genus are unrelated. As an application we answer a question of Hirose, Kalfagianni, and Kin about volumes of mapping tori that are double branched covers.
We then develop geometric tools for analyzing generating sets of cusped mapping tori. Using this framework, we make progress on the rank–genus problem for cusped mapping tori whose monodromy has large translation distance in the curve graph.
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