
TL;DR
This paper introduces CaTherine wheels, a topological construct linking various fields and providing a new framework for understanding hyperbolic 3-manifolds through structures like pseudo-Anosov flows and quasimorphisms.
Contribution
It develops the theory of CaTherine wheels and establishes a canonical correspondence between different structures associated with hyperbolic 3-manifolds.
Findings
Established a bijection between pseudo-Anosov flows and CaTherine wheels.
Connected CaTherine wheels to minimal G-zippers and quasimorphisms.
Extended the theory of fiberings and Thurston norm in hyperbolic 3-manifolds.
Abstract
A CaTherine wheel is a surjective continuous map such that for every closed interval the image is homeomorphic to a disk, and is contained in the boundary of this disk. CaTherine wheels arise in many areas of low-dimensional geometry and topology, including conformal dynamics (expanding Thurston maps, expanding origamis), probability theory (whole plane for , LQG metric trees) and elsewhere. We develop their theory in generality, and explain how CaTherine wheels and their associated structures can serve as a dictionary between these various fields. Our most substantial applications are to the theory of hyperbolic 3-manifolds. If is a closed hyperbolic 3-manifold and , we show that there is a canonical bijection between four kinds of structures associated to : 1.…
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