CaTherine wheels
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866914510994931712 |
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| author | Calegari, Danny Loukidou, Ino |
| author_facet | Calegari, Danny Loukidou, Ino |
| contents | A CaTherine wheel is a surjective continuous map $f:S^1 \to S^2$ such that for every closed interval $I\subset S^1$ the image $f(I)$ is homeomorphic to a disk, and $f(\partial I)$ 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 ${\rm SLE}_κ$ for $κ\ge 8$, 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 $M$ is a closed hyperbolic 3-manifold and $G=π_1(M)$, we show that there is a canonical bijection between four kinds of structures associated to $M$:
1. orbit-equivalence classes of pseudo-Anosov flows on $M$ without perfect fits;
2. $G$-equivariant CaTherine wheels up to conjugacy;
3. minimal $G$-zippers; and
4. connected components of the space of uniform quasimorphisms on $G$.
This generalizes and amplifies the theory of fiberings of hyperbolic 3-manifolds over the circle and the Thurston norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24619 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | CaTherine wheels Calegari, Danny Loukidou, Ino Geometric Topology Dynamical Systems Group Theory Probability A CaTherine wheel is a surjective continuous map $f:S^1 \to S^2$ such that for every closed interval $I\subset S^1$ the image $f(I)$ is homeomorphic to a disk, and $f(\partial I)$ 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 ${\rm SLE}_κ$ for $κ\ge 8$, 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 $M$ is a closed hyperbolic 3-manifold and $G=π_1(M)$, we show that there is a canonical bijection between four kinds of structures associated to $M$: 1. orbit-equivalence classes of pseudo-Anosov flows on $M$ without perfect fits; 2. $G$-equivariant CaTherine wheels up to conjugacy; 3. minimal $G$-zippers; and 4. connected components of the space of uniform quasimorphisms on $G$. This generalizes and amplifies the theory of fiberings of hyperbolic 3-manifolds over the circle and the Thurston norm. |
| title | CaTherine wheels |
| topic | Geometric Topology Dynamical Systems Group Theory Probability |
| url | https://arxiv.org/abs/2604.24619 |