Pierre-Antoine Guiheneuf, Emmanuel Militon
Published 2023-02-16Version 1
In this article we finish the classification of actions of torus homeomorphisms on the fine curve graph initiated by Bowden, Hensel, Mann, Militon, and Webb in [BHM + 22]. This is made by proving that if $f \in \mathrm{Homeo}(\mathbb{T}^2)$, then $f$ acts elliptically on $C^{\dagger}(\mathbb{T} ^2)$ if and only if $f$ has bounded deviation from some $v \in \mathbb{Q}^2 \setminus \left\{ 0 \right\}$. The proof involves some kind of slow rotation sets for torus homeomorphisms.
Torus diffeomorphisms with parabolic and non-proper actions on the fine curve graph and their generalized rotation sets
Toward the classification of cohomology-free vector fields
Classification of horocycle orbit closures in $ \mathbb{Z} $-covers
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