Parabolic isometries of the fine curve graph of the torus

Abstract

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 \cite{BHMMW}. This is made by proving that if $f\in \mathrm{Homeo}({\mathbb{T}}^2)$, then $f$ acts elliptically on $C^{†}({\mathbb{T}}^2)$ 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.