Parabolic isometries of the fine curve graph of the torus

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. This is made by proving that if , then acts elliptically on if and only if has bounded deviations in some direction . The proof involves some kind of slow rotation sets for torus homeomorphisms.