Parabolic isometries of the fine curve graph of the torus

TL;DR

This paper completes the classification of torus homeomorphism actions on the fine curve graph, showing that elliptic actions correspond to bounded deviations from rational vectors, using slow rotation sets.

Contribution

It provides a full classification of actions on the fine curve graph for torus homeomorphisms, linking elliptic actions to bounded deviations from rational vectors.

Findings

Elliptic actions occur if and only if there is bounded deviation from some rational vector.

The proof introduces slow rotation sets for torus homeomorphisms.

Completes the classification initiated in prior work.

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 $C^{\dagger}(\mathbb{T}^2)$ 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.