Rotation sets and axes in the fine curve graph for torus homeomorphisms

Abstract

We expand the dictionary between the action of a torus homeomorphism on the fine curve graph and its rotation set. More precisely, we show that the fixed points at infinity of a loxodromic element determine the rotation set up to scale. A key ingredient is a metric version of the classical WPD property from geometric group theory. As a consequence we find new stable criteria for positive scl, and for two homeomorphisms to generate a free group, and we provide a Tits alternative for groups of torus homeomorphisms.