The fine curve graph is a variant of the curve graph for the homeomorphism group: a Gromov hyperbolic graph on which the homeomorphism group acts. In this talk we present joint work with Frédéric Le Roux linking the shape of the rotation set of a torus homeomorphism (a classical, dynamical conjugacy invariant) to the geometry of its action on the fine curve graph. As a consequence, one can construct homeomorphisms with positive scl close to the identity, and obtain Tits alternatives for (certain) subgroups of the homeomorphism group of the torus.