Hyperbolic isometries of the fine curve graph of higher genus surfaces

Abstract

We prove that for a homeomorphism $f$ that is isotopic to the identity on a closed hyperbolic surface, the following are equivalent:

• $f$ acts hyperbolically on the fine curve graph;
• $f$ is isotopic to a pseudo-Anosov map relative to a finite $f$-invariant set;
• the ergodic homological rotation set of $f$ has non-empty interior.