## Abstract

Building on work of [BHW21], we study the action of the homeomorphism group of a surface S on the fine curve graph Cy(S). While the definition of Cy(S) parallels the classical curve graph for mapping class groups, we show that the dynamics of the action of Homeo(S) on Cy(S) is much richer: homeomorphisms induce parabolic isometries in addition to elliptics and hyperbolics, and all positive reals are realized as asymptotic translation lengths.

When the surface S is a torus, we relate the dynamics of the action of a homeomorphism on Cy(S) to the dynamics of its action on the torus via the classical theory of rotation sets. We characterize homeomorphisms acting hyperbolically, show asymptotic translation length provides a lower bound for the area of the rotation set, and, while no characterisation purely in terms of rotation sets is possible, we give sufficient conditions for elements to be elliptic or parabolic.

When the surface S is a torus, we relate the dynamics of the action of a homeomorphism on Cy(S) to the dynamics of its action on the torus via the classical theory of rotation sets. We characterize homeomorphisms acting hyperbolically, show asymptotic translation length provides a lower bound for the area of the rotation set, and, while no characterisation purely in terms of rotation sets is possible, we give sufficient conditions for elements to be elliptic or parabolic.

Original language | English |
---|---|

Journal | Advances in Mathematics |

Publication status | Accepted/In press - 29 Jun 2022 |