Automorphisms of the fine curve graph

TL;DR

This paper proves that the automorphism group of the fine curve graph of a surface is isomorphic to the surface's homeomorphism group, extending classical results about the curve graph's automorphisms.

Contribution

It establishes an isomorphism between the automorphism group of the fine curve graph and the homeomorphism group of the surface, generalizing Ivanov's classical theorem.

Findings

Automorphism group of the fine curve graph is isomorphic to the surface's homeomorphism group.

Extends classical results from the curve graph to the fine curve graph.

Provides a new characterization of surface symmetries via graph automorphisms.

Abstract

Building on work of Farb and the second author, we prove that the group of automorphisms of the fine curve graph for a surface is isomorphic to the group of homeomorphisms of the surface. This theorem is analogous to the seminal result of Ivanov that the group of automorphisms of the (classical) curve graph is isomorphic to the extended mapping class group of the corresponding surface.