Automorphisms of fine curve graphs of planar surfaces

TL;DR

This paper proves that for boundaryless planar surfaces with at least 7 punctures, the automorphism group of the fine curve graph is isomorphic to the surface's homeomorphism group, revealing a deep symmetry connection.

Contribution

It establishes a natural isomorphism between the automorphism group of the fine curve graph and the homeomorphism group for certain planar surfaces, a new result in surface topology.

Findings

Automorphism group is isomorphic to the homeomorphism group

Result applies to boundaryless planar surfaces with ≥7 punctures

Deepens understanding of surface symmetries and curve graphs

Abstract

The fine curve graph of a surface is the graph whose vertices are simple closed essential curves in the surface and whose edges connect disjoint curves. In this paper, we prove that the automorphism group of the fine curve graph of a surface is naturally isomorphic to the homeomorphism group of the surface for boundaryless planar surfaces with at least 7 punctures.