Automorphisms of the fine 1-curve graph

TL;DR

This paper proves that the automorphism group of the fine 1-curve graph of a surface is isomorphic to the surface's homeomorphism group, establishing a deep connection between graph symmetries and surface topology.

Contribution

It establishes a natural isomorphism between the automorphism group of the fine 1-curve graph and the homeomorphism group of the surface, revealing new insights into surface symmetries.

Findings

Automorphism group of the fine 1-curve graph is isomorphic to the surface's homeomorphism group.

The result holds for closed, orientable surfaces with genus at least one.

Provides a new perspective on the relationship between surface topology and graph automorphisms.

Abstract

The fine 1-curve graph of a surface is a graph whose vertices are simple closed curves on the surface and whose edges connect vertices that intersect in at most one point. We show that the automorphism group of the fine 1-curve graph is naturally isomorphic to the homeomorphism group of a closed, orientable surface with genus at least one.