Automorphisms of some variants of fine graphs

TL;DR

This paper extends the understanding of automorphism groups of fine graphs associated with surfaces, proving their isomorphism to surface homeomorphism groups for a broader class of surfaces including the torus.

Contribution

It generalizes previous results by establishing automorphism group isomorphisms for fine graphs on various surfaces, beyond closed surfaces of genus ≥ 2.

Findings

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

Method applies to more general surfaces, including non-compact and non-orientable cases.

Discussion of a smooth version of the fine graph enhances the theoretical framework.

Abstract

Recently Bowden, Hensel and Webb defined the fine curve graph for surfaces, extending the notion of curve graphs for the study of homeomorphism or diffeomorphism groups of surfaces. Later Long, Margalit, Pham, Verberne and Yao proved that for a closed surface of genus $g\geqslant 2$, the automorphism group of the fine graph is naturally isomorphic to the homeomorphism group of the surface. We extend this result to the torus case $g=1$; in fact our method works for more general surfaces, compact or not, orientable or not. We also discuss the case of a smooth version of the fine graph.