  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.
