A note on commuting diffeomorphisms on surfaces

Let Sigma be a closed surface with nonzero Euler characteristic. We prove the existence of an open neighbourhood V of the identity map of Sigma in the C-1-topology with the following property: if G is an abelian subgroup of Diff(1) (Sigma) generated by any family of elements in V, then the elements of G have common fixed points. This result generalizes a similar result of Bonatti (Bonatti C 1989 Diffeomorphismes commutants des surfaces et stabilite des fibrations en tores Topology 29 101-26).