  In 1980, Albert Fathi asked whether the group of area-preserving
homeomorphisms of the 2-disc that are the identity near the boundary is a
simple group. In this paper, we show that the simplicity of this group is
equivalent to the following fragmentation property in the group of compactly
supported, area preserving diffeomorphisms of the plane: there exists a
constant m such that every element supported on a disc D is the product of at
most m elements supported on topological discs whose area are half the area of
D.
