In an influential article from the 1970s, Albert Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the n-ball is simple for n≥3,asked if the same statement holds in dimension 2. In a joint work with Cristofaro-Gardiner and Humilière, we proved that the group of compactly supported area-preserving homeomorphisms of the 2-disc is not simple. This answers Fathi's question and settles what is known as "the simplicity conjecture" in the affirmative.
In fact, Fathi posed a more general question about all compact surfaces: is the group of "Hamiltonian homeomorphisms" (which I will define) simple? In my talk, I will review recent joint work with Cristofaro-Gardiner, Humilière, Mak and Smith answering this more general question of Fathi. The talk will be for the most part elementary and will only briefly touch on Floer homology which is a crucial ingredient of the solution.