Proof of the simplicity conjecture

Abstract

In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for \hbox $n \ge 3$, asked if the same statement holds in dimension two. We show that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple. This settles what is known as the “simplicity conjecture” in the affirmative. In fact, we prove the a priori stronger statement that this group is not perfect.

Our general strategy is partially inspired by suggestions of Fathi and the approach of Oh towards the simplicity question. In particular, we show that infinite twist maps, studied by Oh, are not finite energy homeomorphisms, which resolves the “infinite twist conjecture” in the affirmative; these twist maps are now the first examples of Hamiltonian homeomorphisms that can be said to have infinite energy. Another consequence of our work is that various forms of fragmentation for volume-preserving homeomorphisms that hold for higher dimensional balls fail in dimension two.

A central role in our arguments is played by spectral invariants defined via periodic Floer homology. We establish many new properties of these invariants that are of independent interest. For example, we prove that these spectral invariants extend continuously to area-preserving homeomorphisms of the disc, and we also verify for certain smooth twist maps a conjecture of Hutchings concerning recovering the Calabi invariant from the asymptotics of these invariants.