PFH spectral invariants on the two-sphere and the large scale geometry of Hofer’s metric

We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer’s metric. Namely: (1) we resolve the Kapovich–Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also find, as a corollary, that the group of area-preserving diffeomorphisms of the open disc, equipped with an area form of finite area, is not perfect. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology.