  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 obtain, 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.
