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We resolve three longstanding questions related to the large scale geometryof the group of Hamiltonian diffeomorphisms of the two-sphere, equipped withHofer's metric. Namely: (1) we resolve the Kapovich-Polterovich question byshowing that this group is not quasi-isometric to the real line; (2) moregenerally, we show that the kernel of Calabi over any proper open subset isunbounded; and (3) we show that the group of area and orientation preservinghomeomorphisms of the two-sphere is not a simple group. We also obtain, as acorollary, 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 ofour proofs are new sequences of spectral invariants over the two-sphere,defined via periodic Floer homology.

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