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

TL;DR

This paper investigates the large scale geometry of Hamiltonian diffeomorphisms on the two-sphere using new spectral invariants, resolving longstanding questions about their quasi-isometry, kernel unboundedness, and group simplicity.

Contribution

It introduces novel spectral invariants via periodic Floer homology to answer key questions about the group's geometry and algebraic structure.

Findings

The group is not quasi-isometric to the real line.

The kernel of Calabi over any proper open subset is unbounded.

The group of area and orientation preserving homeomorphisms is not simple.

Abstract

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.