Relative simplicity of the universal coverings of transformation groups and Tsuboi's metric

Morimichi Kawasaki, Mitsuaki Kimura, Hiroki Kodama, Yoshifumi Matsuda, Takahiro Matsushita, Ryuma Orita
Journal of Topology and Analysis, Online Ready
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Overview

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Many groups of transformations of a manifold are simple, yet passing to their universal covers can destroy simplicity. We study a weaker property called relative simplicity: a group has a proper normal subgroup containing every other proper normal subgroup. Many universal covers of transformation groups have this property, including the universal cover $\widetilde{\mathrm{Ham}}(M,\omega)$ of the Hamiltonian diffeomorphism group of any closed symplectic manifold $(M,\omega)$.

Relative simplicity also lets us extend Tsuboi’s construction of a metric space for a simple group. We use this extension to study the large-scale geometry of these universal covers. In particular, for every closed symplectic manifold, the Tsuboi metric space of $\widetilde{\mathrm{Ham}}(M,\omega)$ is not quasi-isometric to a half-line: even at large scales, its geometry cannot be modeled by a single ray.

@article{kawasaki2026relative,
  author        = {Kawasaki, Morimichi and Kimura, Mitsuaki and Kodama, Hiroki and Matsuda, Yoshifumi and Matsushita, Takahiro and Orita, Ryuma},
  title         = {{Relative simplicity of the universal coverings of transformation groups and Tsuboi's metric}},
  journal       = {Journal of Topology and Analysis},
  year          = {2026},
  doi           = {10.1142/S1793525326500445},
  eprint        = {2412.00839},
  archivePrefix = {arXiv}
}