1. Relative simplicity of the universal coverings of transformation groups and Tsuboi's metric
- Author
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Kawasaki, Morimichi, Kimura, Mitsuaki, Kodama, Hiroki, Matsuda, Yoshifumi, Matsushita, Takahiro, and Orita, Ryuma
- Subjects
Mathematics - Group Theory ,Mathematics - General Topology ,Mathematics - Geometric Topology ,Mathematics - Metric Geometry ,Mathematics - Symplectic Geometry ,20A05, 20E32, 51F30, 53D22, 53D40, 57S05 - Abstract
Many transformation groups on manifolds are simple, but their universal coverings are not. In the present paper, we study the concept of relatively simple group, that is, a group with the maximum proper normal subgroup. We show that many examples of universal coverings of transformation groups are relatively simple, including the universal covering $\widetilde{\mathrm{Ham}}(M,\omega)$ of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,\omega)$. Tsuboi constructed a metric space $\mathcal{M}(G)$ for a simple group $G$. We generalize his construction to relatively simple groups, and study their large scale geometric structure. In particular, Tsuboi's metric space of $\widetilde{\mathrm{Ham}}(M, \omega)$ is not quasi-isometric to the half line for every closed symplectic manifold $(M,\omega)$., Comment: 37 pages, comments are welcome
- Published
- 2024