Relative simplicity of the universal coverings of transformation groups and Tsuboi's metric
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913592334352384 |
|---|---|
| author | Kawasaki, Morimichi Kimura, Mitsuaki Kodama, Hiroki Matsuda, Yoshifumi Matsushita, Takahiro Orita, Ryuma |
| author_facet | Kawasaki, Morimichi Kimura, Mitsuaki Kodama, Hiroki Matsuda, Yoshifumi Matsushita, Takahiro Orita, Ryuma |
| contents | 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 $\widetilde{\mathrm{Ham}}(M,ω)$ of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,ω)$.
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 $\widetilde{\mathrm{Ham}}(M, ω)$ is not quasi-isometric to the half line for every closed symplectic manifold $(M,ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00839 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative simplicity of the universal coverings of transformation groups and Tsuboi's metric Kawasaki, Morimichi Kimura, Mitsuaki Kodama, Hiroki Matsuda, Yoshifumi Matsushita, Takahiro Orita, Ryuma Group Theory General Topology Geometric Topology Metric Geometry Symplectic Geometry 20A05, 20E32, 51F30, 53D22, 53D40, 57S05 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 $\widetilde{\mathrm{Ham}}(M,ω)$ of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,ω)$. 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 $\widetilde{\mathrm{Ham}}(M, ω)$ is not quasi-isometric to the half line for every closed symplectic manifold $(M,ω)$. |
| title | Relative simplicity of the universal coverings of transformation groups and Tsuboi's metric |
| topic | Group Theory General Topology Geometric Topology Metric Geometry Symplectic Geometry 20A05, 20E32, 51F30, 53D22, 53D40, 57S05 |
| url | https://arxiv.org/abs/2412.00839 |