Simplicity of the contactomorphism group of finite regularity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917662821449728 |
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| author | Oh, Yong-Geun |
| author_facet | Oh, Yong-Geun |
| contents | For a given coorientable contact manifold $(M^{2n+1},ξ)$, we consider the group $ \operatorname{Cont}_c^{(r,δ)}(M,α)$ consisting of $C^{r,δ}$ contactomorphisms with compact support which is equipped with $C^{r,δ}$-topology of Hölder regularity $(r,δ)$ for $r \geq 1$ and $0 <δ\leq 1$. We prove that for all Hölder class exponents with $r > n + 2$ or $r = n+1, \, \frac12 < δ\leq 1$ (resp. $r < n+1$ or $r = n+1$ and $ 0< δ<\frac12$), the group is a perfect (and so a simple) group. In particular, $\operatorname{Cont}_c^r(M,ξ)$ is simple for all integer $r \geq 1$. For the case of $\operatorname{Cont}_c^{(r,δ)}(M,α)$ of general Hölder regularity, we prove the simplicity for all pairs $(r,δ)$ leaving only the case of $(r,δ) = (n+1,\frac12)$ open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18261 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simplicity of the contactomorphism group of finite regularity Oh, Yong-Geun Symplectic Geometry Dynamical Systems Geometric Topology 53D35, 53D10, 58D05 For a given coorientable contact manifold $(M^{2n+1},ξ)$, we consider the group $ \operatorname{Cont}_c^{(r,δ)}(M,α)$ consisting of $C^{r,δ}$ contactomorphisms with compact support which is equipped with $C^{r,δ}$-topology of Hölder regularity $(r,δ)$ for $r \geq 1$ and $0 <δ\leq 1$. We prove that for all Hölder class exponents with $r > n + 2$ or $r = n+1, \, \frac12 < δ\leq 1$ (resp. $r < n+1$ or $r = n+1$ and $ 0< δ<\frac12$), the group is a perfect (and so a simple) group. In particular, $\operatorname{Cont}_c^r(M,ξ)$ is simple for all integer $r \geq 1$. For the case of $\operatorname{Cont}_c^{(r,δ)}(M,α)$ of general Hölder regularity, we prove the simplicity for all pairs $(r,δ)$ leaving only the case of $(r,δ) = (n+1,\frac12)$ open. |
| title | Simplicity of the contactomorphism group of finite regularity |
| topic | Symplectic Geometry Dynamical Systems Geometric Topology 53D35, 53D10, 58D05 |
| url | https://arxiv.org/abs/2403.18261 |