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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.11363 |
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| _version_ | 1866913656623595520 |
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| author | Fukui, Kazuhiko Yagasaki, Tatsuhiko |
| author_facet | Fukui, Kazuhiko Yagasaki, Tatsuhiko |
| contents | In this paper we study boundedness of conjugation invariant norms on diffeomorphism groups of manifold pairs. For the diffeomorphism group ${\mathcal D} \equiv {\rm Diff}(M,N)_0$ of a closed manifold pair $(M, N)$ with $\dim N \geq 1$, first we clarify the relation among the fragmentation norm, the conjugation generated norm, the commutator length $cl$ and the commutator length with support in balls $clb$ and show that ${\mathcal D}$ is weakly simple relative to a union of some normal subgroups of ${\mathcal D}$. For the boundedness of these norms, this paper focuses on the case where $N$ is a union of $m$ circles. In this case, the rotation angle on $N$ induces a quasimorphism $ν: {\rm Isot}(M, N)_0 \to {\Bbb R}^m$, which determines a subgroup $A$ of ${\Bbb Z}^m$ and a function $\widehatν : {\mathcal D} \to {\Bbb R}^m/A$. If ${\rm rank}\,A = m$, these data leads to an upper bound of $clb$ on ${\mathcal D}$ modulo the normal subgroup ${\mathcal G} \cong {\rm Diff}_c(M - N)_0$. Then, some upper bounds of $cl$ and $clb$ on ${\mathcal D}$ are obtained from those on ${\mathcal G}$. As a consequence, the group ${\mathcal D}$ is uniformly weakly simple and bounded when $\dim M \neq 2,4$. On the other hand, if ${\rm rank}\,A < m$, then the group ${\mathcal D}$ admits a surjective quasimorphism, so it is unbounded and not uniformly perfect. We examine the group $A$ in some explicit examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_11363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundedness of diffeomorphism groups of manifold pairs -- Circle case -- Fukui, Kazuhiko Yagasaki, Tatsuhiko Geometric Topology Group Theory 57R50, 57R52, 37C05 In this paper we study boundedness of conjugation invariant norms on diffeomorphism groups of manifold pairs. For the diffeomorphism group ${\mathcal D} \equiv {\rm Diff}(M,N)_0$ of a closed manifold pair $(M, N)$ with $\dim N \geq 1$, first we clarify the relation among the fragmentation norm, the conjugation generated norm, the commutator length $cl$ and the commutator length with support in balls $clb$ and show that ${\mathcal D}$ is weakly simple relative to a union of some normal subgroups of ${\mathcal D}$. For the boundedness of these norms, this paper focuses on the case where $N$ is a union of $m$ circles. In this case, the rotation angle on $N$ induces a quasimorphism $ν: {\rm Isot}(M, N)_0 \to {\Bbb R}^m$, which determines a subgroup $A$ of ${\Bbb Z}^m$ and a function $\widehatν : {\mathcal D} \to {\Bbb R}^m/A$. If ${\rm rank}\,A = m$, these data leads to an upper bound of $clb$ on ${\mathcal D}$ modulo the normal subgroup ${\mathcal G} \cong {\rm Diff}_c(M - N)_0$. Then, some upper bounds of $cl$ and $clb$ on ${\mathcal D}$ are obtained from those on ${\mathcal G}$. As a consequence, the group ${\mathcal D}$ is uniformly weakly simple and bounded when $\dim M \neq 2,4$. On the other hand, if ${\rm rank}\,A < m$, then the group ${\mathcal D}$ admits a surjective quasimorphism, so it is unbounded and not uniformly perfect. We examine the group $A$ in some explicit examples. |
| title | Boundedness of diffeomorphism groups of manifold pairs -- Circle case -- |
| topic | Geometric Topology Group Theory 57R50, 57R52, 37C05 |
| url | https://arxiv.org/abs/2501.11363 |