Coarse geometry of stable mixed commutator length I: duality and functional analysis on chains
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917511209943040 |
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| author | Kawasaki, Morimichi Kimura, Mitsuaki Maruyama, Shuhei Matsushita, Takahiro Mimura, Masato |
| author_facet | Kawasaki, Morimichi Kimura, Mitsuaki Maruyama, Shuhei Matsushita, Takahiro Mimura, Masato |
| contents | Let $G$ be a group and $N$ its normal subgroup. On the mixed commutator subgroup $[G,N]$, the mixed stable commutator length $\mathrm{scl}_{G,N}$ and the restriction of the ordinary stable commutator length $\mathrm{scl}_{G}$ are defined. We characterize when they are bi-Lipschitz equivalent by the vanishing of a certain $\mathbb{R}$-linear space $\mathrm{W}(G,N)$ related to invariant quasimorphisms. For the proof, we obtain a refined version of the generalized mixed Bavard duality theorem, and perform functional analysis on the completion of a certain space of $1$-chains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19843 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Coarse geometry of stable mixed commutator length I: duality and functional analysis on chains Kawasaki, Morimichi Kimura, Mitsuaki Maruyama, Shuhei Matsushita, Takahiro Mimura, Masato Group Theory Functional Analysis Metric Geometry Primary 51F30, Secondary 20F65, 46A20, 20F69 Let $G$ be a group and $N$ its normal subgroup. On the mixed commutator subgroup $[G,N]$, the mixed stable commutator length $\mathrm{scl}_{G,N}$ and the restriction of the ordinary stable commutator length $\mathrm{scl}_{G}$ are defined. We characterize when they are bi-Lipschitz equivalent by the vanishing of a certain $\mathbb{R}$-linear space $\mathrm{W}(G,N)$ related to invariant quasimorphisms. For the proof, we obtain a refined version of the generalized mixed Bavard duality theorem, and perform functional analysis on the completion of a certain space of $1$-chains. |
| title | Coarse geometry of stable mixed commutator length I: duality and functional analysis on chains |
| topic | Group Theory Functional Analysis Metric Geometry Primary 51F30, Secondary 20F65, 46A20, 20F69 |
| url | https://arxiv.org/abs/2605.19843 |