Fixed point theorem for cluster modular groups
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914468830642176 |
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| author | Ishibashi, Tsukasa |
| author_facet | Ishibashi, Tsukasa |
| contents | We prove that any finite subgroup $G \subset Γ_{\boldsymbol{s}}$ of the cluster modular group has fixed points in the cluster manifolds $\mathcal{A}_{\boldsymbol{s}}(\mathbb{R}_{>0})$ and $\mathcal{X}_{\boldsymbol{s}}(\mathbb{R}_{>0})$ under a certain condition. This generalizes Kerckhoff's Nielsen realization theorem [Ker83] for the mapping class group action on the Teichmüller space. The condition holds whenever $Γ_{\boldsymbol{s}}$ admits a cluster DT transformation, and it can be also verified for all finite mutation types except for $X_7$. Our proof closely follows Kerckhoff's argument, based on the convexity of log-cluster variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14338 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fixed point theorem for cluster modular groups Ishibashi, Tsukasa Geometric Topology Combinatorics Group Theory We prove that any finite subgroup $G \subset Γ_{\boldsymbol{s}}$ of the cluster modular group has fixed points in the cluster manifolds $\mathcal{A}_{\boldsymbol{s}}(\mathbb{R}_{>0})$ and $\mathcal{X}_{\boldsymbol{s}}(\mathbb{R}_{>0})$ under a certain condition. This generalizes Kerckhoff's Nielsen realization theorem [Ker83] for the mapping class group action on the Teichmüller space. The condition holds whenever $Γ_{\boldsymbol{s}}$ admits a cluster DT transformation, and it can be also verified for all finite mutation types except for $X_7$. Our proof closely follows Kerckhoff's argument, based on the convexity of log-cluster variables. |
| title | Fixed point theorem for cluster modular groups |
| topic | Geometric Topology Combinatorics Group Theory |
| url | https://arxiv.org/abs/2603.14338 |