Fixed point theorem for cluster modular groups

Tsukasa Ishibashi

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arXiv · 2603.14338

Fixed point theorem for cluster modular groups

Tsukasa Ishibashi

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Abstract

We prove that any finite subgroup $G \subset \Gamma_{{\boldsymbol{s}}}$ 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 Teichm\"uller space. The condition holds whenever $\Gamma_{\boldsymbol{s}}$ 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.

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BibTeX RIS

Tsukasa Ishibashi. 2026-03-15. Fixed point theorem for cluster modular groups. https://arxiv.org/abs/2603.14338

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