Fixed point theorem for cluster modular groups

TL;DR

This paper proves that finite subgroups of cluster modular groups have fixed points in associated cluster manifolds under certain conditions, extending a classical theorem in Teichmüller theory.

Contribution

It generalizes Kerckhoff's Nielsen realization theorem to cluster modular groups, establishing fixed points under specific conditions related to cluster DT transformations.

Findings

Finite subgroups have fixed points in cluster manifolds under certain conditions.

The condition holds for all finite mutation types except X7.

The proof adapts Kerckhoff's convexity-based argument to the cluster setting.

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.