Fixed point theorem for cluster modular groups

We prove that any finite subgroup $G\subset {\mathrm{\Gamma }}_{\mathit{s}}$ of the cluster modular group has fixed points in the cluster manifolds ${\mathcal{A}}_{\mathit{s}}({\mathbb{R}}_{>0})$ and ${\mathcal{X}}_{\mathit{s}}({\mathbb{R}}_{>0})$ 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 ${\mathrm{\Gamma }}_{\mathit{s}}$ 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.