Fixed point theorem for cluster modular groups

Abstract

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

25 pages, 2 figures. v2: Appendix A added