On a Nielsen-Thurston classification theory for cluster modular groups

Abstract

We classify elements of a cluster modular group into three types. We characterize them in terms of fixed point property of the action on the tropical compactifications associated with the corresponding cluster ensemble. The characterization gives an analogue of the Nielsen-Thurston classification theory on the mapping class group of a surface.

arXiv comment33 pages, 9 figures. To appear in Annales de l'Institut Fourier