# On a Nielsen-Thurston classification theory for cluster modular groups

## TL;DR

This paper develops a classification theory for elements of cluster modular groups, analogous to Nielsen-Thurston classification for surface mapping class groups, based on fixed point properties in tropical compactifications.

## Contribution

It introduces a new classification framework for cluster modular group elements, extending Nielsen-Thurston theory to this algebraic setting.

## Key findings

- Elements are classified into three types based on fixed point properties.
- The classification parallels Nielsen-Thurston theory for surface mapping class groups.
- Provides a new perspective on the structure of 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.