Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists

TL;DR

This paper develops a method to compute presentations of saturated cluster modular groups and demonstrates that certain groups are generated by cluster Dehn twists, providing finite presentations for specific mutation types.

Contribution

Introduces a new method for computing presentations of saturated cluster modular groups and shows these groups are generated by cluster Dehn twists for several mutation types.

Findings

Finite presentations obtained for groups of types X_6 and X_7.

Cluster modular groups of specified types are virtually generated by cluster Dehn twists.

Verification of generation by Dehn twists for multiple mutation types.

Abstract

We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type $X_6$ and $X_7$. We verify that the cluster modular groups of finite mutation type $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E}_8$, $G_2^{(*,*)}$, $X_6$ and $X_7$ are virtually generated by cluster Dehn twists.