Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 025, 22 pages      arXiv:1711.07785      https://doi.org/10.3842/SIGMA.2020.025

Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists

Tsukasa Ishibashi
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

Received January 01, 2020, in final form March 27, 2020; Published online April 07, 2020

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.

Key words: cluster algebras; cluster modular groups; mapping class groups; quivers of finite mutation type.

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