Are cluster automorphism groups finitely generated?

TL;DR

This paper provides a sufficient condition for the finite generation of cluster automorphism groups and verifies this for all finite mutation and acyclic cluster algebras, simplifying their computation.

Contribution

It introduces a new approach using pseudo N-grading to determine finite generation of automorphism groups in cluster algebras.

Findings

Finite generation verified for all finite mutation cluster algebras.

Finite generation verified for all acyclic cluster algebras.

Method simplifies computation of group presentations in specific cases.

Abstract

This paper investigates the finite generation of cluster automorphism groups. By applying the pseudo $\mathbb{N}$-grading introduced in our previous work, we establish a sufficient condition for a cluster automorphism group to be finitely generated. As applications, we verify the finite generation of the automorphism groups for all finite mutation and acyclic cluster algebras. Furthermore, we illustrate through examples that our approach significantly simplifies the computation of presentations for these groups in certain cases.