Pseudo grading on cluster automorphism group with application to cluster algebras of rank $3$

Abstract: We introduce a pseudo $\mathbb{N}$-grading on the cluster auotmorphism group $\operatorname{Aut}(\mathcal{A})$ with respect to an initial seed of $\mathcal{A}$, which consists of a family of subsets ${G_i}{i\in \mathbb{N}}$ of $\operatorname{Aut}(\mathcal{A})$ such that $\operatorname{Aut}(\mathcal{A})=\bigcup{i\in \mathbb{N}}G_i$ and $G_k\cdot G_l\subset \bigcup_{i=0}^{k+l}G_i$. We prove that $\operatorname{Aut}(\mathcal{A})$ is generated by $G_0\cup G_1$, leading to an elementary approach for calculating cluster automorphism groups of certain cluster algebras. As an application, we completely determined the cluster automorphism groups of cluster algebras of rank $3$ with indecomposable exchange matrices.