Three-dimensional purely quasi-monomial actions

Let $G$ be a finite subgroup of $\mathrm{Aut}_k(K(x_1, \ldots, x_n))$ where $K/k$ is a finite field extension and $K(x_1,\ldots,x_n)$ is the rational function field with $n$ variables over $K$. The action of $G$ on $K(x_1, \ldots, x_n)$ is called quasi-monomial if it satisfies the following three conditions (i) $\sigma(K)\subset K$ for any $\sigma\in G$; (ii) $K^G=k$ where $K^G$ is the fixed field under the action of $G$; (iii) for any $\sigma\in G$ and $1 \leq j \leq n$, $\sigma(x_j)=c_j(\sigma)\prod_{i=1}^n x_i^{a_{ij}}$ where $c_j(\sigma)\in K^\times$ and $[a_{i,j}]_{1\le i,j \le n} \in GL_n(\mathbb{Z})$. A quasi-monomial action is called purely quasi-monomial if $c_j(\sigma)=1$ for any $\sigma \in G$, any $1\le j\le n$. When $k=K$, a quasi-monomial action is called monomial. The main problem is that, under what situations, $K(x_1,\ldots,x_n)^G$ is rational (= purely transcendental) over $k$. For $n=1$, the rationality problem was solved by Hoshi, Kang and Kitayama. For $n=2$, the problem was solved by Hajja when the action is monomial, by Voskresenskii when the action is faithful on $K$ and purely quasi-monomial, which is equivalent to the rationality problem of $n$-dimensional algebraic $k$-tori which split over $K$, and by Hoshi, Kang and Kitayama when the action is purely quasi-monomial. For $n=3$, the problem was solved by Hajja, Kang, Hoshi and Rikuna when the action is purely monomial, by Hoshi, Kitayama and Yamasaki when the action is monomial except for one case and by Kunyavskii when the action is faithful on $K$ and purely quasi-monomial. In this paper, we determine the rationality when $n=3$ and the action is purely quasi-monomial except for few cases. As an application, we will show the rationality of some $5$-dimensional purely monomial actions which are decomposable.