Quasi-monomial actions and some 4-dimensional rationality problems

Let $G$ be a finite group acting on $k(x_1,...,x_n)$, the rational function field of $n$ variables over a field $k$. The action is called a purely monomial action if $\sigma...x_j=\prod_{1\le i\le n} x_i^{a_{ij}}$ for all $\sigma \in G$, for $1\le j\le n$ where $(a_{ij})_{1\le i,j\le n} \in GL_n(\bm{Z})$. The main question is that, under what situations, the fixed field $k(x_1,...,x_n)^G$ is rational (= purely transcendental) over $k$. This rationality problem has been studied by Hajja, Kang, Hoshi, Rikuna when $n\le 3$. In this paper we will prove that $k(x_1,x_2,x_3,x_4)^G$ is rational over $k$ provided that the purely monomial action is decomposable. To prove this result, we introduce a new notion, the quasi-monomial action, which is a generalization of previous notions of multiplicative group actions. Moreover, we determine the rationality problem of purely quasi-monomial actions of $K(x, y)^G$ over $k$ where $k= K^G$.