On the Cohen-Macaulay property of multiplicative invariants

We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group $\mathcal{G}$. By definition, these are $\mathcal{G}$-actions on Laurent polynomial algebras $\Bbbk[x_1^{\pm 1},\dots,x_n^{\pm 1}]$that stabilize the multiplicative group consisting of all monomials in the variables $x_i$. For the most part, we concentrate on the case where the base ring $\Bbbk$ is $\mathbb{Z}$. Our main result states that if $\mathcal{G}$ acts non-trivially and the invariant ring $\mathbb{Z} [x_1^{\pm 1},\dots,x_n^{\pm 1}]^\mathcal{G}$ is Cohen-Macaulay, then the abelianized isotropy groups ${\mathcal{G}}_m^{{ab}}$ of all monomials $m$ are generated by the bireflections in $\mathcal{G}_m$ and at least one ${\mathcal{G}}_m^{{ab}}$ is non-trivial. As an application, we prove the multiplicative version of Kemper's $3$-copies conjecture.