Polynomial crystallographic actions on the plane (Q1851072)

Let \({\mathbf P}(\mathbb{R}^n)\) be the group of polynomial diffeomorphisms of \(\mathbb{R}^n\). The elements of \({\mathbf P}(\mathbb{R}^n)\) are those bijections \(\mu\colon\mathbb{R}^n\to\mathbb{R}^n\), such that both \(\mu\) and \(\mu^{-1}\) are expressed via polynomials. A polynomial action of a group \(\Gamma\) on \(\mathbb{R}^n\) is a morphism \(\rho\colon\Gamma\to{\mathbf P}(\mathbb{R}^n)\). Such an action or the corresponding subgroup \(\rho(\Gamma)\) is said to be of bounded degree if the degrees of all elements of \(\rho(\Gamma)\) are bounded above by some constant. A polynomial action is said to be crystallographic if the action of \(\Gamma\) on \(\mathbb{R}^n\) is properly discontinuous and the quotient \(\Gamma\backslash\mathbb{R}^n\) is compact. The subgroup \(\rho(\Gamma)\) is then called a polynomial crystallographic group. In this paper the author studies polynomial crystallographic actions on the plane. The author proves that any such action of a polycyclic-by-finite group is of bounded degree and conversely that any polynomial crystallographic action of bounded degree comes from a polycyclic-by-finite group. This last result is a generalization of the well known Auslander conjecture.