Let \(n>0\). A polynomial diffeomorphism \(p\colon\mathbb{R}^n\to\mathbb{R}^n\) of \(\mathbb{R}^n\) is a bijective map such that both \(p\) and \(p^{-1}\) can be expressed as polynomials. The group of polynomial diffeomorphisms is denoted by \(\mathcal P(\mathbb{R}^n)\). An action \(\rho\colon G\to\mathcal P(\mathbb{R}^n)\) of a group \(G\) on \(\mathbb{R}^n\) via polynomial diffeomorphisms is called polynomial crystallographic if it is both properly discontinuous and cocompact.
In the article are proved the following two theorems: (I) Let \(N\) be a finitely generated torsion-free nilpotent group, \(\rho\colon N\to\mathcal P(\mathbb{R}^n)\) be a polynomial crystallographic action and \(\varphi\colon N\to\text{GL}(k,\mathbb{R})\) a unipotent \(N\)-module structure for \(\mathbb{R}^k\). Then the cochain map \[ \Omega^*_P(\mathbb{R}^n,\mathbb{R}^k)^{\rho(N),\varphi(N)}\to\operatorname{Hom}_{\mathbb{Z} G}(C_*(\mathbb{R}^n),\mathbb{R}^k) \] induces an isomorphism on cohomology, \[ H^*(\Omega^*_P(\mathbb{R}^n,\mathbb{R}^k)^{\rho(N),\varphi(N)})\cong H^*_\varphi(N,\mathbb{R}^k), \] where \(\Omega^*_P(\mathbb{R}^n,\mathbb{R}^k)^{\rho(N),\varphi(N)}\) is some subspace of all \(k\)-tuples of differential forms on \(\mathbb{R}^n\); (II) Suppose \(M\) is the maximal unipotent submodule of \(\mathbb{R}^k\), with an \(N\)-action given by \(\varphi_U\colon N\to\text{GL}(M)\). Then \(H^*_\varphi(N,\mathbb{R}^k)\cong H^*_{\varphi U}(N,M)\).
The paper is an extention/continuation of an earlier article of the authors [Trans. Am. Math. Soc. 359, No. 6, 2539-2558 (2007; Zbl 1123.20043)]. On the last 4 pages of the article under review are given applications of the above theorems to the calculation of some cohomology of virtually nilpotent and Abelian groups.