Finite extensions and unipotent shadows of affine crystallographic groups
Let \(\Gamma\) be a discrete and cocompact subgroup of \(\text{Aff}(\mathbb{R}^n)\), where \(\mathbb{R}^n\) is \(n\)-dimensional Euclidean space. \(\Gamma\) is called an affine crystallographic group. The paper under review considers the question which abstract virtually polycyclic groups are isomorphic to an affine crystallographic group. A homomorphism \(\rho\colon\Gamma\to A\subset\text{Aff}(\mathbb{R}^n)\) is crystallographic if \(\rho\) is injective and the image \(\rho(\Gamma)\) is a crystallographic subgroup. Let us denote by \(\Hom(\Gamma,A)=\{\rho\colon\Gamma\to A\mid\rho\) crystallographic\}. The group \(A\) acts by conjugation on the above space. Put \(D(\Gamma,A)=\Hom(\Gamma,A)/A\). Moreover, let \(\Delta\) be a finite extension of \(\Gamma\). It is effective, if the homomorphism \(\Delta/\Gamma\to\text{Out}(\Gamma)\) is injective. The main result is the following: Let \(D(\Gamma,A)^{\Delta/\Gamma}\) denote the set of fixed points of the action of \(\Delta/\Gamma\). Let \(\Delta\) be a finite effective extension group of \(\Gamma\). Then the restriction map from \(D(\Delta,A)\) to \(D(\Gamma,A)^{\Delta/\Gamma}\) is a continuous bijection. As corollary the author obtains: \(\Delta\) is isomorphic to an affine crystallographic group of type \(A\) if and only if the induced action of \(\Delta/\Gamma\) on \(D(\Gamma,A)\) has a fixed point. The proofs are not given.
scientific article; zbMATH DE number 1850800