Several new results on quasicrystallographic groups in Novikov's sense (Q757598)

A finitely generated abelian subgroup \(T\subset R^ k\), which generates \(R^ k\) as a linear space is called a quasilattice in R. A subgroup G of the group \(E_ k\) of all isometries of k-dimensional Euclidean space is called k-dimensional quasi-crystallographic group, iff its intersection with the subgroup \(R^ k\subset E_ k\) of all translations is some quasilattice \(T\subset R^ k.\) These groups can contain infinite-order rotations even in dimension two. It is proved that in dimension three these groups can be even infinitely generated. The problem of classification (up to group-theoretical isomorphism) of quasi-crystallographic groups with finite group of linear parts is reduced to problems of representation theory of finite groups. In the majority of particular cases in dimension three these algebraic problems are solved.