Quasicrystallographic groups in the sense of Novikov (Q757597)

A finitely generated abelian subgroup \(T\subset R^ k\), which generates \(R^ k\) as a linear space is called a quasilattice in \(R^ k\). 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 the two-dimensional case. It is proved that a two-dimensional quasi-crystallographic group is always finitely generated. The problem of classification (up to group- theoretical isomorphism) of two-dimensional quasi-crystallographic groups is reduced to problems of algebraic number theory.