Free actions of finite Abelian groups on the 3-torus

Abstract

We classify free actions of finite Abelian groups on the 3-torus, up to topological conjugacy. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal Abelian subgroups of Bieberbach groups of finite index, up to affine conjugacy. All such actions are completely classified, see Theorems 2.1, 3.4, 4.2, 5.1, 6.1 and 7.1.