Classification of free actions of finite groups on the 3-torus

Abstract

We classify free actions of finite groups on the 3-torus, up to topological conjugacy. By works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal free Abelian subgroups of affine Bieberbach groups with finite quotients, up to affine conjugacy. A complete classification of all such groups is obtained by utilizing Lee–Shin–Yokura's method [Topology Appl. 53 (1993) 153–175]. Thereby this gives a complete classification of free actions of finite groups on the 3-torus, up to topological conjugacy.