Free actions of finite groups on the 3-dimensional nilmanifold

Abstract

We study free actions of finite groups on the standard 3-dimensional nilmanifold. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying normal nilpotent subgroups of all almost Bieberbach groups of finite index, up to affine conjugacy. We conclude that if a finite group acts freely on the standard 3-dimensional nilmanifold with the first homology Z 2 , then it is cyclic.