Let N be a connected and simply connected nilpotent Lie group, Lambda a lattice in N, and Lambda N the corresponding nilmanifold. We characterize the countable subgroups of the group Aff (Lambda N) of affine transformations of Lambda N whose action on L-2 (Lambda N) has a spectral gap: these are the groups H for which there exists no proper H-invariant subtorus S of the maximal torus factor T of Lambda N such that the projection of H on Aut(T/S) is a virtually abelian group.
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