The non-existence of a non-abelian nilpotent Lie group with one discrete uniform subgroup up to automorphism

摘要

Let G be a connected, simply connected nilpotent Lie group and S(G) the space of the discrete uniform subgroups of G. In this paper, we give some necessary and sufficient conditions for S(G) under which the group G is abelian. In particular, we prove the non-existence of non-abelian nilpotent Lie groups with one discrete uniform subgroup up to automorphism.