On approximation of Lie groups by discrete subgroups

A locally compact group G is said to be approximated by discrete subgroups (in the sense of Tôyama) if there is a sequence of discrete subgroups of G that converges to G in the Chabauty topology (or equivalently, in the Vietoris topology). The notion of approximation of Lie groups by discrete subgroups was introduced by Tôyama in Kodai Math. Sem. Rep.1 (1949) 36–37 and investigated in detail by Kuranishi in Nagoya Math. J.2 (1951) 63–71. It is known as a theorem of Tôyama that any connected Lie group approximated by discrete subgroups is nilpotent. The converse, in general, does not hold. For example, a connected simply connected nilpotent Lie group is approximated by discrete subgroups if and only if G has a rational structure. On the other hand, if Γ is a discrete uniform subgroup of a connected, simply connected nilpotent Lie group G then G is approximated by discrete subgroups Γn containing Γ. The proof of the above result is by induction on the dimension of G, and gives an algorithm for inductively determining Γn. The purpose of this paper is to give another proof in which we present an explicit formula for the sequence (Γn)n ≥ 0 in terms of Γ. Several applications are given.