On approximation of Lie groups by discrete subgroups

Abstract

A locally compact group G is said to be approximated by discrete subgroups (in the sense of Toyama) 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 Toyama 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 Toyama 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 Γ n containing Γ. The proof of the above result is by induction on the dimension of G, and gives an algorithm for inductively determining Γ n . The purpose of this paper is to give another proof in which we present an explicit formula for the sequence (Γ n ) n ≥ 0 in terms of Γ. Several applications are given.