When are radicals of Lie groups lattice-hereditary? (Q2353157)

The purpose of this note is to describe the situation about one fundamental result in the general study of lattices in Lie groups. A lattice in a connected Lie group \(G\) is a discrete subgroup \(\Gamma\), for which \(G/\Gamma\) has a finite measure (induced by Haar measure on \(G\)). Let \(R\) be the solvable radical in \(G\). The main question in this paper is when the intersection \(R \cap \Gamma\) is a lattice in \(R\). The answer to this question has a long and rather dramatic history. The author clarifies the situation and gives a complete proof with commentaries for the following statement. Let \(G\) be a connected Lie group whose semisimple part \(S\) has no compact factor acting trivially on the radical \(R\) of \(G\) and let \(\Gamma\) be some lattice in \(G\). Then for the nilradical \(N\) of \(G\) the intersection \(N \cap \Gamma\) is a lattice in \(N\). If no compact factor of \(S\) acts trivially on \(R/N\), then \(R \cap \Gamma\) is also a lattice in \(R\). Here the condition ``no compact factor of \(S\) acts trivially on \(R/N\)'' in fact means that there are no compact factors in \(S\) at all.