A note on subgroups of cofinite volume
Let \(G\) be a Lie group. The author studies the question whether a closed cofinite volume subgroup of \(G\) contains lattices. First, he notes that, for positive results, one should look at groups which are the real points of a linear algebraic group defined over \(\mathbb{Q}\). In Section 1, he discusses the real points \(G\) of a linear algebraic \(\mathbb{Q}\)-group which is unipotent and shows a necessary assumption for a positive result. In Section 2, he considers semisimple groups without compact factors and proves tha, if \(G\) is a connected semisimple group without compact factors and \(G\) is represented as the real points of a linear algebraic group defined over \(\mathbb{Q}\), then any closed cofinite volume subgroup \(H\) of \(G\) contains a lattice. Next, he shows that, if \(G\) is any semisimple group represented as the real points of a linear algebraic group defined over \(\mathbb{Q}\), then a closed cofinite volume subgroup \(H\) of \(G\) contains some lattice. In the last section, he turns to a mixed group, that is, a group with a unipotent radical and a reductive part. By the Mostow decomposition theorem, \(G=G_{u} \times R\) (semidirect product), where \(G_{u}\) is a radical and \(R\) is a maximal reductive of \(G\). He gives an example showing that the reductive part distorts the Haar measure on the unipotent radical and obtains the following result.NEWLINENEWLINELet \(G=G_{u}\times R\) (semidirect product) be a connected component of the real points of an algebraic \(\mathbb{Q}\)-group, where \(G_{u}\) is abelian and \(R\) is semisimple. If \(H\) is a closed cofinite volume subgroup of G, then \(H\) contains a lattice of \(G\). If \(G\) is reductive and the connected component of the real points of an algebraic \(\mathbb{Q}\)-group, then the same conclusion holds.
scientific article; zbMATH DE number 6199278