On systems of generators of arithmetic subgroups of higher rank groups
We show that if \(\Gamma\) is a noncocompact irreducible discrete subgroup with finite covolume, of a real semisimple group \(G\) of real rank at least two and \(N_ 1\) and \(N_ 2\) are two maximal unipotent subgroups of \(\Gamma\) such that \(N_ 1 \cap N_ 2 = \{1\}\), then the subgroup generated by \(N_ 1\) and \(N_ 2\) inside \(\Gamma\) has finite index in \(\Gamma\). The proof consists in constructing an extension of the congruence completion of \(G(\mathbb{Q})\) (where \(\Gamma\) is commensurate to \(G(\mathbb{Z})\)) whose open subgroups are groups generated by \(N_ 1\) and \(N_ 2\) (for various \(\Gamma \subset G(\mathbb{Z})\)). We then show that this extension is central and splits over \(G(\mathbb{Q})\). Then, by using well known results on central extensions of \(G( \widehat {\mathbb{Z}})\) we show that the extension is finite which then proves the result quoted above.
noncocompact irreducible discrete subgroup
scientific article; zbMATH DE number 759238