Generations for arithmetic groups. (Q2572571)

It is proved in the paper under review that any non-uniform irreducible lattice in a higher rank real semi-simple Lie group has a subgroup of finite index which is generated by just three elements. The higher rank assumption is used to ensure that if \(U^+\) and \(U^-\) are unipotent radicals of some maximal parabolic \(\mathbb{Q}\)-subgroups of the semi-simple algebraic group \(G\) and \(M\) is their intersection, then there are ``generic'' elements in \(U^\pm(\mathbb{Z})\) and a ``generic'' semi-simple element in \(M(\mathbb{Z})\) whose powers generate subgroup \(\Gamma\) of \(G(\mathbb{Z})\). The genericity assumption implies for most groups \(G\) that \(\Gamma\) intersects \(U^\pm(\mathbb{Z})\) in subgroups of finite index. It follows then that \(\Gamma\) is of finite index in \(G(\mathbb{Z})\). The proof that \(\Gamma\) intersects \(U^\pm(\mathbb{Z})\) in a lattice for most groups is reduced to the existence of a torus in the Zariski closure of \(M(\mathbb{Z})\) whose eigenspaces on the Lie algebra of \(G\) are one dimensional. The existence of such a torus is proved by a case-by-case check using the Tits diagrams of simple algebraic groups over number fields.