Rigid generators of classical groups (Q1264317)

The rigidity criterion has been extensively used to realize finite groups as Galois groups over the rationals. All known realizations have been found by determining rigid triples, i.e. applying the rigidity criterion with \(r=3\). Now the author generalizes the concept of Belyi triples and introduces Thompson tuples which are defined as follows. A Thompson tuple in \(GL_n(K)\) is an \((n+1)\)-tuple \((\sigma_1,\dots, \sigma_{n+1})\) such that \(\sigma_1,\dots, \sigma_{n+1}\) generate an irreducible subgroup of \(GL_n(K)\), their product satisfies \(\sigma_1\cdots \sigma_{n+1}=1\), and \(\sigma_i\) has an eigenspace of dimension \(n-1\) for all \(i=1,\dots, n+1\). The study of Thompson tuples leads to new results on Belyi triples. In particular, the author gives an existence theorem on Belyi triples. As a consequence, \(O_n^\pm(2)\), \(\Omega_n^\pm(2)\) and \(Sp_n(2)\) are realizable as Galois groups over \(\mathbb{Q}\) (even these groups have GAL-realizations).