Random generation in semisimple algebraic groups over local fields.
Recall that a local field is a field which is complete with respect to a discrete valuation such that its field of residue classes is finite. The field is called non-Archimedean if the valuation satisfies the ultrametric inequality. Denote by \(K\) a local non-Archimedean field with characteristic \(p\geq 0\) and \(G\) be a semisimple algebraic group over \(K\). Let \(C\) be a compact open subgroup of \(G(K)\), where \(G(K)\) denotes the \(K\)-points of \(G\). Consider the Haar measure on \(C\). With respect to this measure we may look at the probability that two randomly chosen elements generate topologically an open subgroup of \(G(K)\). The main result of the paper says that this probability is \(1\) except the case where \(p\) divides the degree of the universal covering map on \(G\). In the latter case, \(C\) is not finitely generated.
scientific article; zbMATH DE number 2037085