Let G be a semisimple algebraic group over a local field K of characteristic p. If the order of the fundamental group of G is divisible by p, then no open subgroup of G(K) is topologically finitely generated. Otherwise, for any compact open subgroup C of G(K), with probability 1 two randomly chosen elements of C topologically generate an open subgroup.
Our main tool is a recent theorem by Richard Pink characterizing compact groups linear over a local field (Compact subgroups of linear algebraic groups J. Algebra 206 (1998) 438-504.) The idea is that such groups should not be too far from being open subgroups of the group of points of a (possibly smaller) algebraic group over a (possibly smaller) local field. To prove that two random elements generate the whole group, the idea is to show first that (with probability 1) they generate a Zariski-dense subgroup and then (again with probability 1) that they do not lie in the "same" group over a proper subfield of K. The latter is more difficult and depends on trace arguments.
This theorem can be regarded as a topological analogue of the Dixon conjecture for finite groups, which asserts that the probability that two randomly chosen elements of a finite simple group will generate that group tends to 1 as the order of the group grows without bound. The Dixon conjecture was settled by Bill Kantor, Alex Lubotzky, Martin Liebeck, and Aner Shalev.