Compact groups with high commuting probability of monothetic subgroups

Abstract

If H is a subgroup of a compact group G, the probability that a random element of H commutes with a random element of G is denoted by Pr(H,G). Let 〈g〉 stand for the monothetic subgroup generated by an element g∈G and let K be a subgroup of G. We prove that Pr(〈x〉,G)>0 for any x∈K if and only if G has an open normal subgroup T such that K/CK(T) is torsion. In particular, Pr(〈x〉,G)>0 for any x∈G if and only if G is virtually central-by-torsion, that is, there is an open normal subgroup T such that G/Z(T) is torsion. We also deduce a number of corollaries of this result.