Compact groups with high commuting probability of monothetic subgroups

TL;DR

This paper characterizes compact groups where the probability that a generated monothetic subgroup commutes with the whole group is positive, linking it to the existence of certain open normal subgroups and torsion properties.

Contribution

It establishes a precise criterion for when monothetic subgroups have positive commuting probability in compact groups, connecting it to open normal subgroups and torsion conditions.

Findings

Positive commuting probability implies existence of open normal subgroup with torsion quotient.

G is virtually central-by-torsion if all monothetic subgroups have positive probability.

Several corollaries derived from the main characterization.

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 $\langle g\rangle$ stand for the monothetic subgroup generated by an element $g\in G$ and let $K$ be a subgroup of $G$. We prove that $Pr(\langle x\rangle,G)>0$ for any $x\in K$ if and only if $G$ has an open normal subgroup $T$ such that $K/C_K(T)$ is torsion. In particular, $Pr(\langle x\rangle,G)>0$ for any $x\in G$ 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.