Compact groups with probabilistically central monothetic subgroups

Abstract

If $K$ is a closed subgroup of a compact group $G$, the probability that randomly chosen pair of elements from $K$ and $G$ commute is denoted by $Pr(K,G)$. Say that a subgroup $K\le G$ is $\epsilon$-central in $G$ if $Pr(⟨g⟩,G)\ge \epsilon$ for any $g$ in $K$. Here $⟨g⟩$ denotes the monothetic subgroup generated by $g\in G$. Our main result is that if $K$ is $\epsilon$-central in $G$, then there is an $\epsilon$-bounded number $e$ and a normal subgroup $T\le G$ such that the index $[G:T]$ and the order of the commutator subgroup $[K^e,T]$ both are finite and $\epsilon$-bounded. In particular, if $G$ is a compact group for which there is $\epsilon >0$ such that $Pr(⟨g⟩,G)\ge \epsilon$ for any $g\in G$, then there is an $\epsilon$-bounded number $e$ and a normal subgroup $T$ such that the index $[G:T]$ and the order of $[G^e,T]$ both are finite and $\epsilon$-bounded.

Changes following referee's suggestions. Final version, accepted in Israel J. of Math