Compact groups with probabilistically central monothetic subgroups

TL;DR

This paper investigates the structure of compact groups with subgroups that are probabilistically central, establishing bounds and the existence of normal subgroups with finite index and bounded commutator properties.

Contribution

It introduces the concept of $ ext{epsilon}$-central subgroups and proves structural results linking probabilistic centrality to finite index and bounded commutator subgroups in compact groups.

Findings

Existence of a normal subgroup with finite index and bounded commutator order.

Boundedness of the number $e$ related to probabilistic centrality.

Structural characterization of groups with uniformly positive commuting probability.

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 $K\leq G$ is $\epsilon$-central in $G$ if $Pr(\langle g \rangle,G)\geq \epsilon$ for any $g$ in $K$. Here $\langle g \rangle$ 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 $T\leq G$ 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 $Pr(\langle g \rangle,G)\geq \epsilon$ 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.