Nilpotent probability of compact groups

TL;DR

This paper investigates the probability that randomly chosen elements in a compact group satisfy a nilpotency condition, establishing structural results about the group's open subgroups and connected components.

Contribution

It proves that positive nilpotent probability implies the existence of an open nilpotent subgroup of bounded class in profinite groups and characterizes the structure of connected components.

Findings

Positive nilpotent probability implies an open nilpotent subgroup in profinite groups.

The connected component of the group is abelian.

Existence of a closed normal nilpotent subgroup with specific properties.

Abstract

Let $k$ be any positive integer and $G$ a compact (Hausdorff) group. Let $\mf{np}_k(G)$ denote the probability that $k+1$ randomly chosen elements $x_1,\dots,x_{k+1}$ satisfy $[x_1,x_2,\dots,x_{k+1}]=1$. We study the following problem: If $\mf{np}_k(G)>0$ then, does there exist an open nilpotent subgroup of class at most $k$? The answer is positive for profinite groups and we give a new proof. We also prove that the connected component $G^0$ of $G$ is abelian and there exists a closed normal nilpotent subgroup $N$ of class at most $k$ such that $G^0N$ is open in $G$.