Compact groups with a set of positive Haar measure satisfying a nilpotent law

TL;DR

This paper investigates whether a compact group with a set of positive Haar measure satisfying a nilpotent law necessarily contains an open nilpotent subgroup, confirming this for the case when the nilpotency step is two.

Contribution

The paper provides a positive answer to a previously open question for the case of 2-step nilpotent laws in compact groups.

Findings

Confirmed the existence of an open 2-step nilpotent subgroup under the given conditions.

Extended known results from the case k=1 to k=2.

Contributed to understanding the structure of compact groups with positive measure nilpotent laws.

Abstract

The following question is proposed in [4, Question 1.20]: Let $G$ be a compact group, and suppose that $$\mathcal{N}_k(G) = \{(x1,\dots,x_{k+1}) \in G^{k+1} \;\|; [x_1,\dots, x_{k+1}] = 1\}$$ has positive Haar measure in $G^{k+1}$. Does $G$ have an open $k$-step nilpotent subgroup? The case $k = 1$ is already known. We positively answer it for $k = 2$.