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

Abstract

The following question is proposed in [4, Question 1.20]: Let $G$ be a compact group, and suppose that $${\mathcal{N}}_k(G)=\{(x1,\dots ,x_{k+1})\in G^{k+1}\Vert ;[x_1,\dots ,x_{k+1}]=1\}$$ 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 $k=2$.