Nilpotent probability of compact groups

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 $G$.