Nilpotent probability of compact groups

Let k be any positive integer and G a compact (Hausdorff) group. Let npk(G) denote the probability that k+1 randomly chosen elements x1, . . . , xk +1 satisfy [x1, x2, . . . , xk+1] = 1. We study the following problem: If npk(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 G0 of G is abelian and there exists a closed normal nilpotent subgroup N of class at most k such that G0N is open in G. In particular, a connected compact group G with npk(G) > 0 is abelian.(c) 2023 Elsevier Inc. All rights reserved.