Stable W-length

We study stable W-length in groups, especially for W equal to the n-fold commutator γ_n := [x_1, [x_2,...[x_(n−1), x_n]]....We prove that in any perfect group, for any n ≥ 2 and any element g, the stable commutator length of g is at least as big as 2^(2−n) times the stable γ_n-length of g. We also establish analogues of Bavard duality for words γn and for β_2 := [[x, y], [z,w]]. Our proofs make use of geometric properties of the asymptotic cones of verbal subgroups with respect to bi-invariant metrics. In particular, we show that for suitable W, these asymptotic cones contain certain subgroups that are normed vector spaces.