  We study stable W-length in groups, especially for W equal to the n-fold
commutator gamma_n:=[x_1,[x_2, . . . [x_{n-1},x_n]] . . . ]. We prove that in
any perfect group, for any n at least 2 and any element g, the stable
commutator length of g is at least as big as 2^{2-n} times the stable
gamma_n-length of g. We also establish analogues of Bavard duality for words
gamma_n and for beta_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.
