Stable W-length

TL;DR

This paper investigates stable W-length in groups, focusing on the n-fold commutator, establishing bounds on stable commutator length, and exploring geometric properties of asymptotic cones of verbal subgroups.

Contribution

It provides new bounds relating stable commutator length to stable gamma_n-length and introduces analogues of Bavard duality for specific words.

Findings

Stable commutator length is bounded below by a factor times stable gamma_n-length.

Analogues of Bavard duality are established for gamma_n and beta_2.

Asymptotic cones of verbal subgroups contain subgroups that are normed vector spaces.

Abstract

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.