Stable W-length

Danny Calegari; Dongping Zhuang

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arXiv · 1008.2219

Stable W-length

Danny Calegari·Dongping Zhuang

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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.

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BibTeX RIS

Danny Calegari, Dongping Zhuang. 2011-03-28. Stable W-length. https://arxiv.org/abs/1008.2219

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