Mixed commutator lengths, wreath products and general ranks

抄録

In the present paper, for a pair (G,N) of a group G and its normal subgroup N, we consider the mixed commutator length cl_G,N on the mixed commutator subgroup [G,N]. We focus on the setting of wreath products: (G,N) = . Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. As a byproduct, when an abelian group Γ is not locally cyclic, the ordinary commutator length cl_G does not coincide with cl_G,N on [G,N] for the above pair. On the other hand, we prove that if Γ is locally cyclic, then for every pair (G,N) such that 1 → N → G → Γ → 1 is exact, cl_G and cl_G,N coincide on [G,N]. We also study the case of permutational wreath products when the group Γ belongs to a certain class related to surface groups.