Mixed commutator lengths, wreath products and general ranks

2022/03/08 by Kawasaki, Morimichi, Kimura, Mitsuaki, Maruyama, Shuhei +2
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2203.04048

Abstract

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)=(ℤ\wr Γ, \bigoplus_Γℤ). 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.

Related

• Bavard's duality theorem for mixed commutator length
• Computing commutator length is hard
• The degree of commutativity and lamplighter groups
• The degree of commutativity of wreath products with infinite cyclic top group
• On the group of alternating colored permutations