Commutator Length of Finitely Generated Linear Groups

Abstract

The commutator length “cl(𝐺)” of a group 𝐺 is the least natural number 𝑐 such that every element of the derived subgroup of 𝐺 is a product of 𝑐 commutators. We give an upper bound for cl(𝐺) when 𝐺 is a 𝑑-generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over 𝐂 that depends only on 𝑑 and the degree of linearity. For such a group 𝐺, we prove that cl(𝐺) is less than 𝑘(𝑘+1)/2+12𝑑3+𝑜(𝑑2), where 𝑘 is the minimum number of generators of (upper) triangular subgroup of 𝐺 and 𝑜(𝑑2) is a quadratic polynomial in 𝑑. Finally we show that if 𝐺 is a soluble-by-finite group of Prüffer rank 𝑟 then cl(𝐺)≤𝑟(𝑟+1)/2+12𝑟3+𝑜(𝑟2), where 𝑜(𝑟2) is a quadratic polynomial in 𝑟.