Commutator Length of Finitely Generated Linear Groups

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.