International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 281734, 5 pages
doi:10.1155/2008/281734

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.