Conjugacy classes and growth conditions

In this paper we show using a purely combinatorial argument that a finitely generated infinite group such that f(E) (n) less than or equal to an(s), where a is a constant, admits for every epsilon a sequence {g(i,epsilon)} of non-unit elements whose centralizer contains more than i(1/2-epsilon) elements of length less than i. Of course, the interest of this result is in the fact that it excludes the possibility that the group is a pure torsion group, since otherwise the existence of the sequence {g(i,epsilon)} is obvious. As an application of this result, we show that, in the case where r < 3/2, there exists an element whose centralizer has finite index in G. (C) 2003 Elsevier Inc. All rights reserved.