Conjugacy classes and growth conditions

In this paper we show using a purely combinatorial argument that a finitely generated infinite group such that fE(n) ans, where a is a constant, admits for every ε a sequence {gi,ε} of non-unit elements whose centralizer contains more than i1/2−ε 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 {gi,ε} 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.