ON THE GROWTH-FUNCTION OF DIRECT DECOMPOSITIONS ASSOCIATED WITH HOMOLOGY OF FREE ABELIANIZED EXTENSIONS

Let G be an arbitrary group given by a free presentation G = F/N. We d eal with the homology group H(n)(PHI, Z) where PHI = F/[N, N]. It is k nown that if G has no p-torsion then the p-component of H(n)(PHI, Z) ( p odd) has a natural direct decomposition of the form +k H(nk)(G, Z/pZ ). The number of direct summands is a function of dimension n. We prov e that this function grows faster than n(s) for any s but slower than a(n) for any a > 1. Indeed a more precise asymptotic estimate is given . We also study maximal multiplicity of the group H(G, Z/pZ) in the a bove decomposition and get information on decomposition of two other p eriodic groups related to H(n)(PHI, Z).