On the ranks of homotopy groups of two-cones

Let X be a finite simply-connected CW-complex of dimension nX. If X is a two-cone (that is: X is the homotopy cofiber of a map between suspensions) such that π∗(X)⊗Q is infinite-dimensional, then there exist A,B>0 and α>1 such that for any positive integer k we have Akαk⩽∑j=k+2k+nXrankπj(X)⩽Bkαk. This formula is proved by establishing some analytic property of the Poincaré series ΩX(z)=∑n=0∞ dimHn(ΩX;Q)·zn. These inequalities hold also for some other classes of spaces.