Exponential growth of Lie algebras of finite global dimension

Let $X$ be a finite simply connected CW complex of dimension $n$. The loop space homology $H\mathrm{\_}\ast (\mathrm{\Omega }X;\mathbb{Q})$ is the universal enveloping algebra of a graded Lie algebra $L\mathrm{\_}X$ isomorphic with $ pi_{*-1} (X)\otimes \mathbb Q$\mathrm{.}Let$Q_X \subset L_X$beaminimalgeneratingsubspace,andset$\alpha = \limsup_i \frac{\log{\scriptsize rk} \pi_i(X)}{i}$\mathrm{.}Theorem:If${dim} L_X = \infty$and$\limsup ({dim} (Q_X)_k)^{1/k} < \limsup ({dim} (L_X)_k)^{1/k}$then$$\sum_{i=1}^{n-1} {rk} \pi_{k+i}(X) = e^{(\alpha + \epsilon_k)k} \hspace{1cm} {where} \epsilon_k \to 0 {as} k\to \infty.$$Inparticular$\displaystyle\sum_{i=1}^{n-1} {rk} \pi_{k+i}(X)$growsexponentiallyin$k$.