Are there any relationships between the growth rate of a (finitely generated) group $G$ and its Betti numbers $b_i(G)=b_i(BG)$? I'm particularly interested in upper and lower bounds for the total sum $\sum b_i(G)$ when $G$ grows exponentially. Let's also assume $G$ acts effectively and properly discontinuously on some smooth manifold, if that helps.
For a little background: a theorem by Milnor states that the fundamental group of a negatively curved, compact Riemannian manifold $M$ grows exponentially. On the other hand, a theorem by Gromov gives a linear upper bound for $\sum b_i(M)$ in terms of the volume of $M$, from which one can conclude the same bound for $\sum b_i(\pi_1(M))$ (for cohomology over $\mathbb{R}$).
This is somewhat in line with with my (very poor) intuition that the Betti numbers of a group will be bigger if it has more relations on its generators, which in turn tend to slow down its growth. But i didn't find any references relating these concepts.