Suppose $M$ is a compact negatively pinched Riemannian manifold of dimension $n$. We normalize the metric such that $-1\le K\le -a^2$ for some $0<a\le 1$. Let $G$ be the fundamental group of $M$. We can define the algebraic entropy of $G$ by taking infimum of entropy of $G$ with respect to a generating set $S$. On the othere hand, we can also define the critical exponent of $G$.
Are these two quantities related to each other (via volume entropy of $M$, maybe?) How about noncompact case? A reference of this direction is needed.