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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.

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up vote 2 down vote accepted

There are closed hyperbolic 3-manifolds with arbitrarily large algebraic entropy. However, the critical exponent will always be $2$. So there isn't a relation, although there could possibly be an inequality.

Addendum: Sorry, I should have given justification for the statement above. The point is that for any free group $F_k$, there is a closed 3-manifold $N$ such that there is a surjection $\pi_1(N)\to F_k$ (once one has an example for $k=2$, the rest follow by taking finite-index covers of $F_2$ isomorphic to $F_k$, and taking the induced cover of $N$). The growth of $F_k$ with respect to any generating set is at least the growth of $F_k$ with respect to the standard generators (which is $2k-1$), since among any generating set, there must be an independent set of size $k$. Thus, the growth of $\pi_1(N)$ is at least $2k-1$.

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