Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|cite|improve this question

1 Answer 1

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 $2k-1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.