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Is there a constant $C(n,D)$ such that for any closed Riemannian manifold $M$ with $Ric \ge -(n - 1)$ and $\mathrm{diam} \le D$, the fundamental group $\pi_1(M)$ is generated by at most $C(n,D)$ elements? This statement holds when $\sec \ge -1$.

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It might follow from Kapovitch--Wilking paper, but better ask them directly. (BTW you could formulate the question better). –  Anton Petrunin Jun 24 '13 at 15:24
Is there a constant $C(n,D)$ so that for any compact manifold $M$ with ... –  Ben McKay Jun 24 '13 at 16:38
I edited the question so as to make it clearer. I hope I did not mess up any of the hypotheses. To be safe, I also added the assumption that the manifold is closed. Feel free to revert or change my edits. –  Ricardo Andrade Jun 24 '13 at 21:59
Anton is right. It is Theorem 3 in Kapovitch-Wilking's paper. –  J. GE Jun 27 '13 at 20:38

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