I'm a student of mathematics and I need know about the status of the Milnor conjecture (if there are partial results or if someone solved that). The statement is:

A complete Riemannian manifold with non-negative Ricci curvature has a finitely generated fundamental group.

If someone can help me with references to papers or anything I would be grateful.


According to David Roberts comment and the following paper it is open for dimensions $n\geq 4$.

Pan, Jiayin, A proof of Milnor conjecture in dimension 3, J. Reine Angew. Math. 758, 253-260 (2020). ZBL1432.53053.

There is a nice survey by Shen and Sormani that can be found in author homepage:

Shen, Zhongmin; Sormani, Christina, The topology of open manifolds with nonnegative Ricci curvature, Commun. Math. Anal., Conference 1, 20-34 (2008). ZBL1167.53309.

And a few related and partial attacks to the conjecture:

Paeng, Seong-Hun, On the fundamental group of manifolds with almost nonnegative Ricci curvature, Proc. Am. Math. Soc. 131, No. 8, 2577-2583 (2003). ZBL1040.53042.

Xu, Senlin; Deng, Qintao, The fundamental group of open manifolds with nonnegative Ricci curvature, Acta Math. Sin., Chin. Ser. 49, No. 2, 353-356 (2006). ZBL1120.53021.

Which is stronger than

Sormani, Christina, Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups., J. Differ. Geom. 54, No. 3, 547-559 (2000). ZBL1035.53045.

  • $\begingroup$ Thank's a lot!!!. $\endgroup$ – Marcelo Miranda Aug 31 '20 at 17:48

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