Nonnegative Ricci curvature, stability at infinity and finite generation of fundamental groups

Pan, Jiayin¹

¹ Department of Mathematics, University of California, Santa Barbara, Santa Barbara, CA, United States

Geometry & topology, Tome 23 (2019) no. 6, pp. 3203-3231.

Résumé

We study the fundamental group of an open $n$–manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone whose maximal Euclidean factor has dimension $k$, then ${\pi }_1\left(M\right)$ is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and a unique tangent cone at infinity.

Détail

DOI : 10.2140/gt.2019.23.3203

Classification : 53C20, 53C23, 53C21, 57S30
Keywords: Ricci curvature, fundamental groups

Bibliographie

[1] M T Anderson, On the topology of complete manifolds of nonnegative Ricci curvature, Topology 29 (1990) 41 | DOI

[2] J Cheeger, T H Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996) 189 | DOI

[3] J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, I, J. Differential Geom. 46 (1997) 406 | DOI

[4] J Cheeger, T H Colding, On the structure of spaces with Ricci curvature bounded below, II, J. Differential Geom. 54 (2000) 13 | DOI

[5] T H Colding, Ricci curvature and volume convergence, Ann. of Math. 145 (1997) 477 | DOI

[6] T H Colding, A Naber, Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications, Ann. of Math. 176 (2012) 1173 | DOI

[7] T H Colding, A Naber, Characterization of tangent cones of noncollapsed limits with lower Ricci bounds and applications, Geom. Funct. Anal. 23 (2013) 134 | DOI

[8] K Fukaya, T Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. 136 (1992) 253 | DOI

[9] M Gromov, Almost flat manifolds, J. Differential Geometry 13 (1978) 231 | DOI

[10] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser (2007) | DOI

[11] K Grove, H Karcher, How to conjugate C1–close group actions, Math. Z. 132 (1973) 11 | DOI

[12] P Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. 124 (1986) 1 | DOI

[13] G Liu, 3–Manifolds with nonnegative Ricci curvature, Invent. Math. 193 (2013) 367 | DOI

[14] M Mazur, X Rong, Y Wang, Margulis lemma for compact Lie groups, Math. Z. 258 (2008) 395 | DOI

[15] J Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968) 1 | DOI

[16] R S Palais, Equivalence of nearby differentiable actions of a compact group, Bull. Amer. Math. Soc. 67 (1961) 362 | DOI

[17] J Pan, A proof of Milnor conjecture in dimension 3, J. Reine Angew. Math. (2018) | DOI

[18] C Sormani, Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups, J. Differential Geom. 54 (2000) 547 | DOI

[19] B Wilking, On fundamental groups of manifolds of nonnegative curvature, Differential Geom. Appl. 13 (2000) 129 | DOI