Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups

1998/09/23 by Christina Sormani · 3 citations
Mathematics · #math.DG #msc:53C21

paper · pdf

published as Journal of Differential Geom. 54 (2000), no. 3, 547--559 · 12 pages, figures available from author [email protected]

arxiv created 1998/09/23 · arxiv updated 2009/11/30

Abstract

In 1968, Milnor conjectured that a complete noncompact manifold with nonnegative Ricci curvature has a finitely generated fundamental group. The author applies the Excess Theorem of Abresch and Gromoll (1990), to prove two theorems. The first states that if such a manifold has small linear diameter growth then its fundamental group is finitely generated. The second states that if such a manifold has an infinitely generated fundamental group then it has a tangent cone at infinity which is not polar. A corollary of either theorem is the fact that if such a manifold has linear volume growth, then its fundamental group is finitely generated.

Cited by

• Splitting and Slow Volume Growth for Open Manifolds with Nonnegative Ricci Curvature
• Volume growth and asymptotic cones of manifolds with nonnegative Ricci curvature
• Fundamental groups of open manifolds with nonnegative Ricci curvature and universal cover Euclidean volume growth

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