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

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

1998-09-23

arxiv.org/abs/math/9809133v1

Journal of Differential Geom. 54 (2000), no. 3, 547--559

Mathematics

Differential Geometry

12 pages, figures available from author sormani@math.jhu.edu

Scientific paper

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.

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