[Paper Review] Nonnegative Ricci curvature, small linear diameter growth and finite generation of fundamental groups

Christina Sormani|ArXiv.org|Sep 23, 1998

Geometric Analysis and Curvature Flows10 references4 citations

TL;DR

This paper proves that a complete noncompact Riemannian manifold with nonnegative Ricci curvature and small linear diameter growth has a finitely generated fundamental group. Using the Excess Theorem of Abresch and Gromoll, the author establishes that such manifolds either have finite fundamental group generation or exhibit a non-polar tangent cone at infinity if the fundamental group is infinitely generated, with linear volume growth implying finite generation as a corollary.

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.

Motivation & Objective

• To address Milnor's 1968 conjecture on the finite generation of fundamental groups for complete noncompact manifolds with nonnegative Ricci curvature.
• To investigate the geometric implications of small linear diameter growth on the topology of such manifolds.
• To establish a connection between the structure of tangent cones at infinity and the finite generation of the fundamental group.
• To prove that linear volume growth implies finite fundamental group generation under nonnegative Ricci curvature.

Proposed method

• Application of the Excess Theorem of Abresch and Gromoll to control the geometry of geodesics and diameter growth.
• Analysis of the asymptotic structure of the manifold via tangent cones at infinity.
• Use of metric geometry techniques to relate diameter growth to the algebraic structure of the fundamental group.
• Study of the polar versus non-polar nature of tangent cones at infinity to infer topological constraints on the fundamental group.
• Comparison of volume and diameter growth rates to derive topological finiteness conditions.
• Reduction of the problem to the behavior of rays and asymptotic cones under nonnegative Ricci curvature.

Experimental results

Research questions

• RQ1 Does a complete noncompact Riemannian manifold with nonnegative Ricci curvature and small linear diameter growth have a finitely generated fundamental group?
• RQ2 What geometric conditions at infinity force the fundamental group to be infinitely generated?
• RQ3 Can the structure of the tangent cone at infinity detect whether the fundamental group is finitely or infinitely generated?
• RQ4 Does linear volume growth imply finite generation of the fundamental group under nonnegative Ricci curvature?
• RQ5 Under what conditions is the tangent cone at infinity polar, and how does this relate to the topology of the manifold?

Key findings

• If a complete noncompact manifold with nonnegative Ricci curvature has small linear diameter growth, then its fundamental group is finitely generated.
• If the fundamental group is infinitely generated, then the tangent cone at infinity is not polar.
• Linear volume growth implies finite generation of the fundamental group for such manifolds.
• The Excess Theorem of Abresch and Gromoll is instrumental in linking geometric growth conditions to topological finiteness.
• The existence of a non-polar tangent cone at infinity is a necessary condition for infinite fundamental group generation.
• The results provide partial confirmation of Milnor's conjecture under additional geometric constraints.