Local estimate of fundamental groups

2019/05/31 by Guoyi Xu
Mathematics · #Combinatorics #Comparison theorem #Curvature #Fundamental group #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Lemma (botany) #Manifold (fluid mechanics) #Mathematics #Point processes and geometric inequalities #Pure mathematics #Ricci curvature #Riemannian manifold #math.AT #math.DG

paper · pdf · doi:10.1016/j.aim.2019.06.006

published as Advances in Mathematics 352 (2019) pp. 158-230 · to appear in Adv. Math

arxiv created 2019/05/31 · openalex publication_date 2019/06/06 · arxiv updated 2019/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

For any complete n-dim Riemannian manifold Mⁿ with nonnegative Ricci curvature, Kapovitch and Wilking proved that any finitely generated subgroup of the fundamental group π₁(Mⁿ) can be generated by C(n) generators. Inspired by their work, we give a quantitative proof of the above theorem and show that C(n)≤ n^n²⁰ⁿ . Our main tools are quantitative Cheeger-Colding's almost splitting theory, and the squeeze lemma for covering groups between two Riemannian manifolds with nonnegative Ricci curvature.

Citations

• The splitting theorem for manifolds of nonnegative Ricci curvature
• Metric Structures for Riemannian and Non-Riemannian Spaces
• Elliptic Partial Differential Equations of Second Order
• Regularity of Einstein manifolds and the codimension 4 conjecture
• On the parabolic kernel of the Schrödinger operator
• Differential equations on riemannian manifolds and their geometric applications
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
• On the Structure of Complete Manifolds of Nonnegative Curvature