Local estimate of fundamental groups

Abstract

For any complete $n$-dim Riemannian manifold $M^n$ with nonnegative Ricci curvature, Kapovitch and Wilking proved that any finitely generated subgroup of the fundamental group ${\pi }_1(M^n)$ can be generated by $C(n)$ generators. Inspired by their work, we give a quantitative proof of the above theorem and show that $C(n)\le n^{n^{20n}}$. 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.

to appear in Adv. Math

References in corpus (1)

• Structure of fundamental groups of manifolds with Ricci curvature bounded below

Cited by in corpus (1)

• The first Dirichlet eigenvalue and the width