Local estimate of fundamental groups

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) <= 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. (C) 2019 Elsevier Inc. All rights reserved.