# Local estimate of fundamental groups

## TL;DR

This paper provides a quantitative proof that the fundamental group of a complete nonnegatively Ricci curved manifold can be generated by a number of generators bounded by a function of the dimension, using advanced geometric analysis tools.

## Contribution

It offers a new quantitative bound on the number of generators needed for the fundamental group, improving understanding of its structure in nonnegative Ricci curvature manifolds.

## Key findings

- Bound C(n) ≤ n^{n^{20n}} for generators of the fundamental group.
- Utilizes quantitative Cheeger-Colding's almost splitting theory.
- Employs the squeeze lemma for covering 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)\leq 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.