Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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)\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.

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The categories of this paper are: cmp-lg
28
The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
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The categories of this paper are: acc-phys