Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fundamental Groups of Spaces with Bakry-Emery Ricci Tensor Bounded Below   We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric
measure spaces. Arguments utilizing this extension of the Almost Splitting
Theorem show that if a smooth metric measure space has almost nonnegative
Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental
group is almost abelian. Second, if the smooth metric measure space has
Bakry-Emery Ricci curvature bounded from below then the number of generators of
the fundamental group is uniformly bounded. These results are extensions of
theorems which hold for Riemannian manifolds with Ricci curvature bounded from
below.

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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph