Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Subnormality in $PD_3$-groups and $L^2$-Betti numbers   We reconsider work of Elkalla on subnormal subgroups of 3-manifold groups,
giving essentially algebraic arguments that extend to the case of $PD_3$-groups
and group pairs. However the argument relies on an $L^2$-Betti number
hypothesis which has not yet been shown to hold in general.

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The categories of this paper are: stat
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The categories of this paper are: supr-con
3
The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph