Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solvable Groups of Exponential Growth and HNN Extensions   We prove the following version of Milnor's theorem on solvable groups of
exponential growth: A finitely generated solvable group which is not polycyclic
contains an ascending HNN extension. Consequently, a finitely generated
solvable group is ERF iff it is polycyclic.

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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: supr-con
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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: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin