Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Orderable 3-manifold groups   We investigate the orderability properties of fundamental groups of
3-dimensional manifolds. Many 3-manifold groups support left-invariant
orderings, including all compact P^2-irreducible manifolds with positive first
Betti number. For seven of the eight geometries (excluding hyperbolic) we are
able to characterize which manifolds' groups support a left-invariant or
bi-invariant ordering. We also show that manifolds modelled on these geometries
have virtually bi-orderable groups. The question of virtual orderability of
3-manifold groups in general, and even hyperbolic manifolds, remains open, and
is closely related to conjectures of Waldhausen and others.

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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: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph