Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Intersections and joins of free groups   Let H and K be subgroups of a free group of ranks h and k \geq h. We prove
the following strong form of Burns' inequality: rank(H \cap K) - 1 \leq
2(h-1)(k-1) - (h-1)(rank(H \vee K) -1). A corollary of this, also obtained by
L. Louder and D. B. McReynolds, has been used by M. Culler and P. Shalen to
obtain information regarding the volumes of hyperbolic 3-manifolds. We also
prove the following particular case of the Hanna Neumann Conjecture, which has
also been obtained by Louder. If the join of H and K has rank at least (h + k +
1)/2, then the intersection of H and K has rank no more than (h-1)(k-1) + 1.

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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
3
The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: q-bio