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  "body": "\n## Statement\n\nIf *H*, *K* ≤ *G* are two subgroups of a group *G* and if *a*, *b* ∈ *G* define the same double coset *HaK = HbK* then the [[subgroups|0X1JXT47GA]] *H* ∩ *aKa*\u003csup\u003e−1\u003c/sup\u003e and *H* ∩ *bKb*\u003csup\u003e−1\u003c/sup\u003e are [[conjugate|0DEBXAJ5SA]] in *G* and thus have the same rank. It is known that if *H*, *K* ≤ *F*(*X*) are [[finitely generated|082DWWXMRJ]] subgroups of a finitely generated [[free group|020Z5SEN27]] *F*(*X*) then there exist at most finitely many double coset classes *HaK* in *F*(*X*) such that *H* ∩ *aKa*\u003csup\u003e−1\u003c/sup\u003e ≠ {1}. Suppose that at least one such double coset exists and let *a*\u003csub\u003e1\u003c/sub\u003e,...,*a*\u003csub\u003e*n*\u003c/sub\u003e be all the distinct representatives of such double cosets. The *strengthened Hanna Neumann conjecture*, formulated by her son Walter Neumann (1990), states that in this situation\n\n$$\\sum_{i=1}^n [{\\rm rank}(H\\cap a_iKa_{i}^{-1})-1] \\le ({\\rm rank}(H)-1)({\\rm rank}(K)-1).$$\n\nThe strengthened Hanna Neumann conjecture was proved in 2011 by Joel Friedman. Shortly after, another proof was given by Igor Mineyev.\n",
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  "name": "Strengthened Hanna Neumann conjecture",
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