---
tag: 50WD2CBCT9
kind: statement
role: proposition
name: Strengthened Hanna Neumann conjecture
sources:
  - wikipedia: en:19296207@r1327110966
    locator: Hanna Neumann conjecture, section Strengthened Hanna Neumann conjecture
    note: adopted
    mention: wikipedia://en/19296207#44SZ
    sections: [Statement]
inventory: [2M5CMVV33E]
---

## Statement

If *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*<sup>−1</sup> and *H* ∩ *bKb*<sup>−1</sup> 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*<sup>−1</sup> ≠ {1}. Suppose that at least one such double coset exists and let *a*<sub>1</sub>,...,*a*<sub>*n*</sub> 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

$$\sum_{i=1}^n [{\rm rank}(H\cap a_iKa_{i}^{-1})-1] \le ({\rm rank}(H)-1)({\rm rank}(K)-1).$$

The strengthened Hanna Neumann conjecture was proved in 2011 by Joel Friedman. Shortly after, another proof was given by Igor Mineyev.