Hanna Neumann Conjecture - Partial Results and Other Generalizations

Partial Results and Other Generalizations

• In 1971 Burns improved Hanna Neumann's 1957 bound and proved that under the same assumptions as in Hanna Neumann's paper one has

s ≤ 2mn − 3m − 2n + 4.

• In a 1990 paper, Walter Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).
• Tardos (1992) established the Hanna Neumann Conjecture for the case where at least one of the subgroups H and K of F(X) has rank two. As most other approaches to the Hanna Neumann conjecture, Tardos used the technique of Stallings subgroup graphs for analyzing subgroups of free groups and their intersections.
• Warren Dicks (1994) established the equivalence of the strengthened Hanna Neumann conjecture and a graph-theoretic statement that he called the amalgamated graph conjecture.
• Arzhantseva (2000) proved that if H is a finitely generated subgroup of infinite index in F(X), then, in a certain statistical meaning, for a generic finitely generated subgroup in, we have H ∩ gKg−1 = {1} for all g in F. Thus, the strengthened Hanna Neumann conjecture holds for every H and a generic K.
• In 2001 Dicks and Formanek used this equivalence to prove the strengthened Hanna Neumann Conjecture in the case when one of the subgroups H and K of F(X) has rank at most three.
• Khan (2002) and, independently, Meakin and Weil (2002), showed that the conclusion of the strengthened Hanna Neumann conjecture holds if one of the subgroups H, K of F(X) is positively generated, that is, generated by a finite set of words that involve only elements of X but not of X−1 as letters.
• Ivanov and, subsequently, Dicks and Ivanov, obtained analogs and generalizations of Hanna Neumann's results for the intersection of subgroups H and K of a free product of several groups.
• Wise (2005) showed that the strengthened Hanna Neumann conjecture implies another long-standing group-theoretic conjecture which says that every one-relator group with torsion is coherent (that is, every finitely generated subgroup in such a group is finitely presented).