Hanna Neumann conjecture

History

The subject of the conjecture was originally motivated by a 1954 theorem of Howson{{cite journal :s − 1 ≤ 2mn − m − n.

In a 1956 paper{{cite journal

:s − 1 ≤ 2mn − 2m − n.

In a 1957 addendum, Hanna Neumann further improved this bound to show that under the above assumptions

:s − 1 ≤ 2(m − 1)(n − 1).

She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has

:s − 1 ≤ (m − 1)(n − 1).

This statement became known as the Hanna Neumann conjecture.

Formal statement

Let H, K ≤ F(X) be two nontrivial finitely generated subgroups of a free group F(X) and let L = H ∩ K be the intersection of H and K. The conjecture says that in this case

:rank(L) − 1 ≤ (rank(H) − 1)(rank(K) − 1).

Here for a group G the quantity rank(G) is the rank of G, that is, the smallest size of a generating set for G. Every subgroup of a free group is known to be free itself and the rank of a free group is equal to the size of any free basis of that free group.

Strengthened Hanna Neumann conjecture

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 H ∩ aKa−1 and H ∩ bKb−1 are conjugate in G and thus have the same rank. It is known that if H, K ≤ F(X) are finitely generated subgroups of a finitely generated free group F(X) then there exist at most finitely many double coset classes HaK in F(X) such that H ∩ aKa−1 ≠ {1}. Suppose that at least one such double coset exists and let a1,...,a**n be all the distinct representatives of such double cosets. The strengthened Hanna Neumann conjecture, formulated by her son Walter Neumann (1990),{{cite conference

:\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.

Partial results and other generalizations

• In 1971 Burns improved{{cite journal

:s ≤ 2mn − 3m − 2n + 4.

• In a 1990 paper, Walter Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).
• Tardos (1992){{cite journal
• Warren Dicks (1994){{cite journal
• Arzhantseva (2000) proved{{cite journal
• In 2001 Dicks and Formanek established the strengthened Hanna Neumann conjecture for the case where at least one of the subgroups H and K of F(X) has rank at most three.{{cite journal
• Khan (2002){{cite conference | book-title=Combinatorial and geometric group theory
• Ivanov{{cite journal
• Wise (2005) claimed{{cite journal | archive-date=2014-11-17 | access-date=2021-05-07 | archive-url=https://archive.today/20141117201642/http://blms.oxfordjournals.org/cgi/content/abstract/37/5/697 | url-status=bot: unknown

References