The rank three case of the Hanna Neumann Conjecture

Warren Dicks, Edward Formanek

Research output: Contribution to journalArticleResearchpeer-review

11 Citations (Scopus)


For a free group G, rk(G) denotes the rank of G, and, for each positive integer n, rk-n(G) denotes max{rk(G) - n, 0}. Let H and K be finitely generated subgroups of a free group. Hanna Neumann conjectured that rk-1(H ∩ K) ≤ rk-1(H) rk-1(K). We prove that rk-1(H ∩ K) ≤ rk-1(H) rk-1(K) + rk-3(H) rk-3(K). This extends results of Hanna Neumann, R. G. Burns and G. Tardos, and shows that, if H has rank three or less, then the conjectured inequality holds. Our argument consists of proving the corresponding case of the Amalgamated Graph Conjecture, and therefore applies to Walter Neumann's strengthened version of the Hanna Neumann Conjecture.
Original languageEnglish
Pages (from-to)113-151
JournalJournal of Group Theory
Publication statusPublished - 1 Jan 2001


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