On the Hanna Neumann Conjecture

2003/02/01 by Toshiaki Jitsukawa, Jitsukawa, Toshiaki, Bilal Khan +3
Mathematics · #20E05 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:20E05

paper · pdf · doi:10.48550/arxiv.math/0302009

11 pages, 3 figures

arxiv created 2003/02/01 · arxiv updated 2009/11/30

Abstract

The Hanna Neumann conjecture states that if F is a free group, then for all nontrivial finitely generated subgroups H,K <= F, rank(H intersect K) - 1 <= [rank(H)-1] [rank(K)-1]. Where most papers to date have considered a direct graph theoretic interpretation of the conjecture, here we consider the use of monomorphisms. We illustrate the effectiveness of this approach with two results. First, we show that for any finitely generated groups H,K <= F either the pair H,K or the pair H^-, K satisfy the Hanna Neumann conjecture--Here - denotes the automorphism which sends each generator of F to its inverse. Next, using particular monomorphisms from F to F₂, we obtain that if the Hanna Neumann conjecture is false then there is a counterexample H,K < F₂ having the additional property that all the branch vertices in the foldings of H and K are of degree 3, and all degree 3 vertices have the same local structure or ``type''.

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