We answer a question due to A. Myasnikov by proving that all expected ranks
occur as the ranks of intersections of finitely generated subgroups of free
groups.Comment: 4 pages, 4 figure

Kent IV, Richard P.

English

arXiv

We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups. Let F be a free group. Let H and K be nontrivial finitely generated subgroups of F. It is a theorem of Howson [1] that H ∩K has finite rank. H. Neumann proved in [2] that rank(H ∩K) − 1 ≤ 2(rank(H)  − 1)(rank(K) − 1) and asked whether or not rank(H ∩K) − 1 ≤ (rank(H) − 1)(rank(K) − 1). A. Myasnikov has asked which values between 1 and (m − 1)(n − 1) can be achieved as rank(H ∩K)  − 1 for subgroups H and K of ranks m and n—this is problem AUX1 of [4]. We prove that all such numbers occur by proving the following Theorem. Let F (a, b) be a free group of rank two. Let Hmk, `  = 〈a, bab −1,..., bkab−k, bk+1an−`b−(k+1)

Achievable ranks of intersections of finitely generated free groups

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