Achievable Ranks of Intersections of Finitely Generated Free Groups

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. Mathematics Subject Classification (2000): 20E05 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. Miasnikov 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 H k,l = 〈a, bab , . . . , bab, bab, bab, bab, . . . , bab〉 and let K = 〈b, aba, . . . , aba〉, where 0 ≤ k ≤ m−2 and 0 ≤ l ≤ n−1. Then the rank of H k,l ∩K is k(n− 1) + l. Corollary. Let F be a free group and let m,n ≥ 2 be natural numbers. Let N be a natural number such that 1 ≤ N − 1 ≤ (m − 1)(n − 1). Then there exist subgroups H,K ≤ F , of ranks m and n, such that the rank of H ∩K is N . Proof of the corollary. The theorem produces the desired subgroups for all N with N − 1 ≤ (m− 1)(n− 1)− 1 after passing to a rank two subgroup of F . For N − 1 = (m− 1)(n− 1), simply let H = 〈a, bab, . . . , bab, b〉 and let K = 〈b, aba, . . . , aba, a〉. Proof of the theorem. Let X be a wedge of two circles and base π1(X) at the wedge point. We identify π1(X) with F = F (a, b) by calling the homotopy class of one oriented circle a and the other b. Given a finitely generated subgroup of F ,