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On the Kurosh Rank of the Intersection of Subgroups in Free Products of Groups

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On the Kurosh Rank of the Intersection of Subgroups in Free Products of Groups View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Advances in Mathematics 218 (2008) 465–484 www.elsevier.com/locate/aim On the Kurosh rank of the intersection of subgroups in free products of groups S.V. Ivanov 1 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA Received 3 January 2006; accepted 15 January 2008 Available online 7 March 2008 Communicated by Michael J. Hopkins Dedicated to the Centenary of Alexander Gennadievich Kurosh Abstract ∗ ∈ The Kurosh rank rK(H ) of a subgroup H of a free product α∈I Gα of groups Gα, α I,isde- fined accordingly to the classic Kurosh subgroup theorem as the number of free factors of H . We prove ∗ ¯ ∩ that if H1, H2 are subgroups of α∈I Gα and H1, H2 have finite Kurosh rank, then rK(H1 H2) q∗ ∗ 2 ∗ r¯ (H )r¯ (H ) 6r¯ (H )r¯ (H ),wherer¯ (H ) = max(r (H ) − 1, 0), q is the minimum of or- q −2 K 1 K 2 K 1 K 2 K K ∗ ∈ ∗ := ∞ q := ders > 2 of finite subgroups of groups Gα, α I, q if there are no such subgroups, and q∗−2 1if ∗ q =∞. In particular, if the factors Gα, α ∈ I, are torsion-free groups, then r¯K(H1 ∩H2) 2r¯K(H1)r¯K(H2). © 2008 Elsevier Inc. All rights reserved. MSC: primary 20E06; secondary 20F06, 20F32 Keywords: Free products of groups; Subgroups of free products; Kurosh rank of a subgroup of a free product 1. Introduction In 1954, Howson [9] proved that if H1, H2 are finitely generated subgroups of a free group F , then the intersection H1 ∩ H2 is also finitely generated and, moreover, the rank r(H1 ∩ H2) of E-mail address: [email protected]. 1 Supported in part by the NSF under grant DMS 04-00476. 0001-8708/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2008.01.003 466 S.V. Ivanov / Advances in Mathematics 218 (2008) 465–484 H1 ∩ H2 is bounded in terms of ranks r(H1) and r(H2). The Howson’s bound was improved by Hanna Neumann [16] in 1957 by showing that r¯(H1 ∩ H2) 2r¯(H1)r¯(H2), (1) where r¯(H ) = max(r(H ) − 1, 0) is the reduced rank of H . Hanna Neumann also conjectured in [16] that the coefficient 2 in (1) could actually be changed to 1. This conjecture has received much attention, e.g. see [2,4–6,8,10,17,18,20–22], but is still far from being settled. In 1966, B. Baumslag [1] established an analogous result for a free product A ∗ B of two groups A,B, proving that if A,B have the Howson property (i.e., the intersection of two arbi- trary finitely generated subgroups is again finitely generated), then A ∗ B also has the Howson property. We remark that this result was reproved by Soma [19] and by the author [11] by geo- metric methods. However, B. Baumslag did not give any estimate of type (1). A version of the estimate (1) for free products was obtained by Soma [19] in 1990 and an attempt to improve on the Soma’s estimate was made by Burns, Chau and Kam [3] in 1998. To state results of [3,19], we recall that, according to the classic Kurosh subgroup theorem [13] ∗ proved in 1934, see also [14,15], every subgroup H of a free product α∈I Gα of groups Gα, α ∈ I , itself is a free product ∗ = ∗ −1 H F(H) Sα,β Hα,β Sα,β , (2) ∈ ∗ ∗ where Hα,β is a subgroup of Gα, Sα,β α∈I Gα, and F(H) is a free subgroup of α∈I Gα ∈ ∗ ∈ ∩ −1 such that, for every S α∈I Gα and γ I , the intersection F(H) SGγ S is trivial. Define the Kurosh rank rK(H ) of H to be the sum rK(H ) = r F(H) + rα(H ), (3) α∈I −1 ∈ where rα(H ) is the number of all nontrivial factors Sα,β Hα,β Sα,β in (2) for given α I . It is not difficult to show (e.g., see Lemma 4 in Section 2) that the Kurosh rank rK(H ) is defined correctly. ∗ = We will say that H is a factor-free subgroup of a free product α∈I Gα if H F(H) in the ∈ ∗ ∈ ∩ −1 = foregoing form (2) of H , that is, for every S α∈I Gα and γ I ,wehaveH SGγ S { } ∗ 1 . Since a factor-free subgroup H of α∈I Gα is free, the rank r(H ) is correctly defined and r(H ) = rK(H ). In 1990, Soma [19] essentially proved that if subgroups H1, H2 of a free product A ∗ B have finite Kurosh rank, then the intersection H1 ∩ H2 also has finite Kurosh rank and r¯K(H1 ∩ H2) 18r¯K(H1)r¯K(H2), (4) where r¯K(H ) = max(rK(H ) − 1, 0) is the reduced Kurosh rank of H . In 1998, Burns, Chau and Kam [3] introduced the term of the Kurosh rank, observed that Soma’s results [19] imply the inequality (4) and attempted to improve on (4) and to show that r¯K(H1 ∩ H2) 2r¯K(H1)r¯K(H2) + 2min r¯K(H1), r¯K(H2) 4r¯K(H1)r¯K(H2). (5) S.V. Ivanov / Advances in Mathematics 218 (2008) 465–484 467 However, in 2001, the author [12] constructed an explicit example of two factor-free subgroups H1, H2 of the modular group Z2 ∗Z3, where Zn is cyclic of order n, such that r¯K(H1) =¯rK(H2) = r¯(H1) =¯r(H2) = 2, whereas r¯K(H1 ∩ H2) =¯r(H1 ∩ H2) = 24 = 6r¯K(H1)r¯K(H2). (6) This example proves that the main result (5) of article [3] is incorrect. The aim of this article is to make another attempt to improve on Soma’s bound (4) and to generalize it for a free product ∗ ∈ α∈I Gα of arbitrary family of groups Gα, α I . Recall that it was proved in [11] that if H1,H2 ∗ ∩ are finitely generated factor-free subgroups of α∈I Gα, then the intersection H1 H2 is also finitely generated and r¯(H1 ∩ H2) 6r¯(H1)r¯(H2) (7) which, in view of (6), is a sharp inequality and cannot be improved. Recently, these and other results of [11,12] were strengthened and refined by Dicks and the author [7] as follows. Let G be a group, q(G) denote the minimum of orders > 2 of finite subgroups of G and q(G) := ∞ if there are no such subgroups. Clearly, q(G) = 4orq(G) is an odd prime or q(G) =∞.Let ∗ ∗ q := q( ∈ Gα) = min{q(Gα) | α ∈ I} and H1, H2 be finitely generated factor-free subgroups α I ∗ of a free product α∈I Gα. It is proved in [7] that q∗ r¯(H ∩ H ) 2 r¯(H )r¯(H ), (8) 1 2 q∗ − 2 1 2 ∗ ∗ q := ∗ =∞ q where q∗−2 1ifq .Note1 q∗−2 3, hence the estimate (8) strengthens (7). In ∗ addition, it is shown in [7] that if the free product α∈I Gα contains an involution and a non- ∗ abelian factor-free subgroup (the latter holds unless α∈I Gα either is a trivial free product, i.e. it contains at most one nontrivial factor, or is dihedral, i.e. it contains precisely two non- ∗ trivial factors of order 2), then there exist factor-free subgroups H1, H2 in α∈I Gα such that 0 < r¯(H1), r¯(H2)<∞ and the inequality (8) turns into an equality, thus, in this case, the esti- mate (8) is sharp. In this article, we will prove that an analogous bound holds for the Kurosh rank as well. ∗ Theorem. Suppose that H1, H2 are subgroups of a free product α∈I Gα and H1, H2 have finite Kurosh rank rK(H1), rK(H2). Then the intersection H1 ∩ H2 also has finite Kurosh rank and q∗ r¯ (H ∩ H ) 2 r¯ (H )r¯ (H ) 6r¯ (H )r¯ (H ), (9) K 1 2 q∗ − 2 K 1 K 2 K 1 K 2 where q∗ is the minimum of orders > 2 of finite subgroups of groups G , α ∈ I , q∗ := ∞ if there ∗ α q := ∗ =∞ ∗ are no such subgroups and q∗−2 1 if q . In addition, if a free product α∈I Gα contains ∗ an involution, is not trivial and is not dihedral, then there exist subgroups H1, H2 in α∈I Gα such that 0 < r¯K(H1), r¯K(H2)<∞ and the inequality (9) turns into an equality. In the case q∗ =∞, quite analogously to the Hanna Neumann inequality (1), we obtain the following. 468 S.V. Ivanov / Advances in Mathematics 218 (2008) 465–484 ∗ Corollary. Suppose that a free product α∈I Gα is torsion-free or, more generally, every finite ∈ ∗ subgroup of groups Gα, α I , has order at most 2, and H1, H2 are subgroups of α∈I Gα that have finite Kurosh rank. Then r¯K(H1 ∩ H2) 2r¯K(H1)r¯K(H2) ∗ and, as in the Theorem, this estimate is sharp whenever α∈I Gα contains an involution, is not trivial and is not dihedral. Clearly, the inequality (9) generalizes (8) because r¯K(H ) =¯r(H ) when H is a factor-free sub- group. This generalization, however, is nontrivial and, in Section 5, we will discuss two examples that illustrate the intricacy of interplay between the free and factors’ ingredients of the Kurosh rank under intersection of subgroups.
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