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Failure of the Finitely Generated Intersection Property for Ascending Failure of the finitely generated intersection property for ascending HNN extensions of free groups Jacob Bamberger Department of Mathematics and Statistics McGill University Montreal, Quebec July 2020 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Master of Science. © Jacob Bamberger 1 2 ABSTRACT The main result of this thesis is a proof of the failure of the finitely generated intersection property of ascending HNN extensions of non-cyclic finite rank free groups. This class of group consists of free-by-cyclic groups and properly ascending HNN extensions of free groups. We also give a sufficient condition for the failure of the FGIP in the context of relative hyperbolicity, we apply this to the free-by-cyclic groups of exponential growth. ABREG´ E´ Le r´esultatprincipal de cette th`eseest une d´emonstrationde l'´echec de la propri´et´e Howson dans le cas d'extension HNN de groupes libres non-cyclique de g´en´eration finie. Cette classe de groupe consiste des groupes ayant un sous-groupe normal libre dont le quotient est cyclique, ainsi que les groupes d'extension HNN propre. Nous donnons une condition suffisante pour l'´echec de la propri´et´ede Howson dans le context d'hyperbolicit´erelative, nous utilisons ce resultat dans le cas o`ul'extension HNN est un automorphism de croissance exponentielle. 3 Acknowledgements I would like to express my gratitude to my supervisor, Daniel Wise, for the weekly meetings, for his guidance and for showing me dedication. I thank my family, for their unconditional support. I thank Eduardo Martinez Pedroza, for ping pong help. Lastly, thank you Harrison, for making this experience more enjoyable. Contribution of Authors Unless otherwise stated, the results presented in this thesis constitutes of joint work with my supervisor, Professor Daniel T. Wise. 4 Contents ABSTRACT 2 ABREG´ E´ 2 Acknowledgements 3 Contribution of Authors 3 1. Introduction 5 2. The Finitely Generated intersection property 7 3. Proper ascending HNN extension 8 4. Relative train track maps 11 4.1. Mapping Tori 13 5. Polynomially growing case 13 6. The exponential case 15 7. Failure of the FGIP for certain relatively hyperbolic groups 19 7.1. Ping-pong 19 7.2. Relative hyperbolicity and the FGIP 20 7.3. Exponential growth free-by-cyclic 20 References 21 5 1. Introduction Definition 1.1. A group has the finitely generated intersection property (FGIP) if the intersection of any two finitely generated subgroups is also finitely generated. The most famous class of groups having the FGIP are locally quasiconvex word- hyperbolic groups [27]. This generalizes the fact that free groups have the FGIP which was proven by Howson [17], and indeed the FGIP is sometimes referred to as the Howson property. The purpose of this thesis is to examine the FGIP for ascending HNN extensions of finitely generated free groups. We show the failure of the finitely generated inter- section property (FGIP) for ascending HNN extensions of non-cyclic finite rank free groups. Definition 1.2 (Ascending HNN extension of a free group). Let φ : F ! F be a monomorphism from a free group to itself. Its associated ascending HNN extension is the group G = hF; t j tft−1 = φ(f): 8f 2 F i. If φ is surjective then G is free-by-cyclic and we denote G by F oφ Z. If φ is not surjective then G is a proper ascending HNN extension. Our main goal is the following result which is new for proper HNN extensions: Theorem 1.3. Any ascending HNN extension G of a finitely generated non-cyclic free group F fails to have the FGIP. The failure of the FGIP for F × Z was first proved in [23]. The failure for free-by- cyclic groups was later proved in [8]. The authors were unfortunately oblivious to the work of Burns and Brunner and formulated an argument of the free-by-cyclic case which is separated into the polynomially growing and exponentially growing cases in Sections 5 & 6. These use Bestvina-Feighn-Handel relative train track maps [4, 5] which are recalled in Section 4. We believe our argument to be similar to [8] in nature, but hopefully less oblique as it relies less on combinatorial group theory and 6 more on geometric intuition but at the expense of a powerful tool. In Theorem 3.1, the properly ascending case of Theorem 1.3 is proved using elementary methods. When G is hyperbolic Theorem 1.3 has a much crisper explanation using the ping pong lemma (see Section 7.1). The idea is that for any f 2 F − f1Gg, the ping pong lemma provides a free subgroup K = hf m; tmi and the intersection F \ K is not finitely generated. We develop this idea further in Section 7 and obtain the following sufficient condition for the failure of the FGIP in the relative hyperbolic framework: Theorem 1.4. Let G be hyperbolic relative to a collection of proper subgroups. Let N ⊂ G be a finitely generated subgroup. Suppose tNt−1 ⊂ N for some infinite order t2 = N. Suppose there is an infinite order w 2 N such that t; w do not lie in the same parabolic or virtually cyclic subgroup. Then G fails to have the FGIP. We combine this result with recent powerful statements concerning relative hyper- bolicity of free-by-cyclic groups [12, 13, 14] to provide a unified proof of the failure of the FGIP for exponentially growing free-by-cyclic groups. This explanation is complex since it depends upon a constellation of deep results, but it is interesting to see how the exponential case fits into general framework via Theorem 1.4. We expect this unity to prevail as relatively hyperbolic structures are constructed for general ascending HNN extensions. Note that the exponentially growing case is the main case since failure in the polynomially growing case is a simple consequence of the failure for F × Z. Finitely generated subgroups of ascending HNN extensions of rank 1 free groups are easily shown to be either trivial, cyclic or of finite index (see e.g. [21]). Inter- secting a subgroup H with a finite index subgroup K yields a finite index subgroup of H. Hence if H is finitely generated so is H \ K. We therefore focus on ascending HNN extensions of a free group F with rank(F ) ≥ 2. 7 2. The Finitely Generated intersection property In this section we survey known results on the FGIP [29]. It is also called the Howson property since Howson first proved that free groups have the FGIP [17]. Moldavansky later proved that F ×Z does not have the FGIP and that the Baumslag Solitar group BS(1; n) has the FGIP [23]. Greenberg proved that fundamental groups of compact surfaces have the FGIP [15]. Soma studied the FGIP for geometric 3 manifolds, giving examples of 3- manifolds with and without the FGIP [28]. In particular, hyperbolic 3-manifolds of infinite volume have the FGIP and closed hyperbolic 3-manifolds do not. Polycyclic- by-finite groups have the FGIP [1]. Baumslag proved that the FGIP is preserved under free products [3], this was generalized by Cohen to amalgamated products and HNN-extensions over finite subgroups [9]. Burns proves that amalgamated products of free groups along infinite cyclic subgroups have the FGIP if and only if the cyclic subgroup is maximal cyclic in both factors [7]. A right angled Artin group A(Γ) has the FGIP iff every connected component of Γ is complete [26]. Dahmani proved that limit groups have the FGIP [11]. A large class of groups, including many of the above, having the FGIP are locally quasi-convex groups, a proof can be found in [27]. Wise and McCammond used the previous result to prove that many small cancellation groups have the FGIP [22]. Using these techniques, Schupp proved that many Coxeter groups have the FGIP n [25]. One relator groups of form G = hx1; ··· xmjr = 1i where n ≥ jrj are locally quasi-convex and therefore also have the FGIP [18]. Kapovich gives examples of one- relator groups that fail the FGIP [19], note that many ascending HNN extensions of free groups are one relator groups, for example ha; b; t : tat−1 = b; tbt−1 = ababi =∼ hb; t : tbt−1 = t−1btbt−1btbi. The Grigorchuk group does not have the FGIP [24]. 8 A \generic" m-generator, n-relator group has the FGIP [2]. The wreath product Z o Z does not have the FGIP [20]. The question of whether SL(3; Z) has the FGIP remains open. Note that SL(n; Z) where n > 3 has F × Z subgroups where F is free of rank 2 so fails to have the FGIP. 3. Proper ascending HNN extension In this section we prove the following theorem: Theorem 3.1. Any proper ascending HNN extension of a non-cyclic finitely gen- erated free group fails to have the FGIP. Lemma 3.2. Let H ⊂ F be a finitely generated infinite index subgroup of a free group. There exists f such that hf; Hi = hfi ∗ H. ^ Proof. Regard F as π1(B; b), where B is a bouquet of circles. Let (B;b b) ! (B; b) be ^ a based covering map with H = π1(B;b b). Since H is finitely generated, we can chose ^ ^ a finite based subgraph (C; b) ⊂ (B;b b) whose inclusion induces a π1-isomorphism. We claim that it is possible to add an edge e to C to obtain an immersion D ! B. Note that π1(D; b) = hfi ∗ H for some nontrivial f 2 π1(B; b).
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