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Algebraic and Definable Closure in Free Groups Daniele A.G

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Algebraic and Definable Closure in Free Groups Daniele A.G Algebraic and definable closure in free groups Daniele A.G. Vallino To cite this version: Daniele A.G. Vallino. Algebraic and definable closure in free groups. Group Theory [math.GR]. Université Claude Bernard - Lyon I, 2012. English. tel-00739255 HAL Id: tel-00739255 https://tel.archives-ouvertes.fr/tel-00739255 Submitted on 6 Oct 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Université Claude Bernard - Lyon 1 École Doctorale Infomaths Institut Camille Jordan Università degli Studi di Torino Scuola di Dottorato in Scienze ed Alta Tecnologia, indirizzo Matematica Dipartimento di Matematica ‘Giuseppe Peano’ Tesi presentata per il conseguimento del titolo di Dottore di Ricerca da Thése présentée en vue de l’obtention du titre de Docteur de Recherche par VALLINO Daniele Angelo Giorgio Algebraic and definable closure in free groups Data di discussione/date de soutenance: 5 giugno 2012/5 juin 2012 Direttori/directeurs: OULD HOUCINE Abderezak, Université Claude Bernard Lyon 1 ZAMBELLA Domenico, Università degli Studi di Torino Referees/rapporteurs: SELA Zlil, Hebrew University, Jerusalem TENT Katrin, Westfälische Wilhelms-Universität Münster Membri della commissione/Membres du jury: ALTINEL Tuna, Université Claude Bernard Lyon 1 CONSOLE Sergio, Università degli Studi di Torino LEVITT Gilbert (président), Université de Caen OULD HOUCINE Abderezak, Université Claude Bernard Lyon 1 TENT Katrin, Westfälische Wilhelms-Universität Münster VIALE Matteo, Università degli Studi di Torino WAGNER Frank, Université Claude Bernard Lyon 1 ZAMBELLA Domenico, Università degli Studi di Torino Contents Acknowledgements iii Introduction (English) v Introduction (Français) ix Introduzione xiii 1 Basic notions 1 1.1 Basics of combinatorial group theory . 1 1.1.1 Free groups . 3 1.1.2 Free products . 7 1.1.3 Amalgamated free products and HNN-extensions . 10 1.2 Bass-Serre theory . 14 1.2.1 Graphs; Cayley graphs . 14 1.2.2 Graphs of groups, actions on trees . 17 1.3 Hyperbolic groups . 23 1.3.1 Hyperbolic spaces . 24 1.3.2 Hyperbolic groups . 30 1.4 Basics of model theory . 33 1.5 Asymptotic cones . 45 2 Bestvina-Paulin method 49 2.1 Acylindrical actions . 49 2.2 Main theorem . 50 2.3 Proof of main theorem . 52 3 Limit groups, shortening argument, JSJ decompositions 61 3.1 Limit groups . 61 3.2 Shortening argument . 64 3.3 JSJ decompositions . 70 4 The algebraic closure 75 4.1 Constructibility from algebraic closure . 76 4.2 Algebraic closure in JSJ decomposition . 85 i ii 5 Algebraic and definable closure 89 5.1 Main statement and preliminaries . 89 5.2 Equality in low ranks . 91 5.3 A counterexample in higher ranks . 92 5.4 Results in rank 3 . 95 Bibliography 100 Acknowledgements This research project would not have been possible without the support of many people. I wish to express my gratitude to supervisors, Prof. Abderezak Ould Houcine and Prof. Domenico Zambella, who led me out to geometric group theory and model theory respec- tively, two fields that at the beginning of my PhD program I found as rich of interest as well as unknown. Supervisors have been abundantly helpful and offered assistance, support and guidance during all the program. Deep gratitude is also due to the referees, Prof. Zlil Sela and Prof. Katrin Tent, as well as to the members of the supervisory committee and to Prof. Amador Martin-Pizarro, the first link for me to the Lyon group. I would also like to convey thanks to the Italian Ministry of Instruction and Research, which allowed me to temporarily retire from high school teaching for three years, so to relieve me of finding any financement, to Franco-Italian University Foundation, whose Vinci prize provided me for accommodation and trip facilities to Lyon, and to Faculties of Turin, Lyon and Mons for providing work facilities. Finally, I express my love and gratitude to my families: my parents who several years ago allowed and fostered me to follow my favourite studies; my wife Silvia, who - even by taking on all cares for our newborn daughter, including what should have been up to me, and by utilizing her skills to manage grandparents’ helpful support activity - deeply understood what it meaned for me to take the challenge of starting a research program twelve years after having finished my last degree. iii iv Introduction The notion of algebraic closure originated in field theory. It first appeared in the proof by Gauss of the fundamental theorem of algebra in 1799 and laid the basis of the development of Galois (1811-1832) theory. Model theory generalized the notion of algebraic closure to any first-order theory. The algebraic closure of a subset A in a structure G, denoted aclG(A), is the set of elements g such that there exists a formula φ(x), with parameters from A, such that G satisfies φ(g) and such the set of elements satisfying φ is finite. In model theory, one considers also the notion of definable closure which is the set of elements g such that in the previous definition the set of elements g satisfying φ is a singleton. ∃ Two related notions are those of existential algebraic closure aclG(A) and quantifier- QF free algebraic closure aclG (A), where φ(x) is restricted to be an existential, respectively quantifier-free formula. As a basic reference for the notion of algebraic closure in model theory, see [Hod97, p.138]. The previous definition generalizes the notion of algebraic closure in two ways. First it considers algebraic closure in an arbitrary model G, not just an algebraically closed field. Second, as not every theory has elimination of quantifiers, it allows φ(x) to be any first-order formula, not just an equation. In the context of algebraically closed fields, the above model-theoretic notion coincides with the usual one. Galois theory relies on the fact that algebraic closure coincides with what we call restricted algebraic closure. Restricted algebraic closure, denoted racl(A), is the set of elements g such that the orbit {f(g)|f ∈ AutA(G)} is finite, where AutA(G) is the group of automorphisms of G fixing A pointwise. The corresponding notion of restricted definable closure is defined in a similar way. We investigate here algebraic and definable closure in free groups; the main results can be summarized as follows. 1. We prove a constructibility result for torsion-free hyperbolic groups from algebraic closure of some subset. The property of algebraic closure which is relevant for constructibility is the fact that it coincides with its own existential algebraic closure. On the contrary, existential algebraic closure does not have this property - in other words, existential closure operator is not idempotent. Nevertheless, we have proved constructibility of free groups from existential algebraic closure, thanks to Theorem 1.1.24. 2. We looked for some results about the position of aclG(A) in some decomposition of the group G. At first we focused on raclG(A), since we have disponibility of results v vi linking automorphisms and vertex groups of certain decompositions (see Lemmas 3.3.9 and 3.3.10). For torsion-free hyperbolic groups we obtained that raclG(A) coincides with the vertex group containing A in the generalized malnormal cyclic JSJ decomposition of G relative to A (Definitions 3.3.12 and 3.3.13), obtained after ruling out free factors of G not containing A and making edge groups malnormal in adjacent vertex groups. Moreover, for free groups such vertex coincides with strictly-speaking algebraic closure. 3. We studied the possibility of generalizing Bestvina-Paulin method in another direc- tion by considering finitely generated groups acting acylindrically (in the sense of Bowditch) on hyperbolic graphs. The typical formulation of the method supposes a group G acting on a Cayley graph Γ of a group H. Instead of weakening as- sumptions on G or H, we have dropped the assumption that Γ is a Cayley graph. Informally speaking, a Bowditch-acylindrical action bounds the number of elements of G that move ‘little enough’ vertices of Γ that are ‘far enough from each other’. The results obtained by using this notion seem rather close to those (see [RW10]) obtained making groups act on hyperbolic Cayley graphs. 4. We investigated in relations between the different algebraic closure notions and between algebraic and definable closure. We found that in free groups algebraic closures coincide, the strictly-speaking and the restricted one. Regarding the prob- lem algebraic/definable closure, the knowledge gap between the class of free groups and larger classes is mainly due to Theorem 1.2.39. In 2008, Sela asked whether aclF (A)= dclF (A) for every subset A of a free group F . A positive answer has been given for a free group F of rank smaller than 3: for every subset A ⊆ F we have aclF (A)= dclF (A). Instead, for free groups of rank strictly greater than 3 we found a counterexample. For the free group of rank 3 we found a necessary condition on the form of a possible counterexample. Rapidly browsing contents, Chapter 1 is a survey of introductive notions. We give basics on combinatorial group theory, starting from free groups and proceeding with the fun- damental constructions: free products, amalgamated free products and HNN extensions.
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