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quaderni di matematica volume 11

edited by

Dipartimento di Matematica ` di Napoli Seconda Universita

Published with the support of Seconda Universita` di Napoli

quaderni di matematica Published volumes 1 - Classical Problems in Mechanics (R. Russo ed.) 2 - Recent Developments in Partial Differential Equations (V. A. Solonnikov ed.) ˇ Hol´ 3 - Recent Progress in Function Spaces (G. Di Maio and L. a eds.) 4 - Advances in Fluid Dynamics (P. Maremonti ed.) 5 - Methods of Discrete Mathematics (S. L¨ owe, F. Mazzocca, N. Melone and U. Ott eds.) 6 - Connections between Model Theory and Algebraic and Analytic Geometry (A. Macintyre ed.) 7 - Homage to Gaetano Fichera (A. Cialdea ed.) 8 - Topics in Infinite Groups (M. Curzio and F. de Giovanni eds.) 9 - Selected Topics in Cauchy-Riemann Geometry (S. Dragomir ed.) 10 - Topics in Mathematical Fluids Mechanics (G.P. Galdi and R. Rannacher eds.) 11 - Model Theory and Applications (L. B´ elair, Z. Chatzidakis et al. eds.) 12 - Topics in Diagram Geometry (A. Pasini ed.)

Next issues Dispersive nonlinear problems in Mathematical Physics (P. D’Ancona and V. Georgev eds.) Kinetic Models of Granular and Reacting Flows (G. Toscani ed.) Complexity of computations and proofs (J.Kraj´ıˇ cek ed.) Calculus of Variations: topics from the mathematical heritage of E. De Giorgi (D. Pallara ed.)

Model Theory and Applications

´ lair, Z. Chatzidakis, P. D’Aquino, D. Marker, M. Otero, F. Point L. Be and A. Wilkie.

edited by

Authors’ addresses:

S ¸ erban A. Basarab Institute of Mathematics of the Romanian Academy P.O. Box 1–764, RO–70700 Bucharest Romania e-mail: [email protected]

Luc B´ elair D´ epartement de Math´ ematiques - UQAM C.P. 8888, Succ. Centerville Montr´ eal (Qu´ ebec) H3C 3P8 Canada e-mail: [email protected]

Oleg Belegradek Department of Mathematics Istanbul Bilgi University 80370, Dolapdere–Istanbul, Turkey email: [email protected]

Zo´ e Chatzidakis Universit´ e Paris 7 UFR de Math´ ematiques, Case 7012 75251 Paris Cedex 05 France e-mail: [email protected]

Gregory Cherlin Department of Mathematics, Rutgers University Hill Center, Piscataway, NJ 08854, U.S.A e-mail: [email protected]

Raf Cluckers Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 200B B-3001 Leuven Belgium email: [email protected]

Paola D’Aquino Dipartimento di Matematica Seconda Universit` a degli Studi di Napoli via Vivaldi, 43 81100 Caserta Italy e-mail: [email protected]

Lou van den Dries University of Illinois Department of Mathematics 1409 W. Green Street Urbana, IL 61801 U.S.A. e-mail: [email protected]

Deirdre Haskell Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada email: [email protected]

Ehud Hrushovski Department of Mathematics Hebrew University Givat-Ram 91904 Israel e-mail: [email protected]

Eric Jaligot Equipe de Logique Math´ ematique, Universit´ e de Paris VII 75251 Paris France e-mail: [email protected]

Angus Macintyre School of Mathematical Sciences Queen Mary University of London Mile End Road London E1 4NS U.K. e-mail: [email protected]

Dugald Macpherson Department of Pure Mathematics University of Leeds Leeds LS2 9JT U.K. e-mail: [email protected]

David Marker Department of Mathematics University of Illinois at Chicago 851 S. Morgan St. Chicago, IL 60607 U.S.A. email: [email protected]

Rahim Moosa Faculty of Mathematics 200 university Ave. W. Waterloo, Ontario N2L 3G1 Canada e-mail: [email protected]

Yerulan Mustafin Institut Girard Desargues Universit´ e Lyon 1 21 Avenue Claude Bernard 69622 Villeurbanne Cedex France e-mail: [email protected]

Margarita Otero Universidad Aut´ onoma de Madrid Departamento Matem´ aticas Facultad de Ciencias, C-XV ctra. de Colmenar Viejo, km. 15 28049 Madrid Spain email: [email protected]

Ya’acov Peterzil Department of Mathematics University of Haifa 31905 Haifa Israel e-mail: [email protected]

Anand Pillay University of Illinois Department of Mathematics 1409 W. Green Street Urbana, IL 61801 U.S.A. e-mail: [email protected]

Fran¸coise Point Universit´ e de Mons-Hainaut Institut de Math´ ematiques (Le Pentagone) 6, Avenue du Champ de Mars 7000 Mons Belgium e-mail: [email protected]

Bruno Poizat Institut Girard Desargues Math´ ematiques, Bˆ atiment 101 Universit´ e Claude Bernard - Lyon 1 69622 Villeurbanne cedex France e-mail: [email protected]

Olivier Roche Mathematisches Institut Abteilung f¨ ur Mathematische Logik Albert-Ludwigs-Universit¨ at Freiburg Eckerstr. 1, D-79104 Freiburg Germany e-mail: [email protected]

Thomas Scanlon University of California, Berkeley Department of Mathematics 910 Evans Hall Berkeley, CA 94720-3840 U.S.A. e-mail: [email protected]

Patrick Speissegger Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada e-mail: [email protected]

Katrin Tent School of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT U.K. email: [email protected]

Frank Wagner Institut Girard Desargues Universit´ e Claude Bernard (Lyon-1) 21 av. Claude Bernard 69622 Villeurbanne C´ edex France e-mail: [email protected]

Alex Wilkie 24-29 St Giles Mathematical Institute University of Oxford Oxford, OX1 3LB U. K. e-mail: [email protected]

Boris Zilber 24-29 St Giles Mathematical Institute University of Oxford Oxford, OX1 3LB U. K. e-mail: [email protected]

Received May 13, 2003 c 2002 by Dipartimento di Matematica della Seconda Universit` ° a di Napoli

Photocomposed copy prepared from a TEX file. ISBN 88-7999-411-5

Contents Introduction A Representation Theorem for a Class of Arboreal Groups ¸Serban A. Basarab Semi-Bounded Relations in Ordered Abelian Groups

1

15

Oleg Belegradek

Simple Groups of Finite Morley rank, 2002

41

Gregory Cherlin

Model Theory of Valued Fields

73

Raf Cluckers

O-minimal Preparation Theorems

87

Lou van den Dries and Patrick Speissegger

Definable Sets in Valued Fields

117

Deirdre Haskell and Dugald Macpherson

Pseudo-Finite Fields and Related Structures

151

Ehud Hrushovski

Tame Minimal Simple Groups of Finite Morley Rank and of Odd Type

213

Eric Jaligot

A History of Interactions between Logic and Number Theory

227

Angus Macintyre

The Mordell-Lang Conjecture in Positive Characteristic Revisited

273

Rahim Moosa and Thomas Scanlon

Sous-Groupes de SL2 (K) sur un Corps Superstable Yerulan Mustafin

297

Some Topological and Differentiable Invariants in O-Minimal Structures – A Survey of the Solution to the Torsion Point Problem

307

Ya’acov Peterzil

Two Remarks on Differential Fields

325

Anand Pillay

Quelques Mauvais Corps de Rang Infini

349

Bruno Poizat

Bad Fields with a Torus of Rank 1

367

Olivier Roche and Frank Wagner

O-Minimal Expansions of the Real Field

379

Patrick Speissegger

Recognizing Groups of Finite Morley Rank via BN-pairs

409

Katrin Tent

Model Theory, Geometry and Arithmetic of the Universal Cover of a Semi-Abelian Variety

427

Boris Zilber

Open Problems in Model Theory

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Mathematics subject classification. A Representation Theorem for a Class of Arboreal Groups: 20E08, 05C25; Semi-Bounded Relations in Ordered Abelian Groups: 03C64, 03C60, 03C10, 20F60; Simple Groups of Finite Morley rank, 2002: 20E32, 20D05, 03C45, 20A15,03C60; Model Theory of Valued Fields: 03C07, 03C98, 03C64, 03C07; O-minimal Preparation Theorems: 03C64, 03C10, 32B05; Definable Sets in Valued Fields: 03C60; Pseudo-Finite Fields and Related Structures: 03C60, 03C45, 12F10; Tame Minimal Simple Groups of Finite Morley Rank and of Odd Type: 20E32, 03C60; A History of Interactions between Logic and Number Theory: 03C20; The Mordell-Lang Conjecture in Positive Characteristic. . . : 11G25, 11G10, 11U09, 14G15; Sous-Groupes de SL2 (K) sur un Corps Superstable: 03C45, 03C60, 20G15; Some Topological and Differentiable Invariants in O-Minimal Structures. . . : 03C64, 22E30; Two Remarks on Differential Fields: 12H05, 12L12; Quelques Mauvais Corps de Rang Infini: 03C45; Bad Fields with a Torus of Rank 1: 03C45, 12L12; O-Minimal Expansions of the Real Field: 03C64, 30D60, 58A17; Recognizing Groups of Finite Morley Rank. . . : 20E42, 03C45, 03C60, 20A15, 20E32; Model Theory, Geometry and Arithmetic of the Universal Cover. . . : 03C45, 03C60, 14K15.

Introduction

In the last few years model theorists have found deep connections between their subject and other areas of mathematics. Our goal in organizing the European Conference on Model Theory and Applications was to assess the state of the art, especially for the younger generation. This conference was also the opportunity to acknowledge, on the occasion of his sixtieth birthday, the influence of Angus Macintyre in the development of our subject. This has been both immense and diverse and we feel it appropriate to mention a few highlights here. Angus’ research has links with algebra, geometry, number theory, asymptotics and even to computational complexity theory and randomized algorithms. His early successes included characterizations of uncountably categorical abelian groups and fields as well as results on the elementary theory of Banach algebras. He also completed the proof of a beautiful equivalence (the “if” direction being due to B H Neumann) between model theoretic and recursion theoretic notions in combinatorial group theory: a finitely generated group can be embedded in every algebraically closed group if and only if it can be recursively presented with solvable word problem. Soon after this period,

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while still making deep contributions to the model theory of groups and rings and, indeed, of arithmetic (both to foundational questions involving Peano Arithmetic and its subtheories, and to the more application orientated “nonstandard number theory”), he established his quantifier elimination result for p-adic fields. As we all now know, this was a pivotal moment in applied model theory and, through the work of Denef and many others, Angus’ insights at that time continue to have deep applications in p-adic diophantine geometry via the theory of Poincar´e series. It does seem remarkable that the original work was done almost thirty years ago. In fact, we list below a chronological selection of his papers so that the reader will be able to appreciate more fully the ramifications of Angus’ ideas on what we now call the model theory of fields and his dominant role within it. Almost all of these papers are extremely familiar to the editors of this volume, but to those not brought up under Angus’ influence may we further single out his definitive work on PAC and regularly closed fields, his synthesis of Ax’s methods for finite fields with difficult analytic number-theoretic results to obtain information on primes in nonstandard models of systems of arithmetic and, of course, his work on exponentiation. Many of his well known results on the decidability and axiomatizability of the real exponential function, and on exponential series, occur in joint papers in the nineties, but it should be pointed out that he had already originated the study of the p-adic analogue many years before this. Also, he was the first to truly believe, in the early eighties, that the real case (Tarski’s problem) was ready for a serious attack: subsequent progress would certainly not have been made were it not for his early papers, the input of numerous preprints and his constant encouragement. You will also see that his more recent research is just as wide-ranging. It covers generic automorphisms of algebraically closed fields, the model theory of analytic functions and its application to sigmoidal neural networks, formal models for exponential asymptotics and model-theoretic aspects of the Weil conjectures for finite fields. But his influence in model theory goes beyond any list of his achievements. It is also due to the excitement he conveys, whether it be in his lectures or just in everyday conversations at meetings around the world. He is well known

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in the community for his accessibility to discuss mathematics, and the circle of his friends and students has always benefitted from his exceptional energy, his enthusiasm and generosity. There is no doubt that through this personal contact he has helped turn our subject into the thriving area that it now is. Luc B´elair Zo´e Chatzidakis Paola D’Aquino Dave Marker Margarita Otero Fran¸coise Point Alex Wilkie

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PH.D. STUDENTS OF ANGUS MACINTYRE • Yale 1973-1985 – Peter Winkler – Larry Manevitz – Emily Grosholz (Philosophy– joint with Korner) – Lee Davidson (Philosophy– joint with Fitch) – Steve Garavaglia – Ken McKenna – Kay Smith – Chico Miraglia – George Loullis – David Marker – Zo´e Chatzidakis – Ali Nesin – Stu Smith – Luc B´elair – Laura Mayer • Oxford 1985-1997 – Frank Wagner – Larry Matthews – Margarita Otero – Paola D’Aquino – Tim Gardener – Charles Morgan • Edinburgh 1997-2004 – Ivan Tomasic – James Gray – Antongiulio Fornasiero

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SELECTED PAPERS OF ANGUS MACINTYRE 1. Weil cohomology and model theory, in Connections between model theory and algebraic and analytic geometry, A. Macintyre ed., Quaderni matematica 6 (2000), 179–199. 2. Logarithmic-exponential power series, J. London Math. Soc. (2) 56 (1997), 417–434. With L. van den Dries and D. Marker. 3. Generic automorphism of fields, Annals Pure Applied Logic 88 (1997), 165–180. 4. Polynomial bounds for VC dimension of sigmoidal and general Pfaffian neural networks, J. Comput. System Sci. 54 (1997), 169–176. With M. Karpinski. 5. On the decidability of the real exponential field, in Kreiseilania: about and around Georg Kreisel, P. Odifreddi ed., AK Peters (1996), 441–467. With A. Wilkie. 6. Definable sets over finite fields, J. Reine Angew. Math 427 (1992), 107– 135. With Z. Chatzidakis and L. van den Dries. 7. Rationality of Poincar´e series: Uniformity in p, Annals Pure and Appl. Logic 49 (1990), 31–74. 8. Degrees of recursively saturated models, Trans. Am. Math. Soc. 282 (1984), 539–554. With D. Marker. 9. Constructive logic versus algebraization, in The L.E.J. Brouwer Centen. Symp. Proc. Conf., North-Holland (1982), 217–260. With G. Kreisel. 10. Decidability and undecidability theorems for PAC-fields, Bull. Am. Math. Soc., New Ser. 4 (1981), 101–104. With G. Cherlin and L. van den Dries. 11. Nonstandard number theory, in Proc. Int. Congr. Math. Helsinki 1978, vol. 1 (1980), 253–262. 12. On definable subsets of p-adic fields, J. Symb. Logic 41 (1976), 605–610.

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13. Model-completeness for sheaves of structures, Fund. Math. 81 (1973), 73–89. 14. The word problem for division rings, J. Symb. Logic 38 (1973), 428–436. 15. On algebraically closed groups, Ann. Math (2) 96 (1972), 53-97. 16. On ω1 −categorical theories of fields, Fund. Math. 71 (1971), 1–25.

A Representation Theorem for a Class of Arboreal Groups S¸erban A. Basarab

Contents 1. Introduction (3). 2. A representation theorem for ⊥–groups (5).

A Representation Theorem for a Class of Arboreal Groups

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1. Introduction In his paper [2], the author introduced a class of groups called median (or arboreal) and used them for the investigation of the partially commutative Artin-Coxeter groups. Recall that by a median or arboreal group we mean a group G together with a ternary operation Y on G making it a median set or generalized tree such that uY (x, y, z) = Y (ux, uy, uz) for all u, x, y, z ∈ G. Notice that, though different, the notion above is similar with the notion of median group introduced by M. Kolibiar in [11]. Setting x ∩ y = Y (x, 1, y), it follows by [2] Proposition 2.2.1. that a median group can be equivalently defined as a group G together with a semilattice operation ∩, whose associated order ⊂ given by x ⊂ y iff x ∩ y = x satisfies the following conditions: 1 ⊂ x for all x ∈ G, z −1 y ⊂ z −1 x whenever x ⊂ y and y ⊂ z, and x−1 (x ∩ y) ⊂ x−1 y for all x, y ∈ G. In a median group G, an ordered n-tuple (x1 , . . . , xn ) ∈ Gn , n ≥ 2, is said to be reduced (written x1 . . . xn = x1 • x2 • . . . • xn ) if x1 . . . xi ⊂ x1 . . . xi+1 −1 for i < n, i.e. x−1 i . . . x1 ∩ xi+1 = 1 for i < n. In particular, an ordered pair (x, y) ∈ G2 is reduced iff x ⊂ xy iff x−1 ∩ y = 1. One checks easily that −1 (x1 , . . . , xn ) is reduced iff (x−1 n , . . . , x1 ) is reduced, and for all x, y ∈ G, x ∩ y is the unique element z of G satisfying x = z • (z −1 x), y = z • (z −1 y) and x−1 y = (x−1 z) • (z −1 y). Note also that for reduced pairs (x, y) and (x, z), xy ∩ xz = x(y ∩ z), in particular, xy ⊂ xz iff y ⊂ z. In other words, x−1 y ∩ x−1 z = x−1 (y ∩ z) whenever x ⊂ y and x ⊂ z. The poset (G, ⊂) admits a partially defined least upper bound (l.u.b.) w.r.t. the partial order ⊂ : x ∪ y = Y (x, y, z) = z(z −1 x ∩ z −1 y), whenever x ⊂ z and y ⊂ z. Set by convention x ∪ y = ∞ whenever the pair (x, y) is not bounded above w.r.t. the order ⊂. A median group G is called locally linear, resp. locally boolean, resp. simplicial if its underlying median set is locally linear, resp. locally boolean, resp. simplicial. ∗

The author is grateful for the kind hospitality during his visit supported by a grant of the Royal Society to the Departments of Mathematics of the QMW College of the University of London and of the University of Edinburgh when this research was partly carried out. He also acknowledges a partial support from the Grant 47/2002 awarded by the Romanian Academy.

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The elements x and y of a median group G are said to be orthogonal (write x ⊥ y) if x ∩ y = 1 and x ∪ y 6= ∞. The median group G is called ⊥–group if the following condition is satisfied: (⊥) x ∪ y = xy = yx whenever x ⊥ y. Given a group G and a set S ⊆ G of generators such that 1 6∈ S and S ∩ S −1 = S1 := {s ∈ S | s2 = 1}, it turns out according to [2] Theorem 2.4.1. that the canonical order on (G, S), given by x ⊂ y iff l(x) + l(x−1 y) = l(y), makes G a ⊥–group if and only if the group G admits the presentation G = < S; s2 = 1 for s ∈ S1 , and sts−1 t−1 = 1 for s, t ∈ S, s 6= t subject to the identity st = ts in G >. Such systems (G, S) were called in [2] partially commutative Artin–Coxeter groups. In particular, for a pair (G, S) satisfying S ∩ S −1 = ∅, the canonical order on (G, S) makes G a ⊥–group iff the group G admits the presentation G =< S; sts−1 t−1= 1 for s, t ∈ S, s 6= t subject to the identity st = ts in G >. Such systems (G, S) were introduced by Baudisch in [3, 4] under the name of semi–free groups, and extensively studied in the last few years under various names (right–angled Artin groups, free partially commutative groups, graph groups) by people working in combinatorial and geometric group theory, associative algebras, computer science (see for instance the works [8–10]). Their nice properties were exploited by Bestvina and Brady [5] in their construction of examples of groups which are of type (F P ) but are not finitely presented, as well as by Crisp and Paris [7] in their proof of a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. According to [8], the finitely generated right–angled Artin groups (moreover, the weakly partially commutative Artin–Coxeter groups as defined in [2]) are linear and hence equationally noetherian. On the other hand, any l–group (G, ., ≤, ∧, ∨), not necessarily commutative, has a canonical structure of median group. Indeed, as the underlying lattice of G is distributive, G has a canonical structure of median set with the median defined by Y (x, y, z) = (x ∧ y) ∨ (y ∧ z) ∨ (z ∧ x) = (x ∨ y) ∧ (y ∨ z) ∧ (z ∨ x). Obviously, the median operation is compatible with the multiplication, and hence, in particular, G becomes a median group. Notice that x ⊂ y iff x+ ≤ y+ and x− ≤ y− , (x ∩ y)+ = x+ ∧ y+ , (x ∩ y)− = x− ∧ y− , where x+ = x ∨ 1, x− =

A Representation Theorem for a Class of Arboreal Groups

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(x−1 )+ = (x ∧ 1)−1 . For x, y ∈ G, x ⊥ y iff x and y are orthogonal as elements of the l–group G, i.e. |x| ∧ |y| = 1, where |x| = x ∨ x−1 = x+ x− . Consequently, G is a ⊥–group by [6] Proposition 3.1.3. According to [2] Theorem 2.4.1., the ⊥–group G is simplicial iff G is abelian, freely generated by the minimal positive elements. It is well known that any l–group is isomorphic to a subdirect product of transitive groups [6] Theorem 4.1.7., in particular, any commutative l–group is isomorphic to a subdirect product of totally ordered abelian groups [6] Corollary 4.1.8. As the structure of ⊥–groups is far more intricate than the structure of the l–groups, we cannot expect to obtain a perfect analogue of the representation theorem above. However we will show in this paper that any ⊥–group naturally admits a simple transitive action on a subdirect product of locally linear median sets. The paper has one section. Some basic notions of the theory of l–groups are suitably extended to ⊥–groups in order to prove the main result of the paper (Corollary 2.8.).

2. A representation theorem for ⊥–groups Lemma 2.1. Assume G is a ⊥–group. Given a reduced decomposition x = x1 • . . . • xn of some element x ∈ G, for any y ∈ G such that y ⊂ x there exist yi ∈ G, i = 1, . . . , n, such that y = y1 • . . . • yn , and yi ⊂ xi , i = 1, . . . , n. Proof – We proceed by induction on n, the case n = 1 being trivial. Assuming n ≥ 2, set x0 = x1 • . . . • xn−1 ⊂ x, y 0 = y ∩ x0 ⊂ x0 . By induction hypothesis there exist yi ∈ G, i = 1, . . . , n − 1 such that y 0 = y1 • . . . • yn−1 and yi ⊂ xi , i = 1, . . . , n − 1. Setting yn = (y 0 )−1 y, we get yn ∩ (y 0 )−1 x0 = 1 and yn ⊂ (y 0 )−1 x = ((y 0 )−1 x0 ) • xn , therefore yn ⊥ (y 0 )−1 x0 . As G is a ⊥–group, it follows that yn ⊂ xn and hence y = y 0 • yn = y1 • . . . • yn−1 • yn , as required. Before considering the general case, let us treat the simpler case of the locally boolean ⊥–groups. Recall that a median set X is locally boolean if any cell C of X equals its boundary ∂C, i.e. for all x, y ∈ X and for every z ∈ [x, y] (i.e. Y (x, y, z) = z) there exists a (unique) u ∈ X such that [x, y] = [z, u]. A rooted

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locally boolean median set (X, 0), i.e. a locally boolean median set X together with a fixed element (the root) 0 ∈ X, has a canonical structure of a quasi– boolean algebra with the partial order x ⊂ y if x ∈ [0, y], the least element 0, the lattice operations x ∩ y = Y (x, 0, y), x ∪ y = the (unique) element z ∈ X for which [x, y] = [x ∩ y, z], and the relative complement (negation) ¬x y = the unique element u for which [0, x] = [x ∩ y, u]. Proposition 2.1. Given a rooted locally boolean median set (X, 0), there exists a unique group law + on X with 0 as neutral element such that X becomes a ⊥–group w.r.t. + and the median operation on X. The group law + is commutative, and 2x = 0 for all x ∈ X. Proof – Define x + y = (¬x y) ∪ (¬y x), the symmetrical difference of the elements x, y ∈ X. One checks easily that X is a 2–elementary ⊥–group w.r.t. + and the median Y . Conversely, assuming . is a group operation on X with 0 as neutral element, such that (X, ., Y ) is a ⊥–group, we have to show that x . y = x + y for all x, y ∈ X. First notice that, assuming x ⊥ y, i.e. x ∩ y = 0 since X is locally boolean, we must have x . y = y . x = x ∪ y = x + y. For arbitrary x, y ∈ X we get x ∪ y 6= ∞, as X is locally boolean, and hence x ∪ y = x(x ∩ y)−1 y = y(x ∩ y)−1 x by the property (⊥). Considering the pair (x∩y, ¬x y) of orthogonal elements, we get x = (x∩y)(¬x y) = (¬x y)(x∩y) for all x, y ∈ X, whence x(x∩y)−1 = ¬x y = (x∩y)−1 x, and hence x∪y = (x∩y)−1 xy, i.e. xy = yx = (x ∩ y)(x ∪ y). Thus the group operation . is commutative. Setting y = x−1 , it follows that 0 = x . x−1 = (x ∩ x−1 )(x ∪ x−1 ). On the other hand, setting y = x ∩ x−1 , we get y −1 = x ∪ x−1 . As y ⊂ y −1 , it follows that y −1 = yz, where z = ¬y−1 y ⊂ y −1 . Consequently, y −1 = yz ⊂ y, therefore x ∩ x−1 = y = y −1 = x ∪ x−1 and hence x = x−1 , i.e. x2 = 0 for all x ∈ X. Finally we obtain x + y = (¬x y) ∪ (¬y x) = (¬x y)(¬y x) = x(x ∩ y)y(x ∩ y) = xy(x ∩ y)2 = x . y, as required. Corollary 2.1. Any locally boolean ⊥–group is isomorphic to a subdirect product of a power set (ZZ/2ZZ)I , with the canonical group and median operations. Proof – According to [1] Theorem 5.2.1., any rooted median set (X, 0) is identified with a subdirect product of the power set (ZZ/2ZZ)I , where I = {P ∈

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Spec(X)| 0 ∈ P, P 6= X}, trough the map x 7→ (xP )P ∈I , where ( 0 mod 2 if x ∈ P xP = 1 mod 2 otherwise Recall that Spec(X) = {P ⊆ X|P and X \ P are convex }. Assuming G is a locally boolean ⊥–group, it remains to observe that any P ∈ Spec(G) satisfying 0 ∈ P, P 6= G, is a subgroup of G too, and G/P ∼ = ZZ/2ZZ. As in the case of l–groups, the convex subgroups play a special role in the study of the ⊥–groups. The next lemma provides a first argument in this sense. Lemma 2.2. If G is a ⊥–group and H is a convex subgroup of G, then the quotient set G/H inherits from G a canonical structure of median set on which G acts from the left. Proof – First of all notice that xH = (x ∪ y)H and Hx = H(x ∪ y) whenever x ∈ G, y ∈ H and x ∪ y 6= ∞. Indeed, as G is a ⊥–group, we obtain x ∪ y = x(x ∩ y)−1 y = y(x ∩ y)−1 x ∈ xH ∩ Hx since z := x ∩ y ∈ H, as H is convex and 1, y ∈ H, therefore z −1 y, yz −1 ∈ H, as H is a subgroup of G. To prove the lemma we have to show that Y (x, y, zh)H = Y (x, y, z)H for all x, y, z ∈ G, h ∈ H. We get Y (x, y, zh) = zY (z −1 x, z −1 y, h) = z((z −1 x ∩ z −1 y) ∪ h0 ), where h0 = (z −1 x ∩ h) ∪ (z −1 y ∩ h) ∈ H since 1 ∈ H, h0 ⊂ h ∈ H, and H is a convex subset of G. Thanks to the remark above, it follows that Y (x, y, zh)H = z(z −1 x ∩ z −1 y)H = Y (x, y, z)H as desired. Since the intersection of a family of convex subgroups is a convex subgroup too, we may speak of the convex subgroup generated by any subset A ⊆ G and denote it by C(A). A more precise description of C(A) is provided by the next statement. Proposition 2.2. Given a subset A of the ⊥–group G, the convex subgroup C(A) generated by A is the subgroup generated by [A∪{1}], the convex closure of the subset A ∪ {1}. Proof – It suffices to show that for any A ⊆ G for which 1 ∈ A and y ∈ A whenever y ⊂ x and x ∈ A, the subgroup B :=< A > generated by A is convex, i.e. the following two conditions are satisfied:

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(i) y ⊂ x and x ∈ B imply y ∈ B, and (ii) x ∪ y ∈ B whenever x ∪ y 6= ∞, x ∈ B and y ∈ B. First notice that we have only to verify (i), since assuming (i) and x, y ∈ B such that x ∪ y 6= ∞, we get x ∪ y = x(x ∩ y)−1 y, as G is a ⊥–group, and hence x∪y ∈ B since x∩y ∈ B by (i). Let l denote the length function assigned to the pair (B, A\{1}), i.e. for any x ∈ B, l(x) is the minimal length of any expression x = a1 . . . ad , with ai ∈ (A ∪ A−1 ) \ {1}, i = 1, . . . , d. Assuming x ∈ B and y ⊂ x, we show by induction on l(x) = d that y ∈ B. The case d = 0, i.e. x = 1, is trivial, so assume d ≥ 1, x = a1 . . . ad . Setting x0 = a1 . . . ad−1 , let u = (x0 )−1 ∩ ad , x0 = x00 • u−1 , ad = u • b, x = x00 • b. As G is a ⊥–group, and y ⊂ x, we get by Lemma 2.1. y = y 0 •y 00 , with y 0 = y∩x00 ⊂ x00 ⊂ x0 , and y 00 ⊂ b. Since y 0 ⊂ x0 and l(x0 ) = d − 1 < d, it follows by the induction hypothesis that y 0 ∈ B. On the other hand, b−1 y 00 ⊂ b−1 ⊂ a−1 d . We distinguish two cases: Case (1): a−1 ∈ A \ {1}. Then, by the choice of A, we obtain b−1 ∈ A, and d −1 00 00 b y ∈ A, therefore y ∈ B. Case (2): ad ∈ A \ {1}. Then ad b−1 ⊂ ad b−1 y 00 ⊂ ad , therefore, by the choice of A, ad b−1 ∈ A and (ad b−1 )y 00 ∈ A, and hence y 00 ∈ B. Thus in the both cases y 00 ∈ B. Since y 0 ∈ B too, it follows that y = y 0 y 00 ∈ B, as desired. Corollary 2.2. For all x ∈ G, the convex subgroup C(x) generated by x is the subgroup of G generated by [1, x]. Corollary 2.3. Let H be a convex subgroup of the ⊥–group G, and let x ∈ G. Then C(H ∪ {x}) is the subgroup generated by H ∪ [1, x]. Proof – One checks easily that the convex closure [H ∪ {x}] of H ∪ {x} is the union of the cells [h, x], where h ranges over H. Thus any element y ∈ [H ∪ {x}] has the form y = Y (h, x, y) = (h ∩ y) ∪ (x ∩ y) for some h ∈ H. As H is convex, h ∩ y ∈ H, so [H ∪ {x}] = {h ∪ y | h ∈ H, y ⊂ x, h ∪ y 6= ∞} ⊆ S {yH | y ⊂ x} by the remark from the beginning of the proof of Lemma 2.2. The conclusion of the statement is now immediate by Proposition 2.2. Denote by C(G) the poset with respect to inclusion of all convex subgroups of a ⊥–group G.

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V T W Proposition 2.3. C(G) is a complete lattice with Hi = Hi , Hi = S S < Hi >, the submonoid generated by Hi . Moreover the lattice C(G) is W W distributive, and H ∩ ( Hi ) = (H ∩ Hi ) for arbitrary families of convex subgroups (Hi )i∈I . W S Proof – To prove that Hi =< Hi >, let M denote the subset conS sisting of those x ∈ G for which there exist n ∈ IN and xj ∈ Hi , j = 1, . . . , n S S such that x = x1 • x2 . . . • xn . As Hi ⊆ M ⊆ < Hi >, it suffices to show that M is a convex subgroup. Since M is closed under inversion and G is a ⊥–group, we have to show that y ∈ M whenever y ⊂ x and x ∈ M , and M is closed under multiplication. Assuming x ∈ M of the form above and y ⊂ x, we get y = y1 •. . .•yn with yj ⊂ xj , j = 1, . . . , n by Lemma 2.1.. As the subgroups Hi are convex it follows that y ∈ M . On the other hand, assuming x, y ∈ M , set z = x−1 ∩ y. As xz ⊂ x ∈ M , and y −1 z ⊂ y −1 ∈ M , we get xz, z −1 y ∈ M and hence xy = (xz) • (z −1 y) ∈ M , as required. It remains to prove the latter equality of the statement, implying in particular the distributivity of the latW tice C(G). One inclusion is obvious, so let x ∈ H ∩ ( Hi ), say x = x1 . . . xn , S with xj ∈ Hi , j = 1, . . . , n. Assuming by the first part of the proof that the S decomposition of x above is reduced, we obtain xj ∈ H ∩ ( Hi ), j = 1, . . . , n, S W therefore x ∈< Hi >= Hi , as desired. Remark 2.1 - In general, the operation ∨ is not distributive w.r.t. an infinite intersection. For instance, let G = ZZIN with the arboreal structure induced by the canonical l–group structure on G, H = ZZ(IN) , Hi = {x = (xj ) ∈ G | xi = T T 0}. Then H ∨ ( i∈IN Hi ) = H 6= G = i∈IN (H ∨ Hi ). The following statement provides natural examples of convex subgroups. Proposition 2.4. For any subset A of a ⊥–group G, the set A⊥ :={y ∈ G | y ⊥ x for all x ∈ A} is a convex subgroup of G. Proof – Obviously y ∈ A⊥ whenever y ⊂ x and x ∈ A⊥ . Notice also that A⊥ is closed under inversion by [2] Lemma 2.2.4., so it remains to show that A⊥ is closed under multiplication. Let x, y ∈ A⊥ . Writing xy = x0 • y 0 , where x0 = x(x−1 ∩ y) ⊂ x ∈ A⊥ , (y 0 )−1 = y −1 (x−1 ∩ y) ⊂ y −1 ∈ A⊥ , we get x0 , y 0 ∈ A⊥ , so we may assume from the beginning that xy = x • y. For

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any a ∈ A, we have to show that xy ⊥ a. Setting b = xy ∩ a ⊂ x • y, we get b ⊥ x since b ⊂ a and a ⊥ x. Consequently, b ⊂ a ∩ y = 1, i.e. b = 1, as required. Similarly we get a−1 ∩ xy = 1, therefore axy = a • x • y = x • y • a, i.e. xy ∪ a = axy 6= ∞ and hence xy ⊥ a, as desired. Corollary 2.4. For A ⊆ G, A⊥ = C(A)⊥ . Proof – The inclusion C(A)⊥ ⊆ A⊥ is obvious. Conversely, let b ∈ A⊥ , i.e. A ⊆ b⊥ . As b⊥ ∈ C(G) by Proposition 2.4., we obtain C(A) ⊆ b⊥ , i.e. b ∈ C(A)⊥ , and hence A⊥ ⊆ C(A)⊥ , as required. Corollary 2.5. For A ⊆ G, A ⊆ A⊥⊥ , and A⊥ = A⊥⊥⊥ . Corollary 2.6. A = B ⊥ for some B ⊆ G iff A = A⊥⊥ . Corollary 2.7. Assuming G is a ⊥–group, let x, y ∈ G be such that x∪y 6= ∞. Then C(x) ∩ C(y) = C(x ∩ y) and C(x) ∨ C(y) = C(x ∪ y). In particular, C(x) ∩ C(y) = {1} and C(x) ∨ C(y) = C(xy) = C(yx) whenever x ⊥ y. Proof – The equality C(x) ∨ C(y) = C(x ∪ y) is immediate since x ∪ y = x(x∩y)−1 y. To prove the equality C(x)∩C(y) = C(x∩y), setting z := x∩y, x = z • x0 , y = z • y 0 , we get x0 ⊥ y 0 as x ∪ y 6= ∞. Thus C(x) = C(z) ∨ C(x0 ) and C(y) = C(z) ∨ C(y 0 ), and hence C(x) ∩ C(y) = C(z) ∨ (C(x0 ) ∩ C(y 0 )) by Proposition 2.3., so it remains to show that C(x) ∩ C(y) = {1} whenever x ⊥ y. As x ∈ x⊥⊥ , it follows by Proposition 2.4. that C(x) ⊆ x⊥⊥ , and similarly C(y) ⊆ y ⊥⊥ , so it suffices to check that x⊥⊥ ∩ y ⊥⊥ = {1} whenever x ⊥ y. Now, if u ∈ x⊥⊥ ∩ y ⊥⊥ , it follows that u ∈ y ⊥ since u ∈ x⊥⊥ and y ∈ x⊥ . Consequently, u ⊥ u, i.e. u = 1, as u ∈ y ⊥⊥ . The next lemma is immediate. Lemma 2.3. The following are equivalent for a convex subgroup H of a ⊥– group G : (i) For x, y ∈ G such that x ∪ y 6= ∞, x ∩ y ∈ H implies either x ∈ H or y ∈ H. (ii) For x, y ∈ G, x ⊥ y implies either x ∈ H or y ∈ H. (iii) For x, y, z ∈ G either (x ∩ y)H = (y ∩ z)H or (y ∩ z)H = (z ∩ x)H or (z ∩ x)H = (x ∩ y)H.

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(iv) G/H is a locally linear median set. Proof – (i) −→ (ii) is obvious. (ii) −→ (iii): As x ∩ y, y ∩ z, z ∩ x ⊂ Y (x, y, z), it follows that u−1 (x ∩ y) ⊥ u−1 (y ∩ z), u−1 (y ∩ z) ⊥ u−1 (z ∩ x) and u−1 (z ∩ x) ⊥ u−1 (x ∩ y), where u = x ∩ y ∩ z. The implication (ii) −→ (iii) is now immediate. (iii) −→ (iv): Let x, y, z ∈ G be such that xH ⊂ zH, i.e. xH = (x ∩ z)H, and yH ⊂ zH, i.e. yH = (y ∩ z)H. By assumption it follows that either (x ∩ y)H = (y ∩ z)H = yH or yH = (y ∩ z)H = (z ∩ x)H = xH or xH = (z ∩ x)H = (x ∩ y)H, and hence either xH ⊂ yH or yH ⊂ xH, i.e. the median set G/H is locally linear. (iv) −→ (i): Assuming that x ∪ y 6= ∞ and x ∩ y ∈ H, it follows that xH ∈ [H, zH], yH ∈ [H, zH], where z = x ∪ y, and hence either xH = (x ∩ y)H = H, i.e. x ∈ H, or yH = (x ∩ y)H = H, i.e. y ∈ H, as required. Call prime any convex subgroup H of a ⊥–group G satisfying the equivalent conditions from Lemma 2.3. Notice that a prime convex subgroup H is not necessarily prime as a convex subset of the median set G, i.e. G \ H is not T necessarily a convex subset. Since i∈I Hi is obviously prime whenever (Hi )i∈I is a chain of prime convex subgroups, it follows by Zorn’s lemma that any prime convex subgroup contains a minimal prime. Notice also that any convex subgroup H 0 lying over a prime convex subgroup H is obviously prime. As in the case of l–groups, call regular any convex subgroup H of G such that T H = i∈I Hi , where Hi ∈ C(G), i ∈ I, implies H = Hi for some i ∈ I. Lemma 2.4. Any regular convex subgroup H of a ⊥–group G is prime. Proof – Let x, y ∈ G be such that x ⊥ y, and set H1 = H ∨ C(x), H2 = H ∨ C(y). By Proposition 2.3. and Corollary 2.7. we obtain H1 ∩ H2 = H ∨ (C(x) ∩ C(y)) = H, therefore, by the regularity of H, either H1 = H, i.e. x ∈ H, or H2 = H, i.e. y ∈ H. For any element a ∈ G\{1}, let M(a) denote the set of those convex subgroups H of G which are maximal with the property that a 6∈ H. The next lemma provides an equivalent description for regular convex subgroups.

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Lemma 2.5. The necessary and sufficient condition for a convex subgroup H 6= G to be regular is that there exists a ∈ G \ {1} such that H ∈ M(a). T Proof – Assuming H is regular, set H ∗ = {U ∈ C(G) | H ⊆ U, U 6= H}. As H is regular, H 6= H ∗ , so H ∈ M(a) for all a ∈ H ∗ \ H. Conversely, T assume H ∈ M(a) for some a ∈ G \ {1}, and H = i∈I Hi , Hi ∈ C(G). Let i ∈ I be such that a 6∈ Hi . Since H is maximal amongst the convex subgroups which do not contain a, it follows that H = Hi , and hence H is regular. The next statement is now immediate by Zorn’s lemma. Proposition 2.5. For any convex subgroup H of a ⊥–group G and for any a ∈ G \ H, there exists M ∈ M(a) such that H ⊆ M . In particular, any convex subgroup H is an intersection of regular, resp. of minimal prime convex subgroups. Corollary 2.8. Given a ⊥–group, there exists a canonical simple transitive action of G on a subdirect product of locally linear median sets. Proof – Let Pmin (G) denote the set of the minimal prime convex subgroups of G. By Proposition 2.5., G is naturally embedded as median set Q into the product G/H of locally linear median sets, on which G acts H∈Pmin (G)

canonically from the left.

References [1] Basarab, S ¸ . A.: The dual of the category of generalized trees, An. S¸t. Univ. Ovidius Constant¸a 9 (1) (2001), 1–20. [2] Basarab, S ¸ . A.: Partially commutative Artin–Coxeter groups and their arboreal structure, J. Pure Appl. Algebra 176 (1) (2002), 1–25. [3] Baudisch, A.: Kommutationsgleichungen in Semifreien Gruppen, Acta Math. Acad. Sci. Hungaricae (3–4) 29 (1977), 235–249.

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[4] Baudisch, A.: Subgroups of semifree groups, Acta Math. Acad. Sci. Hungaricae (1–4) 81 (1981), 19–28. [5] Bestvina, M. and Brady, N.: Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 470–495. [6] Bigard, A., Keimel, K. and Wolfenstein, S.: Groupes et Anneaux R´eticul´es, Lecture Notes Math. 608, Springer, Heidelberg (1977). [7] Crisp, J. and Paris, L.: The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group, Invent. Math. 145 (2001), 19–36. [8] Davis, M. and Januszkiewicz, T.: Right–angled Artin groups are commensurable with right–angled Coxeter groups, J. Pure Appl. Algebra 153 (2001), 229–235. [9] Duchamp, G. and Krob, D.: The free partially commutative Lie algebra: bases and ranks, Adv. Math. (1) 95 (1992), 92–126. [10] Duchamp, G. and Krob, D.: Free partially commutative structures, J. Algebra (2) 126 (1993), 318–361. [11] Kolibiar, M.: 73–82.

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