Groups and Model Theory: GAGTA BOOK 2 9783110719710, 9783110719666

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Olga Kharlampovich, Rizos Sklinos (Eds.) Groups and Model Theory

Also of Interest Complexity and Randomness in Group Theory. GAGTA BOOK 1 Frédérique Bassino, Ilya Kapovich, Markus Lohrey, Alexei Miasnikov, Cyril Nicaud, Andrey Nikolaev, Igor Rivin, Vladimir Shpilrain, Alexander Ushakov, Pascal Weil, 2020 ISBN 978-3-11-066491-1, e-ISBN (PDF) 978-3-11-066702-8, e-ISBN (EPUB) 978-3-11-066752-3 Ordinal Computability. An Introduction to Infinitary Machines Merlin Carl, 2019 ISBN 978-3-11-049562-1, e-ISBN (PDF) 978-3-11-049615-4, e-ISBN (EPUB) 978-3-11-049291-0

Elementary Theory of Groups and Group Rings, and Related Topics. Proceedings of the Conference held at Fairfield University and at the Graduate Center, CUNY, November 1-2, 2018 Paul Baginski, Benjamin Fine, Anja Moldenhauer, Gerhard Rosenberger, Vladimir Shpilrain (Eds.), 2020 ISBN 978-3-11-063673-4, e-ISBN (PDF) 978-3-11-063838-7, e-ISBN (EPUB) 978-3-11-063709-0 Sets, Functions, Measures, Volume 1: Fundamentals of Set and Number Theory Valeriy K. Zakharov, Timofey V. Rodionov, 2018 ISBN 978-3-11-055008-5, e-ISBN (PDF) 978-3-11-055094-8, e-ISBN (EPUB) 978-3-11-055021-4 Philosophy of Mathematics Thomas Bedürftig, Roman Murawski, 2018 ISBN 978-3-11-046830-4, e-ISBN (PDF) 978-3-11-046833-5, e-ISBN (EPUB) 978-3-11-047077-2

Groups and Model Theory |

GAGTA BOOK 2 Edited by Olga Kharlampovich, Rizos Sklinos

Editors Prof. Olga Kharlampovich CUNY Graduate Center and Hunter College 695 Park Ave New York 10065 USA [email protected]

Prof. Rizos Sklinos Stevens Institute of Technology School of Engineering & Science Hoboken USA [email protected]

ISBN 978-3-11-071966-6 e-ISBN (PDF) 978-3-11-071971-0 e-ISBN (EPUB) 978-3-11-071976-5 Library of Congress Control Number: 2021930949 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2021 Walter de Gruyter GmbH, Berlin/Boston Cover image: tatadonets / iStock / Getty Images Plus [M] Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Introduction The goal of this book is to show some directions in group theory motivated by logic and model theory. These directions include stable groups and generalizations, model theory of nonabelian free groups and rigid solvable groups, pseudofinite groups, approximate groups, topological dynamics, groups interpreting the arithmetic, decidability/undecidability of theories of groups and Diophantine problems in groups. The title (Groups and Model Theory, GAGTA book 2) reflects the fact that the book follows the course of the GAGTA (Geometric and Asymptotic Group Theory with Applications) conference series. The first book, “Complexity and Randomness in Group Theory. GAGTA book 1,” was published by De Gruyter in 2020. The model theory of groups has always held a special position within the discipline of model theory. As in many other mathematical domains, groups play an important role in model theory. There are two main points of view on the subject that in many cases cross-cut and are not always clearly distinguishable. We give a brief account of the major goals in each case. The “applied” viewpoint is concerned with particular first-order theories, usually important and well-studied in other mathematical disciplines. The objective is to investigate what kind of groups are definable in the given first-order context or ambient structure. It is important to note that groups often come equipped with extra structure, e. g., algebraic groups or Lie groups, so the above study expands the understanding of the additional structure a definable group may have. It is usually the case that one can give a structure to a group that coincides with the expected structure when studied in some ambient model. For example, any definable group in an algebraically closed field can be equipped with the structure of an algebraic group. In many cases this takes some nontrivial effort to prove. Examples of theories that have been thoroughly studied along these lines are the theory of algebraically closed fields and the theory of real closed fields. The “pure” viewpoint centers around properties usually coming from Shelah’s classification theory, e. g., stability, superstability, etc., and studies classes of groups that share these properties. In particular, the study of stable groups was largely initiated by Poizat, borrowing notions, like genericity and connectedness, from algebraic geometry. This has been, indeed, a very fruitful field of research for model theorists. The interest of research on stable groups gradually moved to larger and “wilder” classes, like groups without the independence property or simple (in the sense of model theory) groups, but the guiding force has always been the results in the stable context. In yet another direction of pure model theory, motivated by results and conjectures of Zilber, model theorists studied pregeometries arising from suitable definable sets. This area is known as “geometric stability theory” and on the pure side has contributed many applications to “core” mathematics. In this context, questions of what kind of groups are definable in a first-order theory became relevant and revealed https://doi.org/10.1515/9783110719710-201

VI | Introduction much information about the complexity of the always present (in a stable theory) independence relation. It is hard to put the research collected in this book in a comprehensive chart, but our effort is to present the most current trends in the model theory of groups. A definitely conspicuous example, after the positive answer to Tarski’s question, is the model theory of nonabelian free groups and more generally, as the methods developed allowed, the model theory of hyperbolic groups. This study is incorporated in both points of view mentioned earlier. The first-order theories of torsion-free hyperbolic groups are stable and, in some sense, are prototypical examples of stable group theories (since noncyclic torsion-free hyperbolic groups fail to be superstable). In addition, hyperbolic groups have been the center of study of a still growing mathematical discipline – geometric group theory. It is a “new leaf”, a promising new subject that connects two seemingly, and up to some years ago indeed disconnected, disciplines – model theory and geometric group theory. The current trend shows that this interaction rapidly expands to include the model-theoretic study of more classes of groups that traditionally fall within the scope of geometric group theory. The first survey, by A. Pillay, is an almost complete account of what has happened in the model theory of groups in the last decades, as well as what the current trends are. Some specialized topics, like simple groups of finite Morley rank, are not discussed, but apart from that almost all aspects, developments, underlying ideas, and motivation behind the vast amount of research conducted on the subject are recorded. Both pure and applied points of view are discussed. The second survey, by R. Sklinos, specializes in the model theory of nonabelian free groups, which is the most representative example of a stable group. After the work of Kharlampovich–Myasnikov and Sela answering positively Tarski’s question, as well as the surprising result by Sela that the free group is stable, the common theory of nonabelian free groups has attracted a lot of interest. The article discusses the complexity of forking independence in this particular theory. Although the understanding of definable sets is only partial, certain advances are still possible. In particular, one can prove that forking independence is as complicated as possible, i. e., n-ample for all n. This is typically the situation when an infinite definable field is present, which is not the case in a free group. It was proved by Sela and Kharlampovich–Myasnikov that every formula in the theory of a nonabelian free group F is equivalent to a boolean combination of ∃∀-formulas. Kharlampovich–Myasnikov also proved that the elementary theory of a free group is decidable (there is an algorithm to decide, given a sentence, whether this sentence belongs to Th(F)). In the third survey here, they describe a detailed algorithm for the reduction of a first-order formula over a free group to the equivalent boolean combination of ∃∀-formulas. The survey by Kharlampovich, Myasnikov, and Sohrabi is about the so-called rich groups and algebras. In these groups, the language of the first-order logic has the same expressive power as the language of the weak second-order logic. For example, all

Introduction

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finitely generated subgroups in rich groups are definable. Rich groups (algebras) include GLn (ℤ), SLn (ℤ), Tn (ℤ), n ≥ 3, various finitely generated metabelian groups (for example, the free nonabelian ones), many polycyclic groups, free associative algebras, free group algebras over infinite fields, and many others. A question dominating research on elementary equivalence of finitely generated groups and rings has been whether, and when elementary equivalence between finitely generated groups (rings) implies isomorphism. Recently, Avni, Lubotzky, and Meiri coined the term first-order rigidity: a finitely generated group (ring) A is first-order rigid if any other finitely generated group (ring), which is elementarily equivalent to A, is isomorphic to A. All the rich groups mentioned above are first-order rigid. The survey by Romanovskii describes work (much of it joint with Myasnikov) on algebraic geometry and model-theoretical aspects of the theory of rigid solvable groups. Axioms were found for classes of m-rigid groups and divisible m-rigid groups. The completeness of the theory of divisible m-rigid groups, decidability, and ω-stability was proved; saturated models were described, and other questions were investigated. In many ways, the theory of divisible m-rigid groups turned out to be similar to the classical theory of algebraically closed fields. For m ≥ 2, divisible m-rigid groups are examples of ω-stable solvable groups of infinite Morley rank.

Contents Introduction | V Anand Pillay 1 Model theory and groups | 1 Rizos Sklinos 2 Independence and interpretable structures in nonabelian free groups | 51 Olga Kharlampovich and Alexei Myasnikov 3 Quantifier elimination algorithm to boolean combination of ∃∀-formulas in the theory of a free group | 87 Olga Kharlampovich, Alexei Myasnikov, and Mahmood Sohrabi 4 Rich groups, weak second-order logic, and applications | 127 Nikolay Romanovskii 5 Rigid solvable groups. Algebraic geometry and model theory | 193 Index | 231

Anand Pillay

1 Model theory and groups Abstract: This paper is about various ways in which groups arise or are of interest in model theory. In Section 1.1, I briefly introduce three important classes of first-order theories: stable theories, simple theories, and NIP theories. Section 1.2 is about the classification of groups definable in specific theories or structures, mainly fields, and the relationship to algebraic groups. In Section 1.3, I study generalized stability and definable groups in more detail, giving the theory of “generic types” in the various contexts. I also discuss 1-based theories and groups. Section 1.4 is about the compact Hausdorff group G/G00 attached to a definable group and how it may carry information in various contexts (including approximate subgroups and arithmetic regularity). In Section 1.5, I discuss Galois theory, including the various Galois groups attached to first-order theories, various kinds of strong types, and definable groups of automorphisms. In Section 1.6, I study various points of interaction between topological dynamics and definable groups, in particular “Newelski’s conjecture” relating G/G00 to the “Ellis group”. And in Section 1.7, I touch on the model theory of the free group.

1.1 Introduction and preliminaries I will discuss several points of interaction between model theory and group theory. This is not so much about explicit applications of model theory to group theory, as about the model-theoretic perspective on groups, and some mathematical implications. It is a personal account, influenced by my own interests, preoccupations, and mathematical trajectory, and includes both established material, as well as some fairly recent developments, although there will be many important and topical things that I will say nothing about. Model theory studies first-order theories T, often complete. The study of specific first-order theories, such as set theory, or differentially closed fields, can be identified with “applications” of model theory, whereas the study of broad classes of first-order theories (such as all theories, or stable theories) is what is often considered as “pure” model theory. There are various invariants of a first-order theory T. One is the category Mod(T) of models of T (where the morphisms are elementary embeddings). Most of the early material in a basic model theory course (Lowenheim–Skolem, prime models, saturated models, omitting types, Morley’s theorem) concerns this category. Another invariant is Def(T), the category of definable sets. And this Def(T) can be thought of Note: Supported by NSF grants DMS 1665035 and DMS-1760212. Anand Pillay, University of Notre Dame, Notre Dame, USA, e-mail: [email protected] https://doi.org/10.1515/9783110719710-001

2 | A. Pillay syntactically or semantically. Syntactically the objects are formulas ϕ(x)̄ (in the language of T, and where x̄ denotes the tuple of free variables in ϕ) up to equivalence modulo T. The morphisms are given by formulas ψ(x,̄ y)̄ such that T says that ψ is the graph of a (partial) function. The semantic interpretation is the obvious thing and coincides with the syntactic interpretation when T is complete. In many mathematical categories, such as the categories of algebraic varieties and differentiable manifolds, the group objects (algebraic groups, Lie groups, respectively) play an important role. And the same holds for the category Def(T). The group objects of Def(T) are what we call the groups definable in T. In many cases, especially on the stability side of model theory, definable groups arise naturally. For example, if some definable relationship between definable sets X and Y needs parameters to be seen, then a nontrivial definable group appears (as a “definable automorphism group”). The appearance of such groups was behind Hrushovski’s positive answer to Shelah’s question whether “unidimensional stable theories are superstable.” What we call stable group theory is the machinery of stability theory (independence, forking,…) in the presence of a definable group action or definable group operation. It is considered part of “general stability,” and rather recently the fundamental theorem of stable group theory was seen to translate into a strong “arithmetic regularity” theorem for finite groups G equipped with a distinguished subset A, under a stability assumption on A [10] (see also Section 1.4.5). From the point of view of geometric stability theory and its generalizations, whether or not (infinite) groups are definable in a theory T is one measure of the complexity of T. And assuming there are such infinite definable groups G, the complexity of the class of definable (in the ambient theory) subsets of G, G × G, etc., is another measure of complexity of T. There are many things that I will not explicitly include in this paper. One is automorphism groups of ω-categorical structures, which was an important subject in the 1980s, and has recently come back into prominence via Ramsey theory and topological dynamics. Another is the body of work around the Cherlin–Zilber conjecture that simple groups of finite Morley rank are algebraic groups (over algebraically closed fields), which turned into a highly specialized subject with a tentative connection to the rest of stability theory. One more is the model theory of modules which again became a rather niche area. Groups of finite Morley rank (in the group language or with appropriate additional structure) and R-modules (in the usual language which has functions for scalar multiplication by elements of R) are examples of stable groups. Another excluded topic is Hrushovski’s group configuration theorem which gives an abstract model-theoretic context for recovering definable groups (see Chapter 5 of [55]). The proofs of function field Mordell–Lang conjecture in [19] are also an application of stable group theory, or rather its incarnation in specific stable theories. The really new theorem is in positive characteristic which I will not discuss, although I will mention the characteristic 0 case in Section 1.2.4.

1 Model theory and groups | 3

Pseudofinite groups (more or less, ultraproducts of finite groups) will be touched on in a few ways, although not systematically. (But see Section 1.4.5.) An early application of the model theory of groups definable in pseudofinite fields (and more generally groups definable in simple theories) to algebraic groups over finite fields (strong approximation) was in [27] but will not be discussed in detail here. At the minimum I would hope that this article shows or reveals the flexibility of model theory in dealing with many different mathematical topics and environments, from a common point of view or standpoint. I will assume a basic knowledge of model theory: languages L, formulas, firstorder theories T, structures, definable sets, types, elementary maps, saturated models, as well as “imaginaries.” See [43] or [73], or my course notes [59]. Unless said otherwise, variables x, y range over finite tuples in the case of the traditional case of 1-sorted structures, and over arbitrary sorts in the many-sorted case (including T eq ). Notions such as quantifier elimination and model completeness are connected with the study of specific theories in a specific language or vocabulary; T has quantifier elimination if every formula ϕ(x)̄ is equivalent, modulo T to a quantifier-free for̄ And T is model complete if whenever M is a substructure of N, and both mula ψ(x). are models of T, then M is an elementary substructure of N (M ≺ N). Quantifier elimination implies model completeness. One can always force quantifier elimination by the process of Morleyization: adjoining new relation symbols for all formulas (as well as their tautological definitions). So from the point of view of model theory, in and for itself, we can always assume that the theory in question has quantifier elimination, but not for the study of specific theories in a specific language. Given a complete theory T, we typically work inside a sufficiently saturated ̄ Namely, all models we consider are assumed to be model of T, which we call M. ̄ Sufficiently saturated means κ-saturated and strongly elementary substructures of M. κ-homogeneous for some large κ. Or if the reader doesn’t mind assuming some set theory, just κ-saturated of cardinality κ. We will often identify definable sets with the formulas defining them. Definable sets are defined with parameters unless stated otherwise. We sometimes say A-definable to mean definable with parameters from ̄ For a formula with parameters b, we sometimes write it as a set A (in the model M). ϕ(x, b) to specify the parameters b (where ϕ(x, y) is an L-formula, and ϕ(x, b) is the result of substituting constants for the elements of b for y). Concerning algebraic geometry, we will typically take the naive point of view of algebraic varieties as point sets, that is, sets of points of finite tuples in a field, defined by finite systems of equations. The geometric point of view means considering such point sets in an ambient algebraically closed field. In general, an algebraic variety (in the sense of Weil, for example) is obtained by gluing together finitely many affine varieties along suitable transition maps (analogous to how real topological or differential manifolds are built); the transition maps are isomorphisms in the sense of algebraic geometry between Zariski open subsets of the various charts. This Weil point of view

4 | A. Pillay is thematically close to the model-theoretic point of view of definable sets as point sets in a saturated model, and is given a systematic treatment in [65], repeated in [54]. Alternatively, a variety over a field k means an integral separated scheme of finite type over k. I would like to thank an anonymous referee for some crucial mathematical corrections, as well as many typographical corrections. Thanks also to Daniel Hoffmann, Purbita Jana, and Chieu-Minh Tran for their corrections, comments, and suggestions.

1.1.1 Stability The complete theory T is said to be stable if there do not exist a formula ϕ(x, y), a model M of T, and tuples ai , bi from M for i < ω such that M 󳀀󳨐 ϕ(ai , bj ) iff i ≤ j. By compactness this is equivalent to saying that for each L-formula ϕ(x, y) there is k such that some (any) model M of T the formula ϕ(x, y) (i. e., the bipartitite graph defined by ϕ(M)) omits the k-half graph. In the last case, we say that the formula ϕ(x, y) is k-stable. Stability and stable groups (groups definable in stable theories) will be discussed below, but in the meantime let us mention a certain strengthening that will be relevant. Following Morley, T is said to be total transcendental (t.t.), or in the case of a countable language, ω-stable, if for any model M of T, the Cantor–Bendixson rank on the Boolean algebra of formulas in any given finite number of variables, and with parameters in M, is “defined,” namely ordinal-valued. When M is an ω-saturated model of T, this Cantor–Bendixson rank is precisely the Morley rank of the formula. The ω-stability of a complete theory T (in a countable language) was related to uncountable “categoricity” in Morley’s work [45], where T is said to be κ-categorical if any two models of T of cardinality κ are isomorphic. This motivated (I guess) introduction of the general notion of stability, by Shelah, which played a central role in Shelah’s program to classify theories according to whether or not there is some classification of their models by, roughly speaking, cardinal invariants [69]. In the meantime, stable theories have been understood as the “logically perfect” first-order theories. Moreover, the notion of stability of a formula (or bipartitite graph) is now seen as a pervasive notion in mathematics, appearing in Grothendick’s 1952 thesis, as well as subsequent work on topological dynamics, which subsumes stable group theory. Superstability is a property in-between stability and ω-stability. It is characterized by the existence of a certain ordinal-valued continuous rank on types and formulas: R(θ(x)) ≥ α + 1 if there are unboundedly many complete types p (say over the monster model) which contain θ and with R(p) ≥ α. Countable, nonsuperstable theories have 2κ models of cardinality κ for all κ > ω, hence in the classification problem, T can be assumed to be superstable.

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There are very few “natural” stable theories and/or groups; the theory of an infinite set with only equality, the theory of any algebraically closed field (in the ring language), the theory of any abelian group (in the group language), the theory of differentially closed fields, and recently the theory of the (noncommutative) free group.

1.1.2 Simplicity The machinery used to understand models of stable theories and definable sets in stable theories is called “stability theory.” The invention of this machinery by Shelah represented a level of model-theoretic sophistication which was not present in other parts of model theory, and stability was considered as a kind of singularity. Actually from early on, Shelah was interested in unstable theories, and had defined in [68] the notion of a simple theory as that where every complete type over a set B does not divide over some “small” subset A of B. Dividing and the related notion of forking give rise to the fundamental notion of (in)dependence in stable theories, “a is independent from b over C”, satisfying a number of “algebraic” properties, as well as a “uniqueness” property. In his 1996 thesis [33] Byungham Kim showed that all these algebraic properties also hold for (non)dividing in the broader class of simple theories. This showed that the underlying machinery of stability was not a singularity, and applied to contexts such as the asymptotic theory of finite fields, which up to that point had been studied using only elementary model-theoretic tools (but sophisticated algebraic-geometric tools). Other examples of simple, unstable, theories are (the theory of) the random graph, as well as ACFA, the model companion of the theory of fields with a generic automorphism. Stable group theory extended naturally to groups definable in simple theories, with interesting modifications, which are reflected to some extent in the “stabilizer theorem” for approximate groups (although the latter context is far from simple). The technical notions of dividing, forking, etc., will be given later.

1.1.3 NIP An opposite (from simplicity) generalization of stability is in the direction of NIP (not the independence property) theories. In the context of a complete theory T in language L, an L-formula ϕ(x, y) has (or is) NIP if there does not exist a model M of T and tuples {ai : i ∈ ℕ} and {bs : s ∈ 𝒫 (ℕ)} such that M 󳀀󳨐 ϕ(ai , bs ) iff i ∈ s. This can be finitized using compactness, giving the notion of k-NIP, and corresponds to the family of definable sets ϕ(x, b) as b ranges over M having finite Vapnik–Chervonenkis dimension. As suggested by the last sentence, this notion has appeared independently in a number of mathematical contexts: model theory, learning theory, functional analysis. Then T is NIP if every L-formula ϕ(x, y) is NIP (for T). The NIP theories include the much-studied and well behaved theories with some “topological” character, such as

6 | A. Pillay dense linear orderings, real closed fields, p-adically closed fields, various theories of Henselian-valued fields, ordered abelian groups. And stability implies NIP (formulaby-formula). In fact, T is stable iff T is simple and T is NIP. Further, o-minimal theories are also examples of NIP theories, and the general theory of o-minimality was (at least from the point of view of Steinhorn and mine) meant to generalize aspects of stability theory, such as strong minimality, to an unstable context. In [51] I tried to generalize theorems about groups and fields definable in stable theories (more specifically in theories of finite Morley rank) to groups and fields definable in o-minimal theories. This used notions of dimension analogous to Morley rank. However, in subsequent work (since the early 2000s) the actual notions of stability theory (forking, definability of types, finite satisfiability), rather than analogues, were applied to NIP theories and groups definable therein, with striking consequences. This was another surprise for me, as I had believed that simple theories were the maximum class of theories for which the technical notions of stability theory were meaningful. Again Shelah was at the beginning of these developments.

1.2 Groups definable in specific structures We survey here the attempts to describe or classify groups definable in particular theories or structures. Sometimes, this has independent mathematical interest and/or applications. Usually some kind of relative quantifier elimination is involved in the classification, and sometimes some general theory such as stability, or simplicity, plays a nontrivial role. Of course, any group is a definable group in ZFC. This seems like a meaningless remark, but in later sections we may see that it is not so stupid. Some of what we say in this section will depend on material appearing in later sections. Algebraic groups are defined in the next (sub)section. Algebraic groups (as well as their groups of K-rational points where K is not necessarily algebraically closed) are considered here as “known,” and a general theme will be the relationship of definable groups to algebraic groups in the various examples, and what is specifically new or interesting in the definable category.

1.2.1 Algebraically closed fields The important basic facts about the theory ACF of algebraically closed fields, in the ring language (+, ×, −, 0, 1), are that the completions are obtained by fixing the characteristic, that ACF has quantifier elimination (across the characteristics), and that each completion is strongly minimal (in the home sort), namely all definable (with parameters) subsets of the home sort are finite or cofinite. In particular, each completion of ACF is ω-stable. See [65] for more details about this section.

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Let us now fix the characteristic to be 0, so we are dealing with the complete theory ACF0 . We let K denote a model, namely an algebraically closed field of characteristic 0 such as the complex numbers. One of the basic problems of algebraic geometry is the classification up to birational isomorphism of irreducible algebraic varieties over K. There is a bijection between irreducible varieties V over K, and complete types p(x)̄ over K (p being the generic type over K of V). And V, W are birationally isomorphic iff there are realizations ā of pV and b̄ of pW such that ā and b̄ are interdefinable over K if and only there are definable subsets V1 , W1 of V, W, respectively, of maximal Morley rank and a definable bijection between V1 and W1 . So in this sense the birational classification of algebraic varieties coincides with the classification of definable sets up to definable bijection with respect to ACF0 . However, this observation does not seem to have contributed much to the birational classification problem. An algebraic group is a group object in the category of algebraic varieties. Weil’s theorem that algebraic groups can be recovered from birational data translates (with some work, proved by Hrushovski) into the theorem that definable groups coincide with algebraic groups. Namely, any definable group can be definably equipped with the structure of an algebraic group, and that definable isomorphisms between two definable groups coincide with isomorphisms (in the sense of algebraic groups) between the corresponding algebraic groups. This is somewhat subtle and it is worth paying attention to the precise definitions and notions. In fact, it was via this theorem that I got an inkling of what an abstract algebraic variety is. On the other hand, there is a structure theory for algebraic groups. There are two extreme cases of algebraic groups (over an algebraically closed field K), namely linear algebraic groups (algebraic subgroups of some GL(n, K)) and abelian varieties (algebraic groups whose underlying variety is a projective variety). An arbitrary (irreducible, or connected) algebraic group is an extension of an abelian variety by a linear algebraic group. These two kinds of algebraic groups seem, from the outside, to belong to different parts of mathematics, in the sense that there are very few theorems dealing with all algebraic groups (other than theorems about commutative algebraic groups). In fact, the theory of commutative algebraic groups is deeply connected with the theory of differential forms of the first, second, and third kind, on algebraic curves, as well as “arithmetic algebraic geometry.” In terms of identifying the definable groups in the theory ACF, there is nothing more to be said. One can ask what exactly model theory can contribute to algebraic geometry via the first-order theory ACF. Probably not so much, other than the group configuration theorem. However, aspects of algebraic geometry can be captured in richer “tame” theories, as will be discussed below.

8 | A. Pillay

1.2.2 Real closed fields We are dealing here with the theory RCF of the structure ℝ in the ring language mentioned above, or its definitional expansion RCOF in the ordered ring language (noting that the nonnegative elements are precisely the squares). The standard model is the field of real numbers with the usual ordering. We will focus here on groups definable in the standard model, although things generalize suitably to arbitrary models, i. e., real closed (ordered) fields. Tarski’s quantifier elimination theorem says that the sets definable in the structure (ℝ, +, ×) are the semialgebraic sets, where a semialgebraic set is, by definition, a subset of ℝn defined by a finite disjunction of sets defined by f (x)̄ = 0 ∧ g1 (x)̄ > 0 ∧ ⋅ ⋅ ⋅ ∧ gk (x)̄ > 0, for f , gi polynomials over ℝ. We call groups definable in (ℝ, +, ×) semialgebraic groups (that is, both the universe of the group and the graph of the group operation are semialgebraic sets). On the face of it, there is no continuity in the definition of semialgebraic. I will give a rather extended discussion around the problem of classifying semialgebraic groups. The analogue of the theorem that definable groups in ACF can be definably equipped with the structure of algebraic groups is that a semialgebraic group can be definably equipped with the structure of a Nash group and that, moreover, Nash groups are semialgebraic. (And again there will be an equivalence of categories between semialgebraic groups and Nash groups.) Nash manifolds originated in work of John Nash [47], where he introduced the notion of a real algebraic manifold, which in current parlance is a compact affine Nash manifold. They appeared again in [1] and were systematically studied in [71]. We follow [71], in particular making the distinction between Nash manifolds and locally Nash manifolds. The definition is as follows: a Nash manifold is a real analytic manifold with a covering by finitely many open sets each of which is diffeomorphic to some open semialgebraic subset of some ℝn , and such that the transition functions are Nash, namely both analytic and semialgebraic. So a Nash manifold can be described as a “semialgebraic real analytic manifold.” By a Nash group we mean a group object in the category of Nash manifolds, so the underlying set is a Nash manifold and the group operation is Nash, namely analytic and semialgebraic when read in the semialgebraic charts. A Nash group is also a Lie group. The category of Nash groups is in-between that of real algebraic groups and that of Lie groups (in the sense of inclusions of categories). A more general theorem is that a group definable in an o-minimal expansion of the real field has definably the structure of a Lie group (in particular a topological group), and is proved in [51] by adapting the ideas in the ACF case. This theorem specializes to the statement that semialgebraic groups “equal” Nash groups in the case of (ℝ, +, ×). What we call a real algebraic group can be best described as the group G(ℝ) of real points of an algebraic group defined over ℝ. Such a real algebraic group G(ℝ) is also a topological (in fact Lie) group and has a connected component G(ℝ)0 which is semialgebraic and of finite index. For example, if G = GLn , then G(ℝ)0 is the collection

1 Model theory and groups | 9

of matrices of positive determinant. Any open subgroup of G(ℝ) is semialgebraic and lies in-between G(ℝ) and G(ℝ)0 . With a slight abuse of terminology, we will extend the notion of real algebraic group to include their open subgroups, in particular their topological connected component. Notice that with this terminology, any semialgebraic subgroup of a real algebraic group will be real algebraic, as it will be open in its Zariski closure. In any case, we will take real algebraic groups to be something known. And what remains to be done is to give a classification of Nash groups in terms of real algebraic groups, their covers, and their quotients. Let us be more precise. In [26] it was shown that any Nash group is definably locally isomorphic to a real algebraic group, in the sense that there are open definable neighborhoods of the identity of the Nash group and some real algebraic group and a definable bijection between them which respects the group operations (whenever they are defined). It follows that given any connected Nash group G there is a (connected) real algebraic group H and a locally Nash isomorphism between the universal covers G,̃ H̃ of G, H, where these universal covers are both locally Nash groups. Locally Nash just allows an infinite rather than finite covering by open semialgebraic sets. Hence G itself is a quotient of H̃ by some discrete subgroup, and the question is to classify the discrete subgroups of H̃ such that the quotient, which is a priori locally Nash, has a compatible Nash structure. This was basically solved in the commutative case (by me and Starchenko), a few years ago, but not yet written up. On the other hand, imposing the condition of being affine is a very strong condition on Nash groups. Affineness of a Nash manifold X means that there is a Nash embedding of X into some ℝn . Any real quasiprojective variety is affine, due to stereographic projection. Hence all real algebraic groups are affine. It was “proved” in [26] with a corrected proof in [28] that an affine Nash group G (say connected) is virtually algebraic, namely there is a connected real algebraic group H and a definable surjective homomorphism from G to H, with finite kernel. Namely, affine Nash groups are finite covers of real algebraic groups. The category of finite covers of real algebraic groups goes outside the real algebraic category. As is well-known there are finite covers (as Lie groups) of SL(2, ℝ) which have no linear representation. On the other hand, there do exist nonaffine Nash groups, the simplest being ℝ/ℤ viewed as a Nash group. One should be careful here: the category of Nash groups is inbetween the category of real algebraic groups and real Lie groups. All 1-dimensional compact, connected Nash groups are isomorphic as Lie groups but not as Nash groups. These include SO2 (ℝ), E(ℝ)0 (for elliptic curves E over ℝ, which are all real algebraic, hence affine as Nash groups), and ℝ/ℤ as above (the real unit interval with addition modulo 1 and with 0 and 1 identified) and other examples, which are all isomorphic as Lie groups, but not as Nash groups.

10 | A. Pillay

1.2.3 The p-adics I will focus here on analogous questions for the field ℚp of p-adic numbers to those for the reals discussed in the last section. In spite of the model theory of valued fields being a major area of research, these analogies do not seem to have been explored so much, maybe because the expected answers will be not so interesting. I do not want to go into details of the model theory of the p-adics, other than a brief survey. Macintyre proved a quantifier elimination theorem for Th(ℚp , +, ×, −, 0, 1) after adding predicates for the nth powers, for all n [41]. This is in analogy with Tarski’s quantifier elimination for the (theory of the) real field after adding a predicate for the squares. The topology on ℚp is the valuation topology, where the basic open neighborhoods of a point a are {x : v(x − a) ≥ r}, for r ∈ ℤ. With this topology ℚp is totally disconnected, and the ring ℤp of p-adic integers (valuation ≥ 0) is compact, so profinite. We have the notion of a p-adic analytic function on an open subset of ℚnp , given by locally convergent power series, and hence we obtain the notion of a p-adic analytic manifold, and thus also a p-adic analytic group. In the context of ℚp , “semialgebraic” means definable in the structure (ℚp , +, ×, −, 0, 1) (which can be made more explicit using Macintyre’s quantifier elimination theorem), and we obtain the notion of a p-adic Nash group, in analogy with the real case. And again we have an equivalence of categories between groups definable in the field ℚp and p-adic Nash groups. As in the real case, by a p-adic algebraic group we first mean something of the form G(ℚp ) where G is an algebraic group defined over ℚp . Such a group G(ℚp ) is, of course, a p-adic Nash group. And as in the real case, a problem or question is to find the relationship between p-adic Nash groups and p-adic algebraic groups. In the real case, passing from real algebraic groups to their open subgroups was a mild move; there is a smallest one (the topological connected component) which has finite index and is semialgebraic, and so all open subgroups are semialgebraic. What happens in the p-adic case? Note that ℤp is an (open) semialgebraic subgroup of the p-adic algebraic group ℚp (additively, for example), but with infinite index. Moreover, any semialgebraic subgroup of a p-adic algebraic group is open in its Zariski closure. So in analogy with the real case above, it makes sense (to me) to make a couple of moves: first, as was stated informally in [52]. Problem 1.2.1. Is any open subgroup of a p-adic algebraic group p-adic semialgebraic? And secondly to redefine a p-adic algebraic group as a semialgebraic subgroup of a group of the form G(ℚp ) where G is an algebraic group over ℚp . With this new definition, clearly all p-adic algebraic groups are p-adic Nash, and we believe that, in contrast with the real case, the converse holds too. Problem 1.2.2. Up to definable (or semialgebraic or p-adic Nash) isomorphism, p-adic Nash groups are precisely the p-adic algebraic groups.

1 Model theory and groups | 11

Both RCF and the theory of ℚp are NIP but this does not play much of a role in the classification results above. It was rather a relative quantifier elimination, possibly cell-decomposition, and the fact that model-theoretic and field-theoretic algebraic closures coincide. In fact, the latter, together with a delicate application of the group configuration theorem inside an ambient algebraically closed field, yields a local (in the topological sense) semialgebraic isomomorphism between a given definable group G and a real/p-adic algebraic group, which is the first step in the classification results and problems. We have been discussing definable, rather than interpretable groups (which coincide for RCF), and the classification of the interpretable groups in ℚp must take account of groups living in the value group (ℤ, +, 1. The group of torsion points of an abelian variety is Zariski-dense, and its Kolchin closure (the smallest definable subgroup of A containing the torsion) A♯ is a connected definable subgroup of A(𝒰 ) of finite Morley rank. We also have: Theorem 1.2.4. In Case (iv) of Theorem 1.2.3, A♯ is “modular” or “1-based,” meaning, or at least equivalent to the property, that every definable (in 𝒰 ) subset of any Cartesian product A♯ × ⋅ ⋅ ⋅ × A♯ is a (finite) Boolean combination of cosets (translates) of definable subgroups, where, moreover, these definable subgroups are defined over the algebraic closure of the set of parameters defining A♯ . Mordell–Lang and Manin–Mumford type problems originate with the Mordell conjecture (proved by Faltings) that a curve of genus ≥ 2 over the rationals has only finitely many rational points. Hrushovski proved function field Mordell–Lang conjecture in all characteristics [19]. A very special case of function field Mordell–Lang conjecture, as well as function field Manin–Mumford conjecture, follows quickly from Theorems 1.2.3 and 1.2.4. Theorem 1.2.5. Let k < K be algebraically closed fields of characteristic 0, and A a simple abelian variety defined over K which has k-trace 0 (namely is not isomorphic to an abelian variety defined over k). Let X be an irreducible subvariety of the Cartesian product An (defined over K) such that the intersection of X with the group of torsion points of An is Zariski-dense (in X). Then X is a translate of an abelian subvariety of An . Proof. After extending K, equip K with the structure of a differentially closed field (K, 𝜕) whose field of constants is k. Then by assumption and Theorem 1.2.4, A♯ is modular. Hence X ∩(A♯ )n is a finite union of cosets of subgroups. (A little argument is needed here to show that a Kolchin closed set which is a Boolean combination of cosets of definable subgroups is actually a finite union of such cosets.) As (A♯ )n contains the torsion of An , by our assumptions X ∩ (A♯ )n is Zariski-dense in X, from which it follows that X is itself a translate of a subgroup (abelian subvariety) of An . (Again a little argument is required to show that the Zariski closure of a finite union of cosets is a finite union of cosets of algebraic subgroups, and in the case or irreducibility is a single such coset.) Of course, it takes considerable insight to see the connection between Mordell– Lang-type problems and the model theory of differentially closed fields, although Buium [4] had already made the connection with differential algebra. Nevertheless,

14 | A. Pillay it is simply the fact that model-theoretic and definability considerations are even relevant to or meaningful for algebraic and diophantine geometric problems that is striking for me, and is an instantiation of the “unity of mathematics.” Hopefully other parts of this paper will serve a similar purpose.

1.3 Stability, stable group theory, and generalizations The general theory of stable groups is due to Poizat (building on Cherlin, Shelah, Zilber in special cases). It appears in [65], but the original paper is [63]. Stable group theory is a stability theory in the presence of a transitive definable group action, a special case of which is simply a definable group. A very special case is when the ambient theory is ACF (or rather one of its completions, ACF0 or ACFp ). In this case the definable groups are algebraic groups as mentioned in Section 1.2.1. And much of the language of stable group theory, such as connected components, is borrowed from algebraic group theory. Also there is a rather important aspect of stability theory called “local stability” where an assumption of stability is made only on a single formula ϕ(x, y) rather than all formulas. This also has a group version, local stable group theory, appearing in [26]. This is the appropriate point to introduce, or recall, the key notions of dividing and forking (by Shelah). Let T be an arbitrary complete theory for now, and work in a saturated model M̄ of T. We stick with the conventions in the introduction. A formula ϕ(x, b) is said to divide over a set A of parameters, if there is some A-indiscernible sequence (bi : i < ω) with b0 = b such that {ϕ(x, bi ) : i < ω} is inconsistent, equivalently k-inconsistent for some k < ω. The general idea is that this notion of dividing captures model-theoretically notions of “dependence” from both field theory and linear algebra. The notion of forking was introduced to show that the corresponding notion of independence has a kind of “built-in” extension property: a formula ϕ(x, b) is said to fork over A if ϕ(x, b) implies a finite disjunction of formulas each of which divides over A. What is quite amazing is how many phenomena are captured by this apparently obscure notion. Anyway, we will say that a tuple a is f -independent (where f is for forking) from a set B of parameters over a subset A of B, if no formula ϕ(x, b) in the type of a over B forks over A. When T is stable, this independence notion has some properties which have an “algebraic” character, together with a certain uniqueness property. The algebraic properties include things such as symmetry (a is f -independent from B over A iff for each tuple b from B, b is f -independent from A ∪ {a} over A), and extension or existence (for any a, and A ⊆ B, there is a󸀠 with the same type as a over A such that a is f -independent from B over A). For some time, many model-theorists, at least I, had the impression

1 Model theory and groups | 15

that it was these algebraic properties of f -independence in stable theories that were crucial for, or even characteristic of, stability. But it is now clear that it is the uniqueness aspect which is characteristic. This uniqueness property says that working over a base set A which is algebraically closed in the sense of M̄ eq (for example, if A is an ̄ given a and A ⊆ B, there is a unique type of a󸀠 over B elementary substructure of M), 󸀠 such that a and a have the same type over A and a󸀠 is f -independent from B over A. Another way of expressing this is if p(x) is a complete type over A, and Ip is the collection of formulas ϕ(x, b) with arbitrary parameters from M̄ such that p(x) ∪ {ϕ(x, b)} forks over A, then p(x) ∪ {¬ϕ(x, b) : ϕ(x, b) ∈ Ip } determines or axiomatizes a unique ̄ This is a vast stability-theoretic generalization of the fact complete type p󸀠 (x) over M. that if K is an algebraically closed field and V is an algebraic variety over K which is K-irreducible, then V is absolutely irreducible. ̄ we When the base set A of parameters is a model (elementary substructure of M), have the following: Fact 1.3.1. Suppose T is stable, M is a model, B ⊇ M, and p(x) ∈ Sx (B). Then the following are equivalent: (i) p does not fork over M, (ii) p is definable over M, (iii) p is finitely satisfiable in M. Moreover, p is the unique “nonforking extension” of its restriction p|M to M. Let us give some explanations of (ii) and (iii). The notion of a definable type is central to model theory, and may have originated in work of Gaifman on models of Peano arithmetic. Anyway, given a complete type p(x) over a set B and a subset A of B, p is said to be definable (over A) if for each formula ϕ(x, y) of the underlying language, there is a formula ψϕ (y) over A (i. e., with parameters from A) such that for each tuple b from B, ϕ(x, b) ∈ p iff M̄ 󳀀󳨐 ψϕ (b). In the case, as in Fact 1.3.1 above, that A is a ̄ the “defining schema” which takes ϕ to ψϕ model M (elementary substructure of M), yields a complete type over any set B extending M, which coincides with the unique nonforking extension of p over B. On the other hand, we say that p ∈ Sx (B) is finitely satisfiable in M if every formula ϕ(x) in p (which will, of course, have parameters in B, not necessarily in M) has a solution, or realization, in M.

1.3.1 Stable groups Stable group theory is an adaptation of the above machinery to the context of definable homogeneous spaces (a transitive definable action of a definable group G on a definable set X), although we will focus here on the case where X = G. As mentioned in several places earlier, by a stable group we usually mean a group G definable in a model of stable theory. To understand the model theory we often assume that the

16 | A. Pillay ambient model is saturated. It is basically equivalent to think of a stable group, as a 1-sorted L-structure M (some language L) such that the basic or home sort is a group, where the group operation is the interpretation of a symbol in the language L, and the theory of M is stable. And sometimes one considers the language L to consist only of the group operation (and maybe inversion, and the identity element). One can also think of a stable group syntactically (relative to a given theory), namely as a pair of formulas, one defining the universe of the group, and the other defining the group operation. There are several aspects to stable group theory, including chain conditions, stabilizers, generic types, and connected components. Proposition 1.3.2. Let G be a stable group. Let ϕ(x, y) be an L-formula where x ranges over G and such that for each b, ϕ(x, b) defines a subgroup of G. Then (working in a saturated model) any intersection of subgroups of G defined by instances ϕ(x, b) of ϕ(x, y) is a finite subintersection. When G is t.t in the sense that the Morley rank of the formula defining G is an ordinal (rather than ∞), then we have the DCC on the family of all definable subgroups of G. When G is superstable, we have a weaker descending chain condition: there is no infinite chain G1 ⊃ G2 ⊃ ⋅ ⋅ ⋅ of definable subgroups such that Gi+1 has infinite index in Gi . Given a stable group G and model M, G(M) acts on the space SG (M) of complete types over M containing the formula defining G. The action is gp = tp(ga/M) where a realizes p. From definability of types and Proposition 1.3.2 we have: Proposition 1.3.3. If G is a stable group, M a model, and p(x) ∈ SG (M) (complete types def

over M concentrating on G), then Stab(p) = {g ∈ G(M) : gp = p} is an intersection at most |T| many definable subgroups of G(M). When G is t.t, Stab(p) is outright definable.

In the t.t case (for example, when T is ACF0 and G is just an algebraic group), the above results generalize the DCC on algebraic subgroups of G and the fact that, for any subvariety X of G, Stab(X) = {g ∈ G : g(X) = X} is an algebraic subgroup of G. Another notion from algebraic group theory is “connected component” (of the identity): given an algebraic group G, one can write G as a disjoint finite union of connected (in fact, irreducible) sets, each clopen in the Zariski topology, and G0 denotes that connected component which contains the identity. It is a normal subgroup of G of finite index (and the other components are just translates of G0 ). This extends to the general case when G is t.t (although there is no Zariski topology on G): There is a smallest definable subgroup of G of finite index, which we call again G0 , the connected component of G (strictly speaking, of the identity in G).

1 Model theory and groups | 17

In the general stable case, one should be more careful. For a given formula ϕ(x, y) all of whose instances define subgroups, by Proposition 1.3.2, there is a smallest subgroup of G of finite index and which is definable by a conjunction of instances of ϕ(x, y). We can call this Gϕ0 (if there is no ambiguity). And by G0 we mean the inter-

section of all the Gϕ0 , as ϕ varies. Also G0 is a normal subgroup of G. The Gϕ0 are all

0-definable, so their intersection is 0-type-definable. We will in general only “see” G0 and the quotient G/G0 in a saturated model, and this quotient naturally has the structure of a profinite group (inverse limit of finite groups). Example 1.3.4. When G is (ℤ, +) (as both a group and a structure), G0 is the intersection of the subgroups nℤ, which we only really see in a saturated elementary extension. So (working in a saturated model) G/G0 is the inverse limit of the ℤ/nℤ, namely ̂ ℤ the so-called profinite completion of ℤ. Further, (ℤ, +) is the canonical superstable, non-totally transcendental (t.t.) group. We now discuss what I like to call the fundamental theorem of stable group theory (FTSGT), which combines stabilizers, connected components, and “genericity.” A natural notion of a “large” set X in a group G is one such that finitely many (left) translates of X cover G. In the topological dynamics literature, this kind of set is called syndetic. When G is a definable group and X a definable subset, we (in model theory) call X (left) generic. Fix now a stable group and a model M. Lemma 1.3.5. The collection of definable over M subsets of G which are not (left) generic is a proper ideal in the Boolean algebra of M-definable subsets of G. Hence there are types p ∈ SG (M) which avoid this ideal, namely such that all formulas in p are generic, and we call such p a (left) generic type of G over M. Notice that the collection of generic types is closed in SG (M) (by its definition). Now any p(x) ∈ SG (M) determines a coset of G0 (as it determines a coset of each Gϕ0 ). So we have a natural map ι from the collection Gen of generic types to G/G0 .

Theorem 1.3.6 (With above notation and assuming stability). (i) ι determines a homeomorphism between Gen and (the profinite group) G/G0 . (ii) G(M) acts transitively on Gen and for each p(x) ∈ Gen, Stab(p) = G0 . (iii) Left and right genericity coincide. This induces a (topological) group structure on Gen which we will discuss in more detail later. Let us make a couple of additional comments. The connection of genericity with (non)forking is the following: a definable subset X of G is generic if and only if every translate gX of X does not fork over 0. We call this latter property f -genericity of X (over 0).

18 | A. Pillay

1.3.2 Groups definable in simple theories The understanding of simple theories and groups definable in simple theories developed in tandem with the understanding of specific examples, such as pseudofinite fields or “smoothly approximated” ω-categorical structures. As mentioned earlier, there are very few stable theories, and an impression given in the 1970s and early 1980s, especially to those on the “applied” side of model theory, was of stability as a rather obscure or quixotic subject in model theory, heavily settheoretic in character and with little meaning for the rest of mathematics. Shelah had studied what he called simple theories already in the paper [68], and given two equivalent definitions: that any complete type does not divide over a small subset, and that no formula has the “tree property.” Here ϕ(x, y) has the tree property (with respect to an ambient complete theory T) if there are k and a tree {bη : η ∈ ω ⋅ ⋅ ⋅ > G0 = H such that for each i,

60 | R. Sklinos 0 ≤ i < m, one of the following holds: (i) The group Gi+1 has the structure of a floor over Gi , in which H is contained in one of the vertex groups that generate Gi in the floor decomposition of Gi+1 over Gi . Moreover, the pegs of the abelian flats of the floor are glued along (maximal abelian) groups that are not conjugates of each other and they cannot be conjugated into groups which correspond to abelian flats of any previous floor; (ii) The group Gi+1 is a free product of Gi with a finitely generated free group. The next lemma follows from the definition of a constructible limit group. Lemma 2.4.9. If G has the structure of a tower over a limit group, then G is a limit group. If G has the structure of a tower over a subgroup H it will be useful to collect the information witnessing it, thus we define: Definition 2.4.10. Suppose G has the structure of a tower (of height m) over H. Then the tower corresponding to G, denoted by 𝒯 (G, H), is the following collection of data: ((𝒢 (G1 , G0 ), r1 ), (𝒢 (G2 , G1 ), r2 ), . . . , (𝒢 (Gm , Gm−1 ), rm )) where: – the splitting 𝒢 (Gi+1 , Gi ) is the splitting that witnesses that Gi+1 has the structure of a floor over Gi , respectively the free splitting Gi ∗ 𝔽n for some finitely generated free group 𝔽n ; – the morphism ri+1 : Gi+1 → Gi (or ri+1 : Gi+1 → Gi ∗ ℤ) is the retraction that witnesses that Gi+1 has the structure of a floor over Gi , respectively the retraction ri+1 : Gi ∗ 𝔽n → Gi . Remark 2.4.11. The notation 𝒢 (G) will refer to a splitting of G as a graph of groups. The notation 𝒢 (G, H) will refer either to a free splitting of G as H ∗ 𝔽n or to a splitting that corresponds to a floor structure of G over H. A tower in which no abelian flat occurs in some (any) decomposition of its floors into flats is called a hyperbolic tower (or regular NTQ group in the terminology of Kharlampovich–Myasnikov). Furthermore, if a floor consists only of abelian flats we call it an abelian floor.

2.4.2 Test sequences and the implicit function theorem for hyperbolic towers We begin by giving some examples of groups that have the structure of a tower and we define the notion of a test sequence for them. The simplest cases of groups that admit a structure of a tower are finitely generated free groups and free abelian groups.

2 Independence and interpretable structures in nonabelian free groups | 61

For the rest of this section, we fix a nonabelian free group 𝔽 and a basis of 𝔽 with respect to which we will measure the length of elements of 𝔽. Definition 2.4.12. Let ⟨x1 , . . . , xk ⟩ be a free group of rank k. Then a sequence of mor̄ if hn (x)̄ satisfies the phisms (hn )n ⋅ ⋅ ⋅ > xik be some order on the xi ’s. Then a sequence of morphisms (hn )n xi2 > ⋅ ⋅ ⋅ > xik be some order on the fixed basis of ℤk . Then a sequence of morphisms (hn )n Gm+1 = 1

(5.8)

with abelian factors Gi /Gi+1 , each of which has no ℤ[G/Gi ]-torsion. We now modify this definition by removing the condition that all inclusions in (5.8) are strict. Consider the collection ε = (ε1 , . . . , εm ), consisting of zeros and ones. Suppose the group G has

218 | N. S. Romanovskii a normal series G = G1 ⩾ G2 ⩾ ⋅ ⋅ ⋅ ⩾ Gm ⩾ Gm+1 = 1,

(5.9)

such that Gi > Gi+1 for εi = 1 and Gi = Gi+1 for εi = 0, and all factors of the series Gi /Gi+1 are abelian and without ℤ[G/Gi ]-torsion. Such a series, if it exists, is uniquely determined by the group G and the collection ε. We call the group G a m-graded rigid group with grading ε, and series (5.9) the corresponding graded series. For example, for a nontrivial torsion-free abelian group there are exactly m different gradings: (1, 0, . . . , 0), . . . , (0, 0, . . . , 1). At the same time, for an m-rigid group, only one grading (1, 1, . . . , 1) is possible. The trivial group has grading (0, 0, . . . , 0). Let H be a subgroup of G, then we can consider the intersection of series (5.9) with 󸀠 H. If ε󸀠 = (ε1󸀠 , . . . , εm ) is the corresponding grading, then εi󸀠 ≤ εi for all i. By the homomorphism φ : A → B of m-graded rigid groups we mean the usual homomorphism with the condition Ai φ ⩽ Bi , here Ai and Bi denote the members of the corresponding graded series in the groups A and B. Note that the bijectivity of the mapping φ does not mean that it is an isomorphism of graded groups; the condition Ai φ = Bi is required. Thus, the m-graded rigid groups with the indicated morphisms form a category. Let us state the main results of this section. The first theorem essentially states that in the category of m-graded rigid groups, coproducts exist and factors are embedded in them. Theorem 5.11 ([31]). Let G and H be two m-graded rigid groups. Then there is an m-graded rigid group G ∘ H, which we call the m-rigid product of the groups G and H, satisfying the following conditions: 1) The groups G and H are embedded in G ∘ H as graded groups and generate the last group; 2) Arbitrary homomorphisms γ1 : G → L,

γ2 : H → L

of m-graded rigid groups extend to the homomorphism γ : G ∘ H → L. Remark 5.1. The group G ∘ H is uniquely defined by conditions 1) and 2) up to isomorphism of m-graded rigid groups. Remark 5.2. The operation ∘, as a coproduct operation, is commutative and associative. Remark 5.3. Let F1 , . . . , Fn be infinite cyclic groups with m-gradings (1, 0, . . . , 0). Then their m-rigid product F1 ∘ ⋅ ⋅ ⋅ ∘ Fn is a free solvable group of derived length m and rank n

5 Rigid solvable groups. Algebraic geometry and model theory | 219

since any set of homomorphisms Fi to an arbitrary m-rigid group G (and, in particular, to a free solvable group) can be extended to a homomorphism F1 ∘ ⋅ ⋅ ⋅ ∘ Fn → G. Remark 5.4. Using the construction of an m-rigid product, one can prove that if A and B are rigid groups with m-gradings (0, . . . , 0, 1) and (1, . . . , 1, 0), then the product A ∘ B is isomorphic to the wreath product A ≀ B. Theorem 5.12 ([31]). Let G be an m-rigid group and F be a free solvable group of derived length m, which, in the case of rank 1, is considered with grading (1, 0, . . . , 0), and in other cases, has the natural grading (1, 1, . . . , 1). Then the m-rigid product G ∘ F is G-discriminated by the group G. For the needs of algebraic geometry, we derive Corollary 5.11. Suppose that, under the conditions of the theorem, the group F has rank n and {x1 , . . . , xn } is its basis, then G ∘ F is the coordinate group of the affine space Gn in the variables x1 , . . . , xn .

5.13 Representations of rigid groups by relations A normal subgroup N in a rigid group G is called a prime ideal if the quotient group G/N is rigid. Proposition 5.21 ([40]). In an m-rigid group generated by n elements, the length of any strictly increasing (decreasing) chain of prime ideals is bounded by a function depending on m and n. Proposition 5.21 is deduced from the following two statements. Proposition 5.22 ([40]). Let an m-rigid group G be generated by n elements. Then r1 (G) ≤ n, ri (G) ≤ n − 1 (2 ≤ i ≤ m). Proposition 5.23 ([40]). Let φ : G → H be a proper epimorphism of finitely generated m-rigid groups. Then with respect to the lexicographic order r(G) > r(H). Consider again the class Σm of all rigid groups of length ≤ m. It is clear that every n-generated group from Σm is a quotient group of a free solvable group Fm,n of derived length m with basis {x1 , . . . , xn } by some ideal. Consider the question of the possibility of presentation a group in the class Σm by defining relations. Let R = R(x1 , . . . , xn ) be some set of group words in x1 , . . . , xn . In the classical situation, a group having the presentation ⟨x1 , . . . , xn | R⟩ is the quotient group of the corresponding free group by the least normal subgroup containing R. In our case, the free solvable group Fm,n may not have the least prime ideal containing R. This can be seen from the following example. Let m = 2, n = 3, and let R consist of one element [x1 , x2 ]x3 −1 . If in the group G ∈ Σ2 generated by x1 , x2 , x3 , the relation [x1 , x2 ]x3 −1 = 1 holds, then either [x1 , x2 ] = 1

220 | N. S. Romanovskii or x3 ∈ ρ2 (G). The first possibility is realized by the group which is given in the variety of solvable groups A2 using the generating elements x1 , x2 , x3 and the defining relation [x1 , x2 ] = 1. In it, x3 ∉ ρ2 (G). The second possibility is realized by the group given by the 󸀠 relations [x3 , F2,3 ] = 1. In it, [x1 , x2 ] ≠ 1 and x3 ∈ ρ2 (G). Moreover, there is no group from Σ2 , which would cover both of these groups and satisfy the relation [x1 , x2 ]x3 −1 = 1. We denote by Σm (R) the set of rigid groups from Σm generated by x1 , . . . , xn on which the relations R are satisfied. A group from Σm (R) is called maximal if it does not have a proper covering in Σm (R). It follows from Proposition 5.21 that for every group from Σm (R) there are maximal covers of this group. As a result, the given set of defining relations R on the generators x1 , . . . , xn defines in the class Σm , generally speaking, not one group, but some set of groups – the set of all maximal groups from Σm (R). Theorem 5.13 ([40]). For any R in the set Σm (R), there are only finitely many maximal groups. We say that R is a complete set of defining relations if Σm (R) contains a unique maximal group, which is thus uniquely determined by the relations R. Theorem 5.14 ([40]). Every finitely generated group of the class Σm is fully finitely presented, that is, it can be defined in Σm by a complete finite set of defining relations. In connection with the last theorem, recall a fact from [44], showing the difference with the classic case, that a free solvable group of derived length m − 1 ≥ 2 is not finitely presented in the variety Am of solvable groups of derived length ≤ m. Also, in the classic case, there is a well-known example of a solvable finitely presented group with an algorithmically undecidable word problem, constructed by O. G. Kharlampovich [10]. The following also holds: Theorem 5.15 ([40]). In the class Σm , the word problem is decidable. That is, there is an algorithm that allows one to determine whether the relation v(x1 , . . . , xn ) is a consequence of the given finite set of relations R = R(x1 , . . . , xn ) in the class Σm .

5.14 Model theory of rigid groups First, we repeat two propositions from Section 5.5. Proposition 5.11. There is a recursive system of axioms in the standard group signature defining the class of all divisible m-rigid groups. Proposition 5.12. Let M = M(α1 , . . . , αm ) be a countable divisible m-rigid group. There is a recursive system of axioms in the signature of group theory with constants from M, which defines the class of all divisible m-rigid groups containing M as an independent subgroup.

5 Rigid solvable groups. Algebraic geometry and model theory | 221

In fact, in the proofs of these proposals, specific effective ways of constructing systems of axioms are indicated. We denote the corresponding theories by Tm and Tm (M). Theorem 5.16 ([35]). The theories Tm and Tm (M) are complete, so they are decidable and Tm coincides with the elementary theory of any divisible m-rigid group while Tm (M) coincides with the elementary theory with constants from M of any divisible m-rigid group into which M is independently embedded. Corollary 5.12. Let G ⩽ H be divisible m-rigid groups (M-groups). Then the embedding of G in H is elementary only if it is independent. Let us explain the proof of Theorem 5.16. By the condition, any model of the theory Tm and the theory of Tm (M) is a divisible m-rigid group. In the second case, the group M is independently embedded in it. Using the Keisler–Shelah theorem [9], it suffices to prove that for any two countable models of the theory Tm (the theory Tm (M)), their ultrapower over a nonprinciple ultrafilter on a countable set will be isomorphic (M-isomorphic). This follows from the next two sentences. Proposition 5.24 ([35]). Let G = M(α1 , . . . , αm ) be a countable divisible m-rigid group, U a nonprinciple ultrafilter on a countable set I. Then GI/U ≅ M(2ℵ0 , . . . , 2ℵ0 ). Proposition 5.25 ([35]). Let G and H be divisible m-rigid groups and the divisible m-rigid group M independently embedded in each of them. Suppose that the coranks of G and H over M coincide. Then the groups G and H are M-isomorphic. Theorem 5.17 ([22]). The theories Tm and Tm (M) are ω-stable. Corollary 5.13. In the class of rigid groups, exactly divisible rigid groups are ω-stable. Indeed, in one direction everything follows from Theorem 5.17. Now let an m-rigid group G be ω-stable. Suppose ρm (G), as a ℤ[G/ρm (G)]-module, is not divisible by a nontrivial element γ ∈ ℤ[G/ρm (G)]. Then we obtain an infinite decreasing chain of 2 definable (with parameters) subgroups ρm (G) > ρm (G)γ > ρm (G)γ > ⋅ ⋅ ⋅, which contradicts the ω-stability. Therefore, the module ρm (G) must be divisible. Next, we apply induction on m to the group G/ρm (G). Remark. In the paper [1] the following question was posed: Is it true that every solvable ω-stable group G has a normal nilpotent subgroup N such that G/N is an extension of an abelian group by a finite one? The answer to this question is negative: divisible m-rigid groups with m ≥ 3 are not such. Recall that if the group M(α1 , . . . , αm ) is uncountable, then its cardinality coincides with the maximum αi . Theorem 5.18 ([22]). Let λ be an infinite cardinal number. 1) The group M(β1 , . . . , βm ) is λ-saturated if and only if λ ≤ βi for all indices.

222 | N. S. Romanovskii 2) The group M(β1 , . . . , βm ) is saturated if and only if β1 = ⋅ ⋅ ⋅ = βm is an infinite cardinal. 3) A countable model of the theory Tm (M) is saturated if and only if its corank over M is (ℵ0 , . . . , ℵ0 ) = ℵm 0. 4) Let λ > ℵ0 . The model of the theory Tm (M) of the cardinality λ is saturated if and only if it has the form M(λ, . . . , λ) = M(λm ). An important role in the problems under consideration is played by the divisible m-rigid group M(ℵ0 , . . . , ℵ0 ) = M(ℵm 0 ), a countable saturated model of the theory Tm . It is proved that it will be the limit group of the Fraisse system of all finitely generated m-rigid groups. Let us give definitions adapted to our situation. For a given m-rigid group G, we denote by age(G) the set of all finitely generated independent subgroups of derived length m and by age(G) the corresponding class of groups. Let 𝒦m denote the class of all finitely generated m-rigid groups. We know that every finitely generated m-rigid group is independently embedded in a divisible m-rigid group of finite rank m and hence, in the group M(ℵm 0 ). Therefore, age(M(ℵ0 )) = 𝒦m . Let us call an m-rigid group G the limit for the class 𝒦m if it satisfies the following properties: (i) countable; (ii) age(G) = 𝒦m ; (iii) (homogeneity) if U, V ∈ age(G) and φ : U → V is an isomorphism, then it can be extended to an automorphism of G. Theorem 5.19 ([22]). The limit group for the class 𝒦m is uniquely determined and is isomorphic to M(ℵm 0 ). Intersections of elementary submodels in models of theories Tm and Tm (M) are also studied. Theorem 5.20 ([22]). 1) The intersection of some set of elementary submodels of a model of the theory Tm is an elementary submodel if and only if its derived length equal to m. 2) The intersection of any set of elementary submodels of the theory Tm (M) is again an elementary submodel. The next theorem concerns the quantifier elimination of the theories under study. Theorem 5.21 ([22]). Any formula in the theory Tm or in the theory Tm (M) is equivalent to a Boolean combination of ∀∃-formulas. Recall that the (complete) type of a finite tuple a = (a1 , . . . , an ) of elements of the group G is the set of all formulas in n free variables in the signature of group theory that are true on this tuple. Let us denote this type by tp(a1 , . . . , an ). In [25], the authors call a group G strongly ℵ0 -homogeneous if the equality tp(a1 , . . . , an ) = tp(b1 , . . . , bn ) implies that the tuples (a1 , . . . , an ) and (b1 , . . . , bn ) are conjugate by an automorphism

5 Rigid solvable groups. Algebraic geometry and model theory | 223

of the group. They prove that an arbitrary free group has this property. Let us also introduce the ∃-type of the tuple tp∃ (a1 , . . . , an ), that is, the set of ∃-formulas that are true on the given tuple. Theorem 5.22 ([37]). Let G be a divisible rigid group. Then tp∃ (a1 , . . . , an ) = tp∃ (b1 , . . . , bn ) ⇔ the tuples (a1 , . . . , an ) and (b1 , . . . , bn ) are conjugate by an automorphism of G. Corollary 5.14. Let G be a divisible rigid group. Then the complete type of any finite tuple of elements of G is determined by its ∃-fragment. Corollary 5.15. The divisible rigid group is strongly ℵ0 -homogeneous. Corollary 5.16. For the theory Tm , the statement of Theorem 5.21 can be strengthened: the quantifier elimination to a Boolean combination of ∃-formulas can be done. Corollaries 5.14 and 5.15 are obtained from the theorem in an obvious way, and Corollary 5.16 is deduced from Corollary 1 and Theorem 5.3 from [26], which asserts that if the complete types in the models of given theory are determined by formulas of some set F, then quantifier elimination to a Boolean combination of F-formulas can be done. Note that the theory Tm for m > 1 is not model complete, and thus, by Theorem 8.3.1 from [9], it cannot have quantifier elimination to ∃-formulas (or ∀-formulas), so the bound we obtained for quantifier elimination is fairly accurate. We shall now describe definable subgroups in divisible rigid groups. Theorem 5.23 ([39]). Members of the rigid series of a divisible rigid group and no others are definable without parameters subgroups in the signature of the group theory. We remember that a divisible m-rigid group G can be split into a semidirect product G1 G2 ⋅ ⋅ ⋅ Gm of abelian subgroups. The subgroup Gi from this splitting coincides with the centralizer of its arbitrary nontrivial element gi . Let us fix the set of indices (i1 , i2 , . . . , ik ), where 1 ≤ i1 < i2 < ⋅ ⋅ ⋅ < ik ≤ m, and take the subgroup Gi1 Gi2 ⋅ ⋅ ⋅ Gik . It will be divisible and is determined by the formula with parameters of the free variable x: ∃x1 . . . ∃xk (([x1 , gi1 ] = 1) ∧ ⋅ ⋅ ⋅ ∧ ([xk , gik ] = 1) ∧ (x = x1 x2 ⋅ ⋅ ⋅ xk )). Theorem 5.24 ([39]). A subgroup of a divisible m-rigid group G is definable with parameters from G in the signature of the group theory if and only if it has the form Gi1 Gi2 ⋅ ⋅ ⋅ Gik for some splitting G1 G2 ⋅ ⋅ ⋅ Gm of the group G into a semidirect product of abelian subgroups. Remark. It is clear that in the nonabelian case there will be an infinite number of subgroups definable with parameters. But the subgroups corresponding to the same set of indices (i1 , i2 , . . . , ik ) are conjugate. Therefore, up to conjugacy, there will be exactly 2m subgroups definable with parameters.

224 | N. S. Romanovskii In the proof of Theorem 5.24, we used essentially algebraic geometry over divisible rigid groups, in particular, equational Noetherianness (Theorem 5.6) and the description of coordinate groups of irreducible algebraic sets (Theorem 5.9). Let G = M(α1 , . . . , αm ) and the cardinal λ be strictly greater than ℵ0 and all αi . We will assume that the group G is elementarily embedded in the group 𝔾 = M(λ, . . . , λ). The latter is understood as a large (monster) model for G. It is known that there are no proper subgroups of finite index in G. Then, for example, on the basis of Lemma 7.2.5 from [16], we can assert that G has only one generic type, the type p ∈ S1 (G), whose Morley rank RM(p) coincides with the Morley rank RM(G) of the group G. Accordingly, the element x ∈ 𝔾, with the condition that tp(x/G) is a generic type, is called a generic element. An algebraic description of generic elements will be given below, but one more definition is needed first. According to Proposition 5.8, for any splitting G1 ⋅ ⋅ ⋅ Gm of the group G, there is a unique compatible splitting 𝔾1 ⋅ ⋅ ⋅ 𝔾m of the group 𝔾. In it, 𝔾i equals the centralizer in 𝔾 of the nontrivial element gi ∈ Gi . Definition 5.5. An element x ∈ 𝔾 is said to be independent of G if, for some (any) splitting G1 ⋅ ⋅ ⋅ Gm of the group G and compatible with it splitting 𝔾1 ⋅ ⋅ ⋅ 𝔾m of the group 𝔾 and the corresponding splitting x1 ⋅ ⋅ ⋅ xm of the element x and any i, the following condition is satisfied: The element xi , together with any system of elements from Gi linearly independent over ℤ[G1 ⋅ ⋅ ⋅ Gi−1 ], constitutes a linearly independent system over ℤ[𝔾1 ⋅ ⋅ ⋅ 𝔾i−1 ]. Theorem 5.25 (unpublished). An element of 𝔾 is generic over G if and only if it is independent of G.

5.15 Generalized rigid groups There is a generalization of the concept of a rigid group. We say that a pair (G, A), where A is a proper abelian normal subgroup of G, satisfies condition (r1), if any nontrivial element in A does not commute with any element in G \ A, or, in other words, A is the centralizer of any of its nontrivial elements. We say that the group G satisfies condition (r1) if it contains a normal series G = G1 > G2 > ⋅ ⋅ ⋅ > Gm > Gm+1 = 1

(5.8)

with abelian factors, and each pair (G/Gi+1 , Gi /Gi+1 ) satisfies condition (r1). Theorem 5.26 ([36]). A group G with condition (r1) and the corresponding series (5.8) has the following properties: 1) The degree of solvability of the group is exactly m.

5 Rigid solvable groups. Algebraic geometry and model theory | 225

2) If g1 ∈ G1 \ G2 , g2 ∈ G2 \ G3 , . . . , 1 ≠ gm ∈ Gm , then Gi = {x | [x, gi , gi+1 , . . . , gm ] = 1},

i = 1, . . . , m.

3) The terms of the series (5.8) in the standard group signature can be defined by ∃-formulas, as well as ∀-formulas. In particular, this series consists of characteristic subgroups and is uniquely determined by the group G itself. Proposition 5.26 ([36]). The class of m-step solvable groups with condition (r1) in the standard group signature is determined by some finite system of axioms. Again, let A be an abelian normal subgroup of G. We know that the action of the group G on A by conjugation defines, on A, the structure of a right ℤG-module (even a ℤ[G/A]-module). Let Θ(A) denote the annihilator of the module A in ℤG, it is a twosided ideal. Take the ring R = ℤG/Θ(A). So A can be regarded as an R-module. We say that a pair (G, A), where A is a proper abelian normal subgroup of G, satisfies condition (r) if it satisfies condition (r1) and condition (r2): the module A has no R-torsion, that is, if 0 ≠ a ∈ A, 0 ≠ u ∈ R, then au ≠ 0. The definition immediately implies Proposition 5.27 ([36]). Let the pair (G, A) satisfy condition (r). Then 1) the canonical ring epimorphism ℤ[G/A] → R is injective on G/A; 2) the ring R has no zero divisors. We say that group G satisfies condition (r) or is an r-group if it contains a normal series (5.8) with abelian factors, and each pair (G/Gi+1 , Gi /Gi+1 ) satisfies condition (r). Series (5.8) is called a rigid series for G and, as in the case of rigid groups (they satisfy condition (r)), we use the same notation for its members Gi = ρi (G). Proposition 5.28 ([36]). The class of m-step solvable r-groups in the standard group signature is defined by some recursive system of axioms. It is useful here to state also Proposition 5.29 ([36]). Let G be a solvable group, I be a proper ideal of the group ring ℤG, and let the quotient ring R = ℤG/I have no zero divisors. Then R is an (right and left) Ore domain. For a given r-group G with a rigid series (5.8), consider the set of rings (R1 , . . . , Rm ), where Ri , as above, arises from the pair (G/Gi+1 , Gi /Gi+1 ). These rings are (right and left) Ore domains, denote by (Q1 , . . . , Qm ) the corresponding set of division rings. We call G a divisible r-group if each factor Gi /Gi+1 is a divisible Ri -module. In this situation, this factor can be regarded as a vector space over the ring Qi Proposition 5.30 ([36]). The class of m-step solvable divisible r-groups in the standard group signature is defined by some recursive system of axioms.

226 | N. S. Romanovskii Note that the class of abelian r-groups consists of torsion-free abelian groups and abelian groups of prime period. Divisible abelian r-groups are either direct sums of copies of the additive group of rational numbers or any abelian groups of prime period. Examples of metabelian r-groups are the solvable Baumslag–Soliter groups. Consider another example. For each pair of primes (p, q) with the condition that p divides qn − 1 for some natural number n and for each cardinal α ≥ 1, construct a 2-step solvable r-group of period pq, which we denote by E(p, q, α). Let, as usual, Fq denote a field of q elements, ∗

F q be its algebraic closure. In the multiplicative group F q , choose a cyclic subgroup C of order p, such a subgroup is unique. In the field F q , consider the subring generated by C, in fact, it will be some subfield Fqn . Let T be a vector space over the field Fqn with a base of cardinality α. We set E(p, q, α) to be equal to the matrix group ( TC 01 ). This group decomposes as a semidirect product of C and the additive group of the space T. It is an r-group with a rigid series C T

(

0 1 )>( 1 T

0 ) > 1. 1

Theorem 5.27 ([38]). Periodic r-groups are either abelian groups of prime periods or metabelian groups E(p, q, α). Thus, a fairly natural generalization of the notion of a rigid group has been found and it is natural to try to transfer the results obtained for rigid groups to r-groups. However, in the general case, there is little hope of doing this. In short, it is impossible to apply the fundamental methods that were used to study rigid groups: valuations on group rings and division rings, ring equations, etc. Some results can be obtained in the class of metabelian r-groups. Let us list some results on metabelian r-groups that have been proved so far. Theorem 5.28 ([38]). Let G be a 2-step solvable r-group with ρ2 (G) a divisible R-module, where R is associated with ρ2 (G) ring. Then the subgroup G is a split extension of ρ2 (G), that is, there is a subgroup A ≅ G/ρ2 (G) such that G is a semidirect product A ⋅ ρ2 (G). Any other similar decomposition of the group G has the form Af ⋅ ρ2 (G), where f ∈ ρ2 (G). Every element g ∈ G \ ρ2 (G) is conjugated by an element from ρ2 (G) with some element from A. Corollary 5.17. A divisible metabelian r-group splits into a semidirect product of some of its divisible abelian subgroups and ρ2 (G). Theorem 5.29 ([38]). Every metabelian r-group can be independently embedded in a divisible metabelian r-group. Proposition 5.31 ([38]). The intersection of any set of divisible subgroups in a metabelian r-group is, again, a divisible subgroup.

5 Rigid solvable groups. Algebraic geometry and model theory | 227

Remark 5.5. Based on Proposition 5.31, we can define the divisible closure of any subgroup H of a metabelian divisible r-group G as the intersection of all divisible subgroups containing H. Remark 5.6. For an arbitrary rigid group G, it was established (Theorem 5.2) that there exists a divisible closure of it (in a suitable divisible rigid group) in which G is an independent subgroup and this closure is uniquely determined up to a G-isomorphism. The existence of such a closure for a metabelian r-group is deduced from Theorem 5.29, but the uniqueness is not obtained.

5.16 Problems Let us formulate some problems. Problem 5.1. Describe when the union of several algebraic sets in an affine space over a divisible m-rigid group is again algebraic. Problem 5.2. Calculate the exact value of the Morley rank of a divisible m-rigid group. Problem 5.3. Is every r-group equationally Noetherian? Problem 5.4. Is the universal theory of a finitely generated metabelian r-group decidable? Problem 5.5. Describe the groups whose universal theory coincides with the universal theory of a given solvable Baumslag–Soliter group. Problem 5.6. Is every r-group embedded in a divisible r-group of the same derived length? Problem 5.7. Does every divisible r-group split into a semidirect product of abelian subgroups isomorphic to the corresponding factors of the rigid series? Problem 5.8. Is the elementary theory of an arbitrary divisible r-group decidable?

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Index 1-based 24 1-basedness 67 Σm 200 Σm (R) 220 ∼MAX -equivalence 110 abelian flat 58 abelian size 98 absolutely and effectively rich structure 146 absolutely biinterpretable 139 absolutely rich structure 146 ACF 6 algebraic group 7 algebraic set 88, 202 algebraic variety 3 algebraically closed group in Σm 200 amenability 40 ample hierarchy 67 arithmetic regularity 32 atomic model 150 Bass–Serre theory 54 bi-interpretable 137 block-NTQ group 102 Bohr compactification 30 Borel definable 20 canonical base 23 canonical embedding tree 96 canonical Hom-diagram 89 canonical NTQ group 90 canonical NTQ system 90 certificate 117 compact domination 29 complete system 166 complexity 98 configuration group 117 coordinate group 89, 202 coordinate map 131 corank in rigid groups 196 corrective extension 94 curve complex 76 DCF0 11 definable subset 130 definable type 15 definably amenable 21

depth of equation 198 dimension of the fundamental sequence 95 discriminated 202 discriminating family of homomorphisms 99 divide 14 divisible closure 198 divisible completion 198 divisible rigid group 196 effectively rich structure 146 elimination of imaginaries 69 Ellis semiagroup 39 equationally Noetherian group 202 existentially closed group in Σm 200 externally definable 41 f -generic 18 f -independent 14 finite equivalence relation theorem 35, 66 first restriction on fundamental sequences 94 floor 59 fork 14 formula solution 91 free groups 52 free splitting 209 free variables 89 fully residually free group 88 fully residually free group discriminated by Ψ 99 fundamental sequence of solutions 89 fundamental sequence relative to subgroups 98, 99 G-ambit 39 G-compact 44 G-flow 38 G-group 202 G-ideal 211 GA0 26 GA00 26 GA000 26 GalKP (T ) 33 GalL (T ) 34 GalSh (T ) 33 generalized Bohr compactification 40 generic 17 generic definable set 68 generic family of solutions 91

232 | Index

graded rigid group 218 graph of groups 54 group of equations 202 homogeneity 150 hyperdefinable 27 hyperimaginaries 27 imaginary elements 69 independence theorem 35 independent embedding 196 independent subgroup 196 induced NTQ system 98 induced NTQ system and fundamental sequence 96 internal 36 internality 81 interpretable 131 interpretation code 131 JSJ decompositions 55 k-approximate group 29 Krull dimension 211 Kurosh rank 98 Kurosh subgroup theorem 74 lift to a generic point 91 limit group 88 list superstructure 142 m-rigid group 195 M(α1 , . . . , αm ) 197 Magnus splitting 209 mapping class group 77 minimal flow 39 minimal on all levels 99 model-complete 3 modular 25 Mordell–Lang 13 Nash group 8 nonpinching map 94 normal equation 198 normal solution 198 NTQ group 90 NTQ system 90 p-adic numbers 10

parametric ∀∃-tree 114 Picard–Vessiot 37 prime model 150 projective dimension 211 pseudofinite group 31 pseudoplane 67 QFA model 152 quantifier elimination 3 r-group 225 radical 89 rank of rigid group 196 RCF 8 real algebraic group 8 reducing homomorphism 111 reducing quotient 110 regular interpretation 132 regular NTQ system 93 regular size 98 regular size of the NTQ system 98 rich structure 146 rigid group 195 rigid series 195 ring equation 204 second restriction on fundamental sequences 95 semialgebraic 8 sentence 133 separated 202 simple theory 5 size 98 solid limit group 110 special homomorphism 110 special variables 204 specialization 202 split rigid group 196 splitting 209 stability 64 stable 4 stable group 16, 68 stable QH subgroup 107 strict fundamental sequence 89 strongly f -generic 21 strongly minimal 12 sufficient splitting 110 superstability 4 surface flat 57

Index | 233

tight enveloping NTQ group and fundamental sequence 122 topological dimension 211 topological dynamics 38 tower 59, 90 True(Θ)i 116

uniform interpretation 132 weakly f -generic 21 well-aligned fundamental sequences 96 Zariski topology 202