Algebraic geometry over groups III: Elements of model theory

Remarkable ties between group theory, logic and algebraic geometry have come to light via the positive solution of the Tarski conjecture. This has led to further work on the universal theory of groups. In this paper we describe and survey this material a large body of which is not familiar to most group theorists Contents Introduction. First Order Languages and Model Theory. The Tarksi Problems. Residually Free and Universally Free Groups. Algebraic Geometry over Groups and Applications. The Positive Solution to the Tarski Problems. Discriminating, Co-discriminating and Squarelike Groups. Open Questions. Introduction The elementary theory of a group G consists of all the first-order or elementary sentences (see section 2) which are true in G . Although this is a concept which originated in formal logic, in particular model theory, it arises independently from the theory of equations within groups. Recall that an equation in a group G is a word W ( x 1 ,…, x n , g 1 ,…, g k ) in free variables x 1 ,…, x n and constants g 1 ,…, g k which are elements of G . A solution consists of an n-tuple ( h 1 ,…, h n ) of elements from G which upon substitution for x 1 ,…, x n make the word trivial in G . Hence an equation is a first-order sentence in the language L [ G ] consisting of the elementary language of group theory (again see section 2) augmented by allowing constants from the group G .

Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, of the free group of rank n is either virtually finitely generated abelian or contains a nonabelian free group. In this note we consider the more general situation of the outer automorphism group of a finitely generated free-by-finite group G. We show that is translation discrete and that every subgroup of is either virtually finitely generated abelian or contains a nonabelian free group.

A group G is S-universal if every countable group can be embedded in a quotient of G.In this paper we give conditions under which HNN groups are S-universal.These are then applied to showing the S-universality of certain classes of Kleinian groups (discontinuous subgroups of PSL^C)).Finally some connections are mentioned between Sg-universality and the theory of ends.Introduction.A group G is So-universal if every countable group can be embedded in a quotient of G. Thus all nonabelian free groups are Suniversal [4], while it is known that all nontrivial free products except the infinite dihedral group [10], and large classes of amalgamated free products are also , [11].Sacerdote and Schupp have extended these to certain HNN and one-relator groups [9].In this paper we give a series of results which complement those of Sacerdote and Schupp but are somewhat less technical and more directly applicable to geometrically and arithmetically defined groups [2], [3].This will be applied in 3 to showing the S?-universality of a class of Kleinian groups.Finally we make some observations on the relationship between SG;-universality and the theory of ends.1. Recall that an HNN group is a group of the form G = </" ...,t",K; rel K, t^tr1 = <P,(A)>where {L,} are subgroups of K, and {<p,} are isomorphisms L, - <p,(jL,) < K. K is called the base, {L, <p(L)} associated subgroups and </" . .., i") the free part.Groups of this type were used by Higman, Neumann and Neumann to show the S-universality of free groups of rank 2 [4].The base K can be embedded isomorphically in G, and a subgroup structure theory for HNN groups, paralleling that for amalgamated free products, has been developedby Karrass and Solitar [5].Now G has a free quotient of rank n if the free part has rank n, and since clearly a group with an S-universal quotient is itself S-universal we getLemma.An HNN group with free part of rank > 1 is SQ-universal.

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant KΘ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H1 and H2 have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) KΘ(H1) and KΘ(H2) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ0 be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ0) for different varieties Θ and also the groups of autoequivalences of Θ0. The problem is to describe these groups for different Θ.

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

We give a description of the model-theoretic relation of forking independence in terms of the notion of JSJ decompositions in non-abelian free groups

We shall characterize the complete first order theories of torsion free abelian groups having algebraically prime models, elementarily prime models, minimal models, etc., using the well known invariants from [9], [3]. This study was suggested by the questions raised in [2]. Group here will always mean abelian group. For reference on groups see Fuchs [5], [6], for model theory Chang-Keisler [1]. All theories considered are theories in the first order language of group theory. A group G is minimal (strictly minimal) if H < G implies H = G (H-= G), and elementarily prime (algebraically prime) if G is elementarily (algebraically) embeddable in every H G. p, q always denote prime numbers, P the set of all primes. The p-rank of a group G, p-rk(G), is the dimension of G/pG as a vector space over Z/pZ. We define tf(p, G): = min {p-rk(G), ~o}. By the results of [9], [3], two nontrivial torsion free abelian groups G and H are elementarily equivalent iff for all primes p tf(p, G) = tf(p, H). This gives a natural 1-1 correspondence between the complete theories of nontrivial torsion free abelian groups and the functions /3 : P --~ ~o + 1. /3 will always stand for such a function, and the corresponding complete theory will be denoted by T 0. S o is the set of primes p for which/3(p) ~ 0. For more definitions see section 3. Let 13 be a function from P to o) + 1. We shall study the following questions: Does T o have an elementarily (algebraically) prime model? Does T o have a minimal (strictly minimal) model, and if this is the case, how many such models does T o have? Do the minimal and strictly minimal models of T o coincide? These questions have been studied for the special case of the theory of the

Model theory for universal classes with the amalgamation property: A study in the foundations of model theory and algebra

A group G is called logically generated by a subset S, if every element of G can be defined by a first order formula with parameters from S. We consider the case where G is a direct product of finite nilpotent groups with mutually coprime orders and we show that logical and algebraic generations are equivalent in G. We also prove that in the case when G is a free non-abelian group, if S logically generates G then either it generates G algebraically or 〈 S 〉 is not a free factor of G.

In this paper, we prove that the outer automorphism groups of nonabelian topologically finitely generated free profinite groups are strongly indecomposable. This means that no open subgroup has a nontrivial direct product decomposition.

A probability-measure-preserving transformation has the Weak Pinsker Property (WPP) if for every $ε&gt;0$ it is measurably conjugate to the direct product of a transformation with entropy $

In this paper we study the notion of a D-saturated model, which occupies an intermediate position between the notions of a homogeneous model and a saturated model. Both the homogeneous models and the saturated models play a very important role in model theory. For example, the saturated models are the universal domains of the corresponding theories in the sense that is used in algebraic geometry (one can also notice that the algebraically closed fields of infinite transcendence degree, which are the universal domains in algebraic geometry, are the saturated models of the theory of algebraically closed fields); this means that the saturated models realize all what is necessary for the effective study of the corresponding theory.The D-saturated models proved to be useful in various situations. Naturally the question arises on the conditions under which a model is D-saturated. In this paper we indicate some conditions of such sort for weakly o-minimal models and models of stable theories. Namely, for such models we prove that homogeneity and a certain approximation of Dsaturation imply D-saturation.

A group is n-free if every subgroup generated by n or fewer distinct elements is free. In [8], the authors observed that if G is a finitely generated model of the elementary theory of the non-Abelian free groups, then G is 2-free. The main result here is that such a group is 3-free. The principal tool used is a characterization, due to Hmelevskii [10], of the set of nontrivial solutions in a free group of a three variable word equation without coefficients.

Algorithms for Matrix Groups and the Tits Alternative