Equations in Groups and Some Related Questions

Equations in free groups have become prominent recently in connection with the solution to the well-known Tarski conjecture. Results of Makanin and Rasborov show that solvability of systems of equations is decidable and there is a method for writing down in principle all solutions. However, no practical method is known; the best estimate for the complexity of the decision procedure is P -space. The special case of one-variable equations in free groups has been open for a number of years, although it is known that the solution sets admit simple descriptions. We use cancellation arguments to give a short and direct proof of this result and also to give a practical polynomial-time algorithm for finding solution sets. One-variable equations are the only general subclass of equations in free groups for which such results are known. We improve on previous attempts to use cancellation arguments by employing a new method of reduction motivated by techniques from formal language theory. Our paper is self-contained; we assume only knowedge of basic facts about free groups.

Finding all solutions of equations in free groups and monoids with involution

The Ehrenfeucht Conjecture [5] states that if Μ is a finitely generated free monoid with nonempty subset S, then there is a finite subset T⊂S (a “test set”) such that given two endomorphisms f and g on Μ, f and g agree on S if and only if they agree on T. In[4], the authors prove that the above conjecture is equivalent to the following conjecture: a system of equations in a finite number of unknowns in Μ is equivalent to a finite subsystem. Since a finitely generated free monoid embeds naturally into the free group with the same number of generators, it is natural to ask whether a free group of finite rank has the above property on systems of equations. A restatement of the question motivates the following.

In this paper we study the asymptotic probability that a random equation in a finitely generated free group F is solvable in F. For one-variable equations this probability is zero, but for split equations, i.e., equations of the form v(x1, . . . , xk) = g, g ∈ F, the probability is strictly between zero and one if k ≥ rank(F) ≥ 2. As a consequence the endomorphism problem in F has intermediate asymptotic density, and we obtain the first natural algebraic examples of subsets of intermediate density in free groups of rank larger than two.

Equations in Free Groups

On equations in free groups.

Lyndon's group FZ[x] is the free exponential group over the ring of integral polynomials Z[x]. This group, introduced by Lyndon in the 1960s, continues to be of interest to group theorists due to its importance in the study of first-order properties of free groups, in particular, equations over free groups. One of the crucial results of Lyndon's study was that the group FZ[x] is fully residually F; i.e. for any finite collection of nontrivial elements in FZ[x] there exists a homomorphism φ : FZ[x] → F which is the identity on F and maps the given elements of FZ[x] into nontrivial elements of F. The importance of FZ[x] was further emphasized when Kharlampovich and Myasnikov proved in [3] that a finitely generated group is fully residually free if and only if it is embeddable into FZ[x]. Lyndon's group and its subgroups play a vital role in the technique employed by O. Kharlampovich and A. Myasnikov in their solution of the famous Tarski problem on the decidability of the elementary theory of a free group (see [4, 5]). In this paper, we show that Lyndon's group is conjugately residually free, i.e. it is possible to map FZ[x] to the free group F preserving the nonconjugacy of two elements. This result is a further step towards the understanding of the properties of FZ[x]; moreover, it is closely related to the problem of "lifting solutions" of equations from F to FZ[x], since our result implies that the solutions can indeed be "lifted" from F to FZ[x] for equations of the type x-1 c1 x = c2. The structure of Lyndon's group, described by A. Myasnikov and V. Remeslennikov in [8], involves an infinite sequence of free constructions of a specific type, called free extensions of centralizers. For more results on residual properties of certain types of free constructions, see also the works of Ribes, Segal and Zalesskii (for example [9]).

This chapter continues the development of group theory begun in Chapter IV, the main topics being the use of generators and relations, representation theory for finite groups, and group extensions. Representation theory uses linear algebra and inner-product spaces in an essential way, and a structure-theory theorem for finite groups is obtained as a consequence. Group extensions introduce the subject of cohomology of groups.Sections 1–3 concern generators and relations. The context for generators and relations is that of a free group on the set of generators, and the relations indicate passage to a quotient of this free group by a normal subgroup. Section 1 constructs free groups in terms of words built from an alphabet and shows that free groups are characterized by a certain universal mapping property. This universal mapping property implies that any group may be defined by generators and relations. Computations with free groups are aided by the fact that two reduced words yield the same element of a free group if and only if the reduced words are identical. Section 2 obtains the Nielsen-Schreier Theorem that subgroups of free groups are free. Section 3 enlarges the construction of free groups to the notion of the free product of an arbitrary set of groups. Free product is what coproduct is for the category of groups; free groups themselves may be regarded as free products of copies of the integers.Sections 4–5 introduce representation theory for finite groups and give an example of an important application whose statement lies outside representation theory. Section 4 contains various results giving an analysis of the space C(G, ?) of all complex-valued functions on a finite group G. In this analysis those functions that are constant on conjugacy classes are shown to be linear combinations of the characters of the irreducible representations. Section 5 proves Burnside’s Theorem as an application of this theory—that any finite group of order paqb with p and q prime and with a+b > 1 has a nontrivial normal subgroup.Section 6 introduces cohomology of groups in connection with group extensions. If N is to be a normal subgroup of G and Q is to be isomorphic to G/N, the first question is to parametrize the possibilities for G up to isomorphism. A second question is to parametrize the possibilities for G if G is to be a semidirect product of N and Q.

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

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 .

The theory of groups is one of the oldest and richest branches of abstract algebra. Groups of transformations play an important role in geometry, and finite groups are fundamental in Galois’ discoveries in the theory of equations. These two fields provided the original impetus to the development of the theory of groups.

Elliptic operators arise naturally in several different mathematical settings, notably in the representation theory of Lie groups, the study of evolution equations, and the examination of Riemannian manifolds. This book develops the basic theory of elliptic operators on Lie groups and thereby extends the conventional theory of parabolic evolution equations to a natural non-commutative context. In order to achieve this goal, the author presents a synthesis of ideas from partial differential equations, harmonic analysis, functional analysis, and the theory of Lie groups. He begins by discussing the abstract theory of general operators with complex coefficients before concentrating on the central case of second-order operators with real coefficients. A full discussion of second-order subellilptic operators is also given. Prerequisites are a familiarity with basic semigroup theory, the elementary theory of Lie groups, and a firm grounding in functional analysis as might be gained from the first year of a graduate course.

Systematization of dimensionless quantities by group theory

Let Fn be a free group of rank n ≥ 2. Two elements g, h in Fn are said to be translation equivalent in Fn if the cyclic length of φ(g) equals the cyclic length of φ(h) for every automorphism φ of Fn. Let F(a, b) be the free group generated by {a, b} and let w(a, b) be an arbitrary word in F(a, b). We prove that w(g, h) and w(h, g) are translation equivalent in Fn whenever g, h ∈ Fn are translation equivalent in Fn, and thereby give an affermative solution to problem F38b in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edition. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.].

L systems generalize context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper, we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives and bases in the free group on two generators, the Bridson–Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk’s group can be generated by L systems. To complement the result on primitives in rank 2 free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L languages of intermediate growth.