Let G be a group, and let F-n = < x(1), ... , x(n)> be the free group of rank n. For every word w is an element of F-n, the word map is defined as (w) over tilde : G(n) -> G, where (w) over tilde (g(1), ... , g(n)) = w(g(1), ... , g(n)). The set W-w := (w) over tilde (-1)(e), where e is the identity of G, is the variety of representations of the finitely generated group Gamma(w) = with one relation w. In this paper we consder some properties of such varieties for the case when G is a simple algebraic group, in particular, G = SL2(C). Also, we consider here the question of the surjectivity of a word map (w) over tilde : SL2(C)(2) -> SL(C)) for w is an element of F-2.

The paper is devoted to model-theoretic properties of Kac–Moody groups with the focus on elementary equivalence of Kac–Moody groups. We show that elementary equivalence of (untwisted) affine Kac–Moody groups implies coincidence of their generalized Cartan matrices and the elementary equivalence of their ground fields. We study also the Diophantine problem in affine Kac–Moody groups. We show that for the loop group the Diophantine problem is polynomial time equivalent (more precisely, Karp equivalent) to the problem in the ground ring. Finally, we show that in affine Kac–Moody groups over finite fields the Diophantine problem is undecidable.

The main result of the present paper is bounded elementary generation of the Steinberg groups $\mathrm{St}(\Phi,R)$ for simply laced root systems $\Phi$ of rank $\ge 2$ and arbitrary Dedekind rings of arithmetic type. Also, we prove bounded generation of $\mathrm{St}(\Phi,\mathbb F_{q}[t,\,t^{-1}])$ for all root systems $\Phi$, and bounded generation of $\mathrm{St}(\Phi,\mathbb F_{q}[t])$ for all root systems $\Phi\neq\mathsf A_1$. The proofs are based on a theorem on bounded elementary generation for the corresponding Chevalley groups, where we provide uniform bounds.

In this paper, we establish a definitive result which almost completely closes the problem of bounded elementary generation for Chevalley groups of rank >= 2 over arbitrary Dedekind rings R of arithmetic type, with uniform bounds. Namely, we show that for every reduced irreducible root system Phi of rank >= 2, there exists a universal bound L = L (Phi) such that the simply connected Chevalley groups G (Phi, R) have elementary width <= L for all Dedekind rings of arithmetic type R .

In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R), where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1.We establish a variant of double centralizer theorem for elementary unipotents xα(1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup Xα, α∈Φ, is Diophantine in G. Then we prove that the Diophantine problem in Gπ(Φ,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.

We prove that Chevalley groups over polynomial rings 𝔽_q[t] and over Laurent polynomial 𝔽_q[t,t^-1] rings, where 𝔽_q is a finite field, are boundedly elementarily generated. Using this we produce explicit bounds of the commutator width of these groups. Under some additional assumptions, we prove similar results for other classes of Chevalley groups over Dedekind rings of arithmetic rings in positive characteristic. As a corollary, we produce explicit estimates for the commutator width of affine Kac–Moody groups defined over finite fields. The paper contains also a broader discussion of the bounded generation problem for groups of Lie type, some applications and a list of unsolved problems in the field.

In this paper, we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding defining relations. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a linear space and establish the corresponding structure theorems. We also provide a revision of a concept of Gröbner basis for our rings and establish a greedy algorithm for the Ideal Membership Problem.

In this paper, we give a general insight into the ideas that make ground for the developing of universal algebraic geometry and logical geometry. We specify the role of algebraic logic as one of the major instruments of the whole theory. The problem of the sameness of geometries of algebraic and definable sets for different algebras is considered as ans illuminating example how algebra, geometry, model theory, and algebraic logic work together.

In the present paper, we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three specific axioms for corresponding defining relations that provide the small cancellation properties of the obtained ring. We show that this ring is nontrivial. It is called a small cancellation ring.

The paper is devoted to model-theoretic properties of Kac-Moody groups with the focus on elementary equivalence of Kac-Moody groups. We show that elementary equivalence of (untwisted) affine Kac-Moody groups implies coincidence of their generalized Cartan matrices and the elementary equivalence of their ground fields. We also show that elementary equivalence of arbitrary Kac-Moody groups over finite fields implies coincidence of these fields and an isomorphism of their twin root data. The similar result is established for Kac-Moody groups defined over infinite subfields of the algebraic closures of finite fields.

The paper is devoted to model-theoretic properties of Kac-Moody groups with the focus on elementary equivalence of Kac-Moody groups. We show that elementary equivalence of (untwisted) affine Kac-Moody groups implies coincidence of their generalized Cartan matrices and the elementary equivalence of their ground fields. We also show that elementary equivalence of arbitrary Kac-Moody groups over finite fields implies coincidence of these fields and an isomorphism of their twin root data. The similar result is established for Kac-Moody groups defined over infinite subfields of the algebraic closures of finite fields.

The theory of small cancellation groups is well known. In this paper we study the notion of the Group-like Small Cancellation Ring. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial and enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out that the defined ring possesses a kind of Gröbner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory, which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory.

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out that the defined ring possesses a kind of Gröbner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory. This is a short version of paper arXiv:2010.02836

The paper is a short survey of recent developments in the area of first order descriptions of linear groups. It is aimed to illuminate the known results and to pose the new problems relevant to logical characterizations of Chevalley groups and Kac-Moody groups.

We apply small cancellation methods originating from group theory to investigate the structure of a quotient ring Z(2) F/I where Z(2) F is the group algebra of the free group F over the field Z(2), and the ideal I is generated by a single trinomial 1 + v + vw, where v is a complicated word depending on w. In Z(2) F/I we have (1 + w)-1 = v, so 1+ w becomes invertible. We construct an explicit linear basis of Z(2) F/I (thus showing that Z(2) F/I not equal 0). This is the first step in constructing rings with exotic properties.

We extend Borel's theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel's theorem to some words with constants. We also consider the surjectivity problem for particular words and groups, give a brief survey of recent results, present some generalizations and variations and discuss various approaches, with emphasis on new ideas, constructions and connections.