Boris Kunyavskii

Identities for finite solvable groups and equations in finite simple groups

We characterise the class of finite solvable groups by two-variable identities in a way similar to the characterisation of finite nilpotent groups by Engel identities. Let u1 = x −2 y −1 x, and un+1 =[ xunx −1 ,y uny −1 ]. The main result states that a finite group G is solvable if and only if for some n the identity un(x, y) ≡ 1h olds inG. We also develop a new method to study equations in the Suzuki groups. We believe that, in addition to the main result, the method of proof is of independent interest: it involves surprisingly diverse and deep methods from algebraic and arithmetic geometry, topology, group theory, and computer algebra (Singular and MAGMA).

Two-Variable Identities in Groups and Lie Algebras

We study two-variable Engel-like relations and identities characterizing finite-dimensional solvable Lie algebras and, conjecturally, finite solvable groups and introduce some invariants of finite groups associated with such relations. Bibliography: 29 titles.

Geometry and arithmetic of verbal dynamical systems on simple groups

We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engel-like sequences of words in two variables which characterize finite solvable groups. They also give rise to some new phenomena and concepts in the arithmetic of dynamical systems.

A commutator description of the solvable radical of a finite group

We are looking for the smallest integer k> 1providing the following characteri- zation of the solvable radical R.G/ of any finite group G: R.G/ coincides with the collection of all g 2 G such that for any k elements a1 ;a 2 ;:::;a k 2 G the subgroup generated by the elements g; ai ga � 1 i , i D 1; :::;k , is solvable. We consider a similar problem of finding the smallest integer `>1 with the property that R.G/ coincides with the collection of all g 2 G such that for anyelements b1 ;b 2 ;:::;b ` 2 G the subgroup generated by the commutators Œg; bi � , i D 1; :::;` , is solvable. Conjecturally, k DD 3. We prove that both k andare at most 7. In particular, this means that a finite group G is solvable if and only if every 8 conjugate elements of G generate a solvable subgroup.

Word equations in simple groups and polynomial equations in simple algebras

We give a brief survey of recent results on word maps on simple groups and polynomial maps on simple associative and Lie algebras. Our focus is on parallelism between these theories, allowing one to state many new open problems and giving new ways for solving older ones.

Word maps and word maps with constants of simple algebraic groups

In the present paper, we consider word maps w: Gm → G and word maps with constants wΣ: Gm → G of a simple algebraic group G, where w is a nontrivial word in the free group Fm of rank m, wΣ = w1σ1w2 ··· wrσrwr + 1, w1, …, wr + 1 ∈ Fm, w2, …, wr ≠ 1, Σ = {σ1, …, σr | σi ∈ GZ(G)}. We present results on the images of such maps, in particular, we prove a theorem on the dominance of “general” word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the representation variety R(Γw, G) of the group Γw = Fm/.

CHARACTERIZATION OF SOLVABLE GROUPS AND SOLVABLE RADICAL

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.