Geometry of word equations in simple algebraic groups over special fields

This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on algebraic groups defined over special fields: complex, real, p-adic (or close to such), or finite.Comment: 44 page

Word maps in Kac-Moody setting

The paper is a short survey of recent developments in the area of word maps evaluated on groups and algebras. It is aimed to pose questions relevant to Kac--Moody theory.Comment: 16 pag

Equations in simple Lie algebras

Given an element $P(X_1,...,X_d)$ of the finitely generated free Lie algebra, for any Lie algebra $g$ we can consider the induced polynomial map $P: g^d\to g$. Assuming that $K$ is an arbitrary field of characteristic $\ne 2$, we prove that if $P$ is not an identity in $sl(2,K)$, then this map is dominant for any Chevalley algebra $g$. This result can be viewed as a weak infinitesimal counterpart of Borel's theorem on the dominancy of the word map on connected semisimple algebraic groups. We prove that for the Engel monomials $[[[X,Y],Y],...,Y]$ and, more generally, for their linear combinations, this map is, moreover, surjective onto the set of noncentral elements of $g$ provided that the ground field $K$ is big enough, and show that for monomials of large degree the image of this map contains no nonzero central elements. We also discuss consequences of these results for polynomial maps of associative matrix algebras.Comment: 22 page

From Thompson to Baer-Suzuki: a sharp characterization of the solvable radical

We prove that an element $g$ of prime order $>3$ belongs to the solvable radical $R(G)$ of a finite (or, more generally, a linear) group if and only if for every $x\in G$ the subgroup generated by $g, xgx^{-1}$ is solvable. This theorem implies that a finite (or a linear) group $G$ is solvable if and only if in each conjugacy class of $G$ every two elements generate a solvable subgroup.Comment: 28 page

Elementary equivalence of Kac-Moody groups

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.Comment: 10 page

On first order rigidity for linear groups

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

The Diophantine problem in Chevalley groups

In this paper we study the Diophantine problem in Chevalley groups $G_\pi (\Phi,R)$, where $\Phi$ is an indecomposable root system of rank $> 1$, $R$ is an arbitrary commutative ring with $1$. We establish a variant of double centralizer theorem for elementary unipotents $x_\alpha(1)$. This theorem is valid for arbitrary commutative rings with $1$. The result is principle to show that any one-parametric subgroup $X_\alpha$, $\alpha \in \Phi$, is Diophantine in $G$. Then we prove that the Diophantine problem in $G_\pi (\Phi,R)$ 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.Comment: 44 page