Tatiana Bandman

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).

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.

SURJECTIVITY AND EQUIDISTRIBUTION OF THE WORD xayb ON PSL(2, q) AND SL(2, q)

We determine the integers a, b ≥ 1 and the prime powers q for which the word map w(x, y) = xayb is surjective on the group PSL(2, q) (and SL(2, q)). We moreover show that this map is almost equidistributed for the family of groups PSL(2, q) (and SL(2, q)). Our proof is based on the investigation of the trace map of positive words.

On the surjectivity of Engel words on PSL(2,q)

We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2,q) and SL(2,q). For SL(2,q), we show that this map is surjective onto the subset SL(2,q)\{-id} provided that q>Q(n) is sufficiently large. Moreover, we give an estimate for Q(n). We also present examples demonstrating that this does not hold for all q. We conclude that the n-th Engel word map is surjective for the groups PSL(2,q) when q>Q(n). By using the computer, we sharpen this result and show that for any n<5, the corresponding map is surjective for all the groups PSL(2,q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the n-th Engel word map is almost measure preserving for the family of groups PSL(2,q), with q odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group SL(2,q).

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

Jordan groups, conic bundles and abelian varieties

A group

JORDAN PROPERTIES OF AUTOMORPHISM GROUPS OF CERTAIN OPEN ALGEBRAIC VARIETIES

G

Affine surfaces with AK(S)=C

is called Jordan if there is a positive integer

Two-variable identities for finite solvable groups

J=J_G

Equations in simple Lie algebras

such that every finite subgroup