Local-global invariants of finite and infinite groups: around Burnside from another side

This expository essay is focused on the Shafarevich-Tate set of a group $G$. Since its introduction for a finite group by Burnside, it has been rediscovered and redefined more than once. We discuss its various incarnations and properties as well as relationships (some of them conjectural) with other local-global invariants of groups.Comment: 18 page

The Bogomolov multiplier of finite simple groups

The subgroup of the Schur multiplier of a finite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost simple, its Bogomolov multiplier is trivial.Comment: 8 page

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