The main theme of this project is further applications and development of the geometric machinery of graded diagrams created by the principal investigator (PI) to solve one of the most celebrated algebraic problems of the 20th century, the Burnside problem for periodic groups. In particular, the PI will work on a modification of this machinery suitable for free products of groups, will investigate finite, locally finite, free, normal subgroups of free Burnside groups of large exponent and study Burnside extensions of groups given by defining relations. In addition, the PI will work on a generalized version of the Hanna Neumann conjecture on subgroups of free groups, on the Whitehead asphericity conjecture for aspherical and almost aspherical presentations of groups, and on strengthening of the classical embedding theorems for groups discovered by Higman, B. Neumann and H. Neumann in the 1940's. The PI will also continue to study algorithmic problems of 3-dimensional topology, in particular, computational complexity of the recognition problem for the 3-sphere. The proposed activity should result in a more profound understanding of such fundamental objects of modern algebra as periodic groups and, more generally, groups defined by generators and relations.
This research project is in the area of the theory of groups that investigates groups, defined by means of generators and defining relations, and lies at the intersection of the theory of groups with low-dimensional topology, geometry and mathematical logic. The theory of groups is a mathematical theory of symmetries of spaces which interacts with many other disciplines, for example, physics and chemistry outside of mathematics, coding theory, number theory, topology and geometry inside mathematics.
