Free Burnside Groups and Related Topics

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Abstract for award DMS-0400746 of Ivanov:The main theme of this project is further applications anddevelopment of the geometric machinery of graded diagrams createdby the principal investigator (PI) to solve one of the mostcelebrated algebraic problems of the 20th century, the Burnsideproblem for periodic groups. In particular, the PI will work on amodification of this machinery suitable for free products ofgroups, will investigate finite, locally finite, free, normalsubgroups of free Burnside groups of large exponent and studyBurnside extensions of groups given by defining relations. Inaddition, the PI will work on a generalized version of the HannaNeumann conjecture on subgroups of free groups, on the Whiteheadasphericity conjecture for aspherical and almost asphericalpresentations of groups, and on strengthening of the classicalembedding theorems for groups discovered by Higman, B. Neumann andH. Neumann in the 1940's. The PI will also continue to studyalgorithmic problems of 3-dimensional topology, in particular,computational complexity of the recognition problem for the3-sphere. The proposed activity should result in a more profoundunderstanding of such fundamental objects of modern algebra asperiodic groups and, more generally, groups defined by generatorsand relations.This research project is in the area of the theory of groups thatinvestigates groups, defined by means of generators and definingrelations, and lies at the intersection of the theory of groupswith low-dimensional topology, geometry and mathematical logic.The theory of groups is a mathematical theory of symmetries ofspaces which interacts with many other disciplines, for example,physics and chemistry outside of mathematics, coding theory,number theory, topology and geometry inside mathematics.