Presentations of Groups by Generators and Defining Relations

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This project will focus on some outstanding problems of geometric group theory such as the Andrews-Curtis conjecture on balanced presentations of the trivial group, the Burnside problem for periodic groups, the Hanna Neumann conjecture on the rank of the intersection of subgroups in free groups. The notorious Andrews-Curtis conjecture claims that a balanced presentation of the trivial group can be transformed into the standard presentation by a finite sequence of extended Nielsen operations, also called elementary AC-moves. Andrews and Curtis speculated that one type of their elementary AC-moves, which is conjugation, could be replaced by a much more restrictive operation of cyclic permutation, thus making a hypothesis that their conjecture is equivalent to its presumably stronger "cyclic" version. The Principal Investigator (PI) will attempt to establish this equivalence. The PI will work on other questions related to balanced presentations of the trivial group, such as the stabilized version of a conjecture of Magnus and asymptotic functions associated with the presentations. Another goal of the project is to find new applications of the geometric machinery of graded diagrams created by the PI to solve one of the most influential algebraic problems of the 20th century, the Burnside problem on periodic groups for large even exponents. In addition, the PI will work on questions related to the Hanna Neumann conjecture on the intersection of subgroups in free groups, on algorithmic and computational complexity issues in group theory and 3-dimensional topology.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.