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.
This award supports research in geometric group theory that studies abstract algebraic structures, called groups, by means of different techniques from algebra, geometry, topology, operation research, and computer science. The main results and findings of the activity supported in part by this award include the following. A construction of nontrivial pairs of zero divisors in integral groups rings of free Burnside groups of sufficiently large odd exponent is given. A proof of a long-standing hypothesis of Andrews and Curtis, put forward in 1966, on balanced presentations of the trivial group is found. Several different upper bounds for reduced rank of the intersection of factor-free subgroups in free products of groups are obtained. An algorithm, based on linear programming, to compute the Walter Neumann coefficient for a factor-free subgroup in a free product of finite groups is devised. The bounded word problem and the precise word problem for some group presentations, including Baumslag-Solitar one-relator groups, are algorithmically solved in polylogarithmic space. The width problem for elements of a free group is algorithmically solved in cubic logarithmic space. A diagram problem for free groups with no relations is also algorithmically solved in cubic logarithmic space. It is proved that every finite or infinite countable group can be embedded into a 2-generated group in such a way that the solvability of all quadratic equations of bounded length is preserved. These results contribute to more profound understanding of fundamental objects of modern algebra such as free groups, free products of groups and their subgroups, Burnside groups, and, more generally, groups presented by generators and defining relations. Solutions of several long-standing problems, resulted from the supported activity, are clear indications of substantial advances in this understanding. Another impact of this award comes through partial support of a graduate student who successfully received his PhD degree.
