Presentations of Groups by Generators and Defining Relations

本课题将重点研究几何群论中一些突出的问题，如关于平凡群的平衡表示的Andrews-Curtis猜想，关于周期群的Burnside问题，关于自由群中子群的交点秩的Hanna Neumann猜想。著名的Andrews-Curtis猜想声称，平凡群的平衡表示可以通过扩展Nielsen运算的有限序列转化为标准表示，也称为初等ac -移动。Andrews和Curtis推测，他们的一种基本交流移动，即共轭，可以被一种更严格的循环置换操作所取代，从而假设他们的猜想等价于其可能更强的“循环”版本。首席研究员（PI）将尝试建立这种等效性。PI将研究与平凡群的平衡表示有关的其他问题，例如马格努斯猜想的稳定版本以及与这些表示相关的渐近函数。该项目的另一个目标是找到PI创建的分级图的几何机制的新应用，以解决20世纪最具影响力的代数问题之一，即大偶数指数周期群的Burnside问题。此外，PI将研究与Hanna Neumann猜想有关的问题，如自由群中子群的交集，以及群论和三维拓扑中的算法和计算复杂性问题。本研究项目属于群理论领域，研究通过生成器和定义关系来定义的群，是群理论与低维拓扑、几何和数理逻辑的交叉。群理论是一种空间对称性的数学理论，它与许多其他学科相互作用，例如数学之外的物理和化学，数学内部的编码理论，数论，拓扑和几何。

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.