Presentations of Groups by Generators and Defining Relations

本项目将重点研究几何群论中的一些突出问题，如关于平凡群的平衡表示的Andrews-Curtis猜想，周期群的伯恩赛德问题，关于自由群中子群交的秩的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有限序列的扩展尼尔森运算（也称为初等AC-移动）转换为标准表示。安德鲁斯和柯蒂斯推测，他们的一种基本AC移动，即共轭，可以被一种更具限制性的循环置换操作所取代，从而提出一个假设，即他们的猜想等价于其可能更强的“循环”版本。主要研究者（PI）将尝试确定这种等效性。PI将研究与平凡群的平衡演示相关的其他问题，例如马格努斯猜想的稳定版本和与演示相关的渐近函数。该项目的另一个目标是找到新的应用程序的几何机械的分级图创建的PI解决一个最有影响力的代数问题的20世纪世纪，伯恩赛德问题的周期群大甚至指数。此外，PI还将研究与自由群中子群相交的Hanna Neumann猜想相关的问题，群论和三维拓扑中的算法和计算复杂性问题。本研究项目涉及群论领域，研究通过生成元和定义关系定义的群，并且处于群论与低维拓扑学、几何学和数理逻辑的交叉点。群论是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，编码理论，数论，数学内部的拓扑和几何。

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.