Free Burnside Groups and Related Topics

伊万诺夫DMS-0400746奖摘要：这个项目的主题是由首席研究员(PI)为解决20世纪最著名的代数问题之一--周期群的伯恩赛德问题--而创造的分级图几何机制的进一步应用和发展。特别是，PI将致力于修改适合于群的自由积的这一机制，将研究大指数自由Burnside群的有限的、局部有限的、自由的、正规子群，并研究通过定义关系给出的群的Burnside扩张。此外，PI还将研究自由群的子群上的HannaNeumann猜想的一个推广版本，群的非球面和几乎非球面表示的Whitehead非球面猜想，以及对Higman，B.Neumann和dh发现的群的经典嵌入定理的加强。诺伊曼在1940年代的S。PI还将继续研究三维拓扑的算法问题，特别是三维球面识别问题的计算复杂性。这项活动将使我们更深入地了解现代代数的基本对象，如周期群，以及更广泛地说，由生成器和关系定义的群。这项研究项目是在群论领域，通过生成元和定义关系来定义群，位于群论与低维拓扑、几何和数理逻辑的交叉点上。群论是一种关于空间对称性的数学理论，它与数学、编码论、数论、拓扑学和数学内部的几何等许多其他学科相互作用。

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.