Free Burnside Groups and Related Topics

伊万诺夫DMS-0400746奖摘要：该项目的主要主题是进一步应用和发展由首席研究员（PI）创建的梯度图的几何机制，以解决20世纪最著名的代数问题之一，周期群的Burnsideproblem。特别是，PI将致力于改进这种机制，使其适合于群的自由积，将研究大指数自由Burnside群的有限，局部有限，自由，正规子群，并研究通过定义关系给出的群的Burnside扩展。此外，PI将研究关于自由群的子群的HannaNeumann猜想的广义版本，关于群的非球面和几乎非球面表示的whitehead非球面猜想，以及加强由Higman， B. Neumann和h . Neumann发现的群的经典嵌入定理。40年代的诺伊曼。PI还将继续研究三维拓扑的算法问题，特别是三维球体识别问题的计算复杂性。所提出的活动应该导致对现代代数的基本对象，如周期群，更一般地说，由生成器和关系定义的群，有更深刻的理解。本课题属于群理论领域，研究通过生成器和定义关系定义的群，处于群理论与低维拓扑、几何和数理逻辑的交叉领域。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。

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.