Geometric and computational group theory, in and around three-manifolds

从一个角度来看，三流形是对宇宙可能形状的描述。从另一个角度来看，它是这样一个宇宙的一个可能的对称群，推广了化学家和其他人使用的晶体学群。在这个项目中，主要研究者使用抽象和实用的算法，以及更传统的数学技术，来研究与3流形群相关的对称群。将产生的定理和软件应该对其他3流形和几何群论的研究者有用。首席研究员将从事几何群论的两个项目。第一个应用计算机算法来理解(1)自由群中的虚拟几何词集和(2)简单环路猜想的可能反例空间。第二个项目将完善和扩展我们对群理论Dehn填充的理解。

A 3-manifold is, from one point of view, a description of a possible shape of the universe. From another point of view, it is a possible symmetry group of such a universe, generalizing the crystallographic groups used by chemists and others. In this project, the Principal Investigator uses abstract and practical algorithms, as well as more traditional mathematical techniques, to study symmetry groups related to 3-manifold groups. The theorems and software that will be produced should be useful to other investigators of 3-manifolds and geometric group theory.The Principal Investigator will work on two projects in Geometric Group Theory. The first applies computer algorithms to understand (1) the set of virtually geometric words in free groups and (2) the space of possible counterexamples to the Simple Loop Conjecture. The second project will refine and expand our understanding of group theoretic Dehn filling.