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

点击翻译按钮获取中文摘要

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.