Geometry and algorithms in topology and group theory

在这个项目的过程中，PI将与合作者一起解决一些引人注目的问题，在三流形拓扑和几何群论之间的接口。 其中一个主要的项目旨在理解类的三流形组算法，在类的群体与可解的字的问题。另一个项目是从双曲群和相对双曲群的边界的角度来理解群论“德恩填充”的过程。最后，PI将研究（同样在算法上）自由群中的“虚拟几何”词类。三维流形是局部三维的拓扑对象，就像我们宇宙的空间部分。 几何群论是利用几何学和拓扑学的工具研究无限群。 最近解决的几何化猜想的一个结果是（大多数）三流形是由它们的基本群决定的。 因此，所有的拓扑结构（和几何结构！）一个三流形的基本群被编码，然后可以使用几何和算法群论进行研究。这项工作的一个具体产品将是软件，用于探索类的三个流形的介绍。

In the course of this project, the PI will work with collaborators to solve a number of compelling problems at the interface between three-manifold topology and geometric group theory.  One of the main projects aims at understanding the class of three-manifold groups algorithmically, within the class of groups with solvable word problem. Another project is to understand the procedure of group theoretic "Dehn filling" from the perspective of boundaries of hyperbolic and relatively hyperbolic groups. Finally, the PI will study (again algorithmically) the class of "virtually geometric" words in free groups.Three-manifolds are topological objects which are locally three-dimensional, like the spatial part of our universe.  Geometric group theory is the study of infinite groups using tools from geometry and topology.  A consequence of the recently resolved Geometrization Conjecture is that (most) three-manifolds are determined by their fundamental groups.  It follows that all the topology (and geometry!) of a three-manifold is encoded in its fundamental group, which then can be studied using geometric and algorithmic group theory. One concrete product of this work will be software useful for exploring the class of three-manifold presentations.