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三流形拓扑和几何群论之间的一些引人注目的问题。其中一个主要项目旨在通过算法理解三流形群的类，在具有可解字问题的群的类中。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“Dehn填充”的过程。最后，PI将研究(再次通过算法)自由群中的一类“虚拟几何”字。三-流形是局部三维的拓扑对象，就像我们宇宙的空间部分。几何群论是使用几何学和拓扑学的工具来研究无限群。最近解决的几何化猜想的一个结果是，(大多数)三维流形是由它们的基本群决定的。由此推论，所有的拓扑(和几何！)在它的基本群中编码，然后可以用几何和算法群论来研究它。这项工作的一个具体产品将是有助于探索这类三重形式演示的软件。

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.