Geometry and algorithms in topology and group theory

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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.