Topology and geometry of Cayley graphs for groups

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The objective of the proposed research is to investigate the interplay between topological/geometric properties of groups and algorithms for solving the word, conjugacy, and related problems. Major directions for this research include: (1) Applications of autostackable structures to 3-manifold groups, Thompson's group F, and metabelian groups; these structures are characterized using a combination of topological and formal language theory properties of the Cayley graph, and provide algorithms for solving the word problem and for building Dehn diagrams for the group. (2) A study of the asymptotic growth of conjugacy classes and of geodesic words and elements (up to conjugacy) of groups. (3) Improving understanding of filling invariants, by studying refinements of the isodiametric (i.e. intrinsic diameter) inequality and its extrinsic analog. (4) Applications of group invariants to problems in knot theory; local moves, in combination with the standard Reidemeister operations, will be studied to determine when these local moves are (or are not) unknotting or unlinking operations.The proposed work is centered around mathematical problems at the interface between group theory, geometry, and computer science. Group theory is the mathematical study of symmetries of objects; this research area has its beginnings in the study of crystal structures in chemistry, but now has wide applications across many disparate areas of mathematics and the sciences. Knot theory also has seen wide application in recent years, and in particular the study of unknotting via local moves is used to model enzyme actions on DNA molecules. The methods proposed make essential use of tools from one or more of these fields to shed light on important problems in another field. In particular geometric methods are used in the pursuit of computationally effective and efficient algorithms for groups, and algorithmic methods are used to determine geometric properties of groups, including applications to knot theory. This project includes mentoring of undergraduate and graduate students in mathematics research.