Topology and geometry of Cayley graphs for groups

本研究的目的是探讨群的拓扑/几何性质与解决字、共轭和相关问题的算法之间的相互作用。本研究的主要方向包括：(1)自堆叠结构在3流形群、Thompson’s群F和亚串群中的应用；这些结构使用Cayley图的拓扑和形式语言理论特性的组合来表征，并为解决字问题和为群构建Dehn图提供算法。(2)群的共轭类和测地线字元（直至共轭）渐近增长的研究。(3)通过研究等径（即内在直径）不等式及其外在类比的改进，提高对填充不变量的理解。(4)群不变量在结理论问题中的应用；局部移动，结合标准的Reidemeister操作，将被研究，以确定这些局部移动何时是（或不是）解结或解连接操作。建议的工作是围绕在群论，几何和计算机科学之间的接口数学问题。群论是对物体对称性的数学研究；这一研究领域起源于对化学晶体结构的研究，但现在在数学和科学的许多不同领域都有广泛的应用。近年来，结理论也得到了广泛的应用，特别是通过局部移动解开结的研究被用来模拟酶对DNA分子的作用。所提出的方法充分利用这些领域中的一个或多个领域的工具来阐明另一个领域中的重要问题。特别是几何方法用于追求计算有效和高效的群算法，算法方法用于确定群的几何性质，包括应用于结理论。该项目包括指导本科生和研究生进行数学研究。

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.