Topology and geometry of Cayley graphs for groups

这项研究的目的是研究群的拓扑/几何性质和解字、共轭及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)将可自堆叠结构应用于3-流形群、Thompson群F和亚交换群；利用Cayley图的拓扑和形式语言理论性质来刻画这些结构，并给出了解决字问题和建立群的Dehn图的算法。(2)研究群的共轭类的渐近增长以及测地字和群的元素(直到共轭)的渐近增长。(3)通过研究等径(即内径)不等式及其外在类比的改进，提高了对填充不变量的理解。(4)群不变量在纽结理论问题中的应用研究局部移动与标准的Reidomeister运算相结合，以确定这些局部移动何时是(或不是)解结或解链操作。建议的工作集中在群论、几何学和计算机科学之间的数学问题上。群论是对物体对称性的数学研究；这一研究领域始于化学中的晶体结构研究，但现在已广泛应用于数学和科学的许多不同领域。纽结理论近年来也得到了广泛的应用，特别是通过局部运动解结的研究被用来模拟酶对DNA分子的作用。所提出的方法主要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，几何方法被用来追求计算上有效和高效的群的算法，算法方法被用来确定群的几何性质，包括在纽结理论中的应用。该项目包括对本科生和研究生进行数学研究的指导。

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.