Research in Geometric Group Theory

本项目涉及几何群论和非正曲率空间的问题，重点研究Artin群和Garside群。Artin组的范围很广，包括编织组、自由组和自由阿贝尔组。项目的第一部分涉及直角Artin群的自同构群。它们是有限生成的群，其表示只涉及生成子之间的换向子关系。通过改变交换子关系的数目，直角Artin群可以看作是自由群和自由阿贝尔群之间的“插值”。自由群的自同构群是近年来研究的热点。它们已被证明与线性群，特别是与自由阿贝尔群的自同构群有许多共同之处。本项目旨在通过研究一般直角Artin群的自同构群，将这些结果置于更广泛的背景下。关键思想是构造一个可收缩空间，类似于Culler和Vogtmann的“外层空间”，自同构群作用于其上。项目的第二部分涉及加赛德集团。Garside组是具有类似于辫子组的算法属性的组。群上的Garside结构为研究群的组合性质和几何性质提供了一个非常有力的工具。这种方法已广泛应用于有限型Artin群的研究中。虽然对于无限型Artin群不存在这样的结构，但似乎在这些群中至少有一些可能存在类似Garside结构的东西。该项目将考虑Garside群及其性质的各种概括。几何物体的对称性自古以来就被研究，并在数学和科学的许多领域发挥了重要作用。它们构成了“群”这个抽象数学概念的模型。在数学的几乎每个领域都出现了群体。虽然不是所有的群都自然地成为对称群，但总是有可能构造出几何物体，使给定的群在其上充当对称。近年来，群与几何之间的这种相互作用被越来越多地用于理解各种类型的无限群。适合这种技术的一类特别有趣的群体是“编织群体”。n链编织组，顾名思义，编码了一组n个字符串的不同编织方式。它还描述了n个粒子的集合在平面上运动的不同方式。编织组在拓扑学、数学物理和密码学中都有应用。另一类基本群体是“自由群体”。这些群构成了研究群的算法和组合性质的基础。在这个项目中，我们将研究一大类群体，称为Artin群体，其中包括辫状群体、自由群体和许多其他群体。该项目旨在通过研究与之相关的几何物体，以及寻找新的算法技术，更好地了解Artin群体。

This project involves problems in geometric group theory and spaces of non-positive curvature with a focus on Artin groups and Garside groups. Artin groups span a wide range of groups including braid groups, free groups, and free abelian groups. The first part of the project concerns automorphism groups of right-angled Artin groups. These are finitely generated groups whose presentations involve only commutator relations between generators. By varying the number of commutator relations, right-angled Artin groups may be viewed as "interpolating" between free groups and free abelian groups. The automorphism groups of free groups have been the focus of much research in recent years. They have been shown to have much in common with linear groups, in particular with the automorphism groups of free abelian groups. This project aims to put these results in a broader context by studying the automorphism group of a general right-angled Artin group. The key idea is to construct a contractible space, analogous to Culler and Vogtmann's "outer space", on which the automorphism group acts.The second part of the project concerns Garside groups. Garside groups are groups with algorithmic properties similar to those of braid groups. A Garside structure on a group gives a very powerful tool for studying the combinatorial and geometric properties of the group. This approach has been used extensively in the study of finite type Artin groups. While no such structure exists for infinite type Artin groups, it appears that something close to a Garside structure may exist for at least some of these groups. The project will consider various generalizations of Garside groups and their properties.Symmetries of geometric objects have been studied since ancient times and have played an important role in many areas of mathematics and the sciences. They form the model for the abstract mathematical notion of a "group". Groups arise in nearly every field of mathematics. While not all groups occur naturally as symmetry groups, it is always possible to constructgeometric objects on which a given group acts as symmetries. In recent years, this interplay between groups and geometry has been increasingly exploited to understand various types of infinite groups. A particularly interesting class of groups that lends itself to such techniques are the "braid groups". The n-strand braid group, as the name suggests, encodes the different ways that a set of n strings can be braided. It also describes the different ways that a collection of n particles can move around in a plane. Braid groups have applications to topology, mathematical physics, and cryptography. Another fundamental class of groups are the "free groups". These groups form the foundation for studying algorithmic and combinatorial properties of groups in general. In this project we will study a large class of groups, known as Artin groups, that includes the braid groups, the free groups, and many others. The project aims to reach a better understanding of Artin groups by studying geometric objects associated with them, as well as searching for new algorithmic techniques.