Research in Geometric Group Theory

这个项目涉及几何群论和非正曲率空间的问题，重点是Artin群和Garside群。Artin基团跨越广泛的基团，包括辫子基团、自由基团和自由阿贝尔基。该项目的第一部分涉及直角Artin群的自同构群。这些群是有限生成的群，它们的表示只涉及生成元之间的交换子关系。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“内插”。自由群的自同构群是近年来研究的热点。它们与线性群有许多共同之处，特别是与自由阿贝尔群的自同构群。这个项目旨在通过研究一般直角Artin群的自同构群，将这些结果放在更广泛的背景下进行研究。其核心思想是构造一个可收缩的空间，类似于卡勒和沃格特曼的“外层空间”，自同构群作用于这个空间上。Garside群是具有类似于辫子群的算法性质的群。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。这种方法在有限型Artin群的研究中得到了广泛的应用。虽然无限类型Artin群不存在这样的结构，但似乎对于这些群中的至少一些，可能存在类似Garside结构的东西。该项目将考虑Garside群的各种推广及其性质。几何对象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着重要作用。它们构成了抽象的数学概念“群”的模型。群体几乎出现在数学的每一个领域。虽然不是所有的群都自然地以对称群的形式出现，但总是可以构造几何对象，在这些几何对象上给定的群充当对称性。近年来，群和几何之间的相互作用越来越多地被用来理解各种类型的无限群。采用这种技术的一类特别有趣的群体是“辫子群”。顾名思义，n股编织组对一组n股编织的不同方式进行了编码。它还描述了n个粒子的集合可以在平面上四处移动的不同方式。辫子群在拓扑学、数学物理和密码学中都有应用。另一类基本的群体是“自由群体”。这些群构成了研究一般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，称为Artin群，它包括辫子群、自由群和许多其他群。该项目旨在通过研究与Artin组相关的几何对象以及寻找新的算法技术来更好地理解Artin组。

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.