Research in Geometric Group theory: Artin groups and Automorphism Groups

这个项目有两个主要部分。第一个是关于直角Artin群的自同构群。直角Artin群是仅有交换子的有限表示群。它们可以看作是自由群和自由阿贝尔群之间的内插。因此，它们的(外)自同构群介于OUT(F_N)和GL(n，Z)之间，并为理解这些群之间的异同提供了一个背景。目的是推广自由群的自同构群的研究方法，如构造一个具有适当的外(F_N)作用的可压缩的“外层空间”，并用这些方法研究任意直角Artin群的自同构群的性质。该项目的第二部分涉及与无限Coxeter群相关的更一般的Artin群。这个项目将解决一些关于这些群的粗略几何的问题。几何群论可以被视为研究几何对象的对称群。一类特别丰富的几何对象是具有良好局部结构的立方体复合体。这些复合体作为机器人学中各种问题的模型，并应用于数学的其他领域。直角Artin群是某些立方复形的对称群。此外，这些群本身具有有趣的对称性，称为自同构。这个项目将研究这些群的性质，它们的自同构和相关的几何。除了直角Artin群外，该项目还将研究更一般的Artin群的大规模几何。一些Artin群，如辫子群，是很好理解的，并在拓扑学、密码学和其他领域有重要的应用。其他Artin群要神秘得多，群论中一些最基本的问题，如“词”和“共轭”问题，对于这些群来说仍然没有答案。通过研究它们的大规模几何学，该项目旨在深入了解其中一些更神秘的群体。

This project has two main parts. The first concerns automorphism groups of right-angled Artin groups. Right-angled Artin groups are finitely presented groups whose only relators are commutators. They may be viewed as interpolating between free groups and free abelian groups. Thus, their (outer) automorphism groups interpolate between Out(F_n) and GL(n,Z) and provide a context for understanding the similarities and differences between these groups. The goal is to generalize techniques used in the study of automorphism groups of free groups---such as the construction of a contractible "outer space" with a proper Out(F_n) action - and use these techniques to study properties of automorphism groups of arbitrary right-angled Artin groups. The second part of the project concerns more general Artin groups associated to infinite Coxeter groups. The project will address a number of questions regarding the coarse geometry of these groups.Geometric group theory may be viewed as the study of symmetry groups of geometric objects. A particularly rich class of geometric objects are cubical complexes with nice local structure. These complexes serve as models for a variety of problems in robotics and have applications to other areas of mathematics. Right-angled Artin groups arise as groups of symmetries of certain cubical complexes. Moreover, the groups themselves have interesting symmetries known as automorphisms. This project will investigate properties of these groups, their automorphisms and the associated geometries. In addition to right-angled Artin groups, the project will study the large scale geometry of more general Artin groups. Some Artin groups, such as braid groups, are well understood and have important applications to topology, cryptography, and other fields. Other Artin groups are much more mysterious and some of the most fundamental problems in group theory, such as the "word" and "conjugacy" problems, remain unanswered for these groups. By studying their large scale geometry, the project aims to gain insight into some of these more mysterious groups.