Research in Geometric Group theory: Artin groups and Automorphism Groups

该项目有两个主要部分。 第一个是直角Artin群的自同构群。 直角Artin群是唯一的关系子是平行子的平行表示群。 它们可以看作是自由群和自由阿贝尔群之间的插值。 因此，它们的（外）自同构群内插在Out（F_n）和GL（n，Z）之间，并为理解这些群之间的相似性和差异性提供了背景。 目的是推广研究自由群的自同构群所使用的技巧-例如构造一个具有适当Out（F_n）作用的可收缩“外层空间”-并利用这些技巧来研究任意直角Artin群的自同构群的性质。 该项目的第二部分涉及更一般的阿廷组与无限Coxeter组。 该项目将解决一些关于这些群的粗糙几何的问题。几何群论可以被看作是对几何对象的对称群的研究。 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。 这些复合体可以作为机器人中各种问题的模型，并应用于其他数学领域。 直角阿廷群作为某些立方复形的对称群而出现。 此外，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组，该项目将研究更一般的Artin组的大规模几何。 一些阿廷群，如辫群，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。 其他阿廷集团是更加神秘和一些最基本的问题，在群论，如“字”和“共轭”的问题，仍然没有答案，为这些团体。 通过研究它们的大规模几何形状，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.