Geometry of Artin Group Actions

群是一种对对称的数学概念进行编码的结构。虽然人们可能认为对称性是一个几何概念，例如在雪花或墙纸的对称性中，但数学家研究的是一个更抽象的对称性概念，即保持物体特征不变的可能的变换。这类更大的对称性包括从洗一副牌到编辫子时摆弄头发。对这种被称为群论的广义对称性概念的研究在现代数学中起着关键作用，因为了解物体的对称性是深入理解该物体的垫脚石。几何群论是数学领域，旨在通过将这些更抽象的对称群实现为新几何对象的对称性来理解它们。这样做可以让人们使用几何方法来研究这些群的结构，由此产生的代数和几何之间的对话在最近几年被证明是特别有成效的，无论是在数学内外。这个项目集中在一类被称为Artin群的群上，这是涉及编辫的群的广泛推广，在数学的许多领域和更远的领域有分支。虽然辫子群的结构已被较好地了解，但对于一般的Artin群来说，情况要神秘得多，许多重要的和自然的问题仍然悬而未决。这个项目将引入一个新的几何框架来研究一般Artin群。近年来，从几何的角度，特别是从满足某种形式的非正曲率的空间上的作用的角度，研究了来自不同视界的大类群，取得了很大的成功。这就是将在这个项目中进行的这样一种方法。更准确地说，这个项目将通过它们在双曲空间上的作用来研究大类Artin群，并将使用这种作用的动力学来理解这些群的结构。该项目还将突出与其他重要类别群体的结构相似之处。这项工作代表了在代数、组合几何和负曲率动力学之间的十字路口的一个令人兴奋的项目。它将涉及与来自加拿大和法国的研究人员的合作。中途将组织一个研讨会，以便将从不同角度研究ARTIN群体的专家聚集在一起：算法群体理论、组合学等。

Groups are a structure that encode the mathematical idea of symmetry. While one may think of symmetry as a geometric notion, such as in the symmetries of a snowflake or of a wall-paper, mathematicians investigate a more abstract notion of symmetry, namely the possible transformations that leave features of an object unchanged. This larger class of symmetries ranges from shuffling a deck of cards to the manipulation of strings of hair when making a braid. The study of this generalised idea of symmetry, known as group theory, plays a key role in modern mathematics, as understanding the symmetries of an object is a stepping stone towards a deeper understanding of that object.Geometric group theory is the field of mathematics that aims to understand these more abstract symmetry groups by realising them as symmetries of new geometric objects. Doing so allows one to use geometric methods to investigate the structure of these groups, and the resulting dialogue between algebra and geometry has proved particularly fruitful in recent years, both within and outside mathematics.This project focuses on a class of groups known as Artin groups, a vast generalisation of the groups involved in making braids, which have ramifications in many areas of mathematics and beyond. While the structure of braid groups is relatively well understood, the situation is much more mysterious for general Artin groups, and many important and natural questions remain open.This project will introduce a new geometric framework to study general Artin groups. In recent years, large classes of groups from various horizons have been studied with great success from a geometric viewpoint, and particularly from the point of view of actions on spaces satisfying some form of non-positive curvature. This is such an approach that will be carried out in this project. More precisely, this project will study large classes of Artin groups through their actions on hyperbolic spaces, and will use the dynamics of such actions to understand the structure of these groups in great generality. This project will also highlight structural similarities with other important classes of groups. This work represents an exciting project at the crossroads between algebra, combinatorial geometry, and dynamics in negative curvature. It will involve collaborations with researchers from Canada and France. A workshop will be organised halfway through, in order to bring together experts studying Artin groups from various perspectives: algorithmic group theory, combinatorics, etc.