Discrete and continuous geometry in group theory

群是几何群理论家经常研究的一大群。它们与Coxeter群有关，它们包含并推广了辫群，它们有一个涉及复超平面补的自然定义。尽管与这些已知的对象有关联，但很少有Artin群体目前被理解，即使是在基本层面上，而且那些属于高度限制的家族，主要是由现有技术可以应用的情况来定义的。这里描述的方法从根本上来说是新的和不同的。该提案的目标是理解自然的、连续的几何物体（最近由首席研究员和他的合作者介绍），这些物体有可能作为以统一方式研究所有Artin群体的几何基础。这些几何对象被称为因子几何，它们本身就很有趣。例如，存在一个以正交群为商，包含辫群为子群的自然连续群G，它是有限维度量简单复合体的基本群，该复合体的普遍覆盖是具有G作用于其上的类建筑结构的可收缩因子几何。这个不寻常的不可数群和空间，通过将球面等距的最小因式分解集合定义为反射，是李群和度量简单复合体的混合，每个Artin群都作用于类似定义的连续几何对象。最终的结构理论应该与李群相似，早期的迹象表明，这些类型的复合体具有几何和组合结构，足以解决许多长期存在的关于任意Artin群的猜想。数学对象，就像许多物理对象一样，当我们完全理解它们所具有的对称性时，我们就能更好地理解它们。记录这些对称如何相互作用的代数结构被称为“群”，这里考虑的群是一类由“反射”产生的群（一种像镜子反射图像一样的对称）。这个项目的主要目标是加深我们对对称群（特别是Coxeter群）和第二类对称群（Artin群）之间关系的理解。Artin群中最著名的例子是编织群，这种群保持了几股绳子可以以不同的方式编织在一起。涉及Coxeter群、Artingroups和braid群的结构在整个数学中大量出现，包括一些最近的数学物理。在许多这样的案例中，严格地说，直接的联系是与Artin群体而不是Coxeter群体，但这些联系没有被追究，部分原因是Artin群体的理论不发达。一旦这种情况在几何群论中得到纠正，下一步将是将得到的结构理论导出到邻近的领域

Artin groups are a large class of groups commonly studied by geometricgroup theorists. They are related to Coxeter groups, they include andgeneralize the braid groups, and they have a natural definitioninvolving complexified hyperplane complements. Despite theseaffiliations with well-known objects, very few Artin groups arecurrently understood, even at a basic level, and those that are belongto highly restrictive families defined primarily by the situationswhere existing techniques can be applied. The approach described hereis fundamentally new and different. The goal of this proposal is tounderstand the natural, continuous geometric objects (recentlyintroduced by the principal investigator and his collaborators) thathave the potential to serve as a geometric foundation for the study ofall Artin groups in a uniform fashion. These geometric objects,called factor geometries, are of interest in their own right. Thereis, for example, a natural continuous group G that has the orthogonalgroup as a quotient, contains the braid group as a subgroup, and isthe fundamental group of a finite-dimensional metric simplicialcomplex whose universal cover is the contractible factor geometry witha building-like structure on which G acts. This unusual uncountablegroup and space, defined via the collection of minimal factorizationsof spherical isometries into reflections, is a hybrid mix of Liegroups and metric simplicial complexes, and every Artin group acts ona similarly defined continuous geometric object. The eventualstructure theory should resemble that of Lie groups and earlyindications are that these types of complexes carry geometric andcombinatorial structures sufficient to resolve many longstandingconjectures about arbitrary Artin groups.Mathematical objects, like many physical objects, can be betterunderstood when we fully understand the symmetries they possess. Thealgebraic structure that records how these symmetries interact iscalled a ``group'' and the groups under consideration here are a classof groups generated by ``reflections'' (a symmetry like the reflectedimage one sees through a mirror). The main goal of this project is todeepen our understanding of the relationship between symmetry groupsbuilt from reflections (specifically Coxeter groups) and a secondclass of symmetry groups, called Artin groups. The most famousexample of an Artin group is the braid group, the group that keepstrack of the distinct ways in which several strands of string can bebraided together. Constructions involving Coxeter groups, Artingroups and braid groups proliferate throughout mathematics, includingsome recent mathematical physics. In many of these cases, theimmediate connections are, strictly speaking, to Artin groups ratherthan Coxeter groups, but these connections have not been pursuedpartly because the theory of Artin groups is underdeveloped. Oncethis situation is rectified within geometric group theory, the nextstep will be to export the resulting structure theory to theseneighboring domains