The Geometry and Topology of Groups Generated by Involutions

在这笔拨款下，PI将研究由对合产生的群的几何和拓扑问题。第一个问题涉及一个新的几何准则，当Coxeter群是Kleinian群。具体来说，如果直角Coxeter群的定义图可以实现为2球锐角三角剖分的1-骨架，那么Coxeter群是双曲的吗？这将几何群论和经典三维双曲几何的技术与三角化的组合问题联系起来。作为本研究的一部分，PI将研究2球的锐角三角剖分的模空间。特别是，这个空间是连通的吗？第二个问题是在双曲三维空间中由反射生成的群中是否包含结群。这与理解结补语的可通约性密切相关。更广泛地说，PI试图解决在任何给定的通约性类中最多有三个双曲结互补的猜想。Boileau, Boyer， Cebanu和PI在最近的工作中对这一猜想取得了重大进展，本项目旨在解决剩余的情况。在第三个项目中，PI为整数的四个拷贝的自由积的外自同构群寻找一个CAT(0)空间。这组外部自同构特别令人愉悦，并且与提案相关，因为它是由对合产生的。由对合产生的群长期以来一直是进一步理解群几何的指导例子。它们本质上是几何的，与现代轨道理论有关。Coxeter、Thurston、Davis和Moussong以及许多其他著名数学家都以不同的方式研究了这些群体，并且它们继续为当前的研究提供信息。施加一些条件，比如反射的存在，可以让人们在一个难题上获得牵引力，并最终看到全貌。这里提出的项目对其他领域有潜在的影响。例如，三角测量的研究，特别是急性三角测量，在计算机辅助设计和科学计算中是重要的。

Under this grant, the PI will work on problems regarding the geometry and topology of groups generated by involutions. The first problem involves a new geometric criterion for when a Coxeter group is a Kleinian group. Specifically, if the defining graph of a right-angled Coxeter group can be realized as the 1-skeleton of an acute triangulation of the 2-sphere, is the Coxeter group hyperbolic? This relates the techniques of geometric group theory and classical 3-dimensional hyperbolic geometry to combinatorial problems concerning triangulations. As part of this study, the PI will investigate the moduli space of an acute triangulation of the 2-sphere. In particular, is this space connected? A second question asks when a group generated by reflections in hyperbolic 3-space contains a knot group. This is deeply related to understanding commensurability of knot complements. More broadly, the PI seeks to address the conjecture that there are at most three hyperbolic knot complements in any given commensurability class. Boileau, Boyer, Cebanu, and the PI have made significant progress on the conjecture in recent work, and this project aims to address the remaining case. In a third project, the PI seeks a CAT(0) space for the group of outer automorphisms of the free product of four copies of the integers mod 2. This group of outer automorphisms is particularly pleasing and related to the proposal as it is generated by involutions.Groups generated by involutions have long been guiding examples in furthering the understanding of the geometry of groups. They are inherently geometric and relevant to the modern theory of orbifolds. These groups have been studied in various ways by Coxeter, Thurston, Davis, and Moussong, as well as by many other celebrated mathematicians, and they continue to inform current research. Imposing some condition, such as the presence of a reflection, allows one to gain traction on a difficult problem and eventually see the whole picture. The projects proposed here have potential impact in other fields. For example, the study of triangulations, particularly acute triangulations, is important in computer-aided design and scientific computing.