The Geometry and Topology of Groups Generated by Involutions

根据这项补助金，PI将致力于对合所产生的几何和拓扑群的问题。 第一个问题涉及一个新的几何准则时，Coxeter组是Kleinian组。具体地说，如果直角Coxeter群的定义图可以实现为2-球面的锐角三角剖分的1-骨架，那么Coxeter群是双曲的吗？ 这涉及到几何群论和经典的三维双曲几何的技术组合问题有关三角剖分。作为本研究的一部分，PI将研究2-球面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3-空间中由反射生成的群何时包含一个纽结群。 这与理解纽结补语的可互换性密切相关。更广泛地说，PI试图解决猜想，即在任何给定的可并行性类中最多有三个双曲纽结补。 Boileau，Boyer，Cebanu和PI在最近的工作中对该猜想取得了重大进展，该项目旨在解决剩余的情况。 在第三个项目中，PI为整数模2的四个副本的自由积的外自同构群寻找CAT（0）空间。 这组外自同构是特别令人愉快的，并与建议有关，因为它是由对合生成的。对合生成的群长期以来一直是进一步理解群几何的指导性例子。 它们本质上是几何的，与现代的轨道褶皱理论有关。这些群体已经研究了各种方式的考克斯特，瑟斯顿，戴维斯，和Moussong，以及许多其他著名的数学家，他们继续通知目前的研究。强加一些条件，比如反射的存在，可以让一个人在一个困难的问题上获得牵引力，并最终看到整个画面。 这里提出的项目在其他领域有潜在的影响。 例如，三角剖分的研究，特别是锐角三角剖分，在计算机辅助设计和科学计算中很重要。

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.