Classifying subgroups of the modular group using Wicks forms

使用 Wicks 形式对模群的子群进行分类

基本信息

批准号：
2125193
负责人：
金额：
--
依托单位：
Newcastle University
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-2125193
关键词：
Classifying subgroups modular group using

项目摘要

The modular group (denoted by M) is a special mathematical object which has peculiar geometrical properties, and our aim is to use these to classify so-called subgroups of M, structures which can reveal information about the group itself.The modular group can be thought of as a machine for moving points on a two-dimensional grid around according to specific rules. Much of this geometrical behaviour is well-known, and studies date back to Poincaré, Dehn and Cayley in the early 20th century. More recently, however, Vdovina (1995) has innovated the use of combinatorial objects called Wicks forms to shed new light on some of the modular group's properties.Wicks forms have their roots in an altogether distinct branch of mathematics: that of topology. To construct a Wicks form, we think about how to turn "flat" polygons (e.g. sheets of paper) into surfaces of a more interesting shape, like the sphere or torus (doughnut). We make use of the rigorous notion of bending and gluing. Take, for example, a rectangular sheet of paper, and write clockwise around its edges the letters ABAB, such that opposite edges are labelled with the same letter. We aim to glue together all pairs of edges with the same label. If we glue the A edges first, we obtain a cylinder, the circular ends of which are both labelled B. Then, we wrap the cylinder around into a doughnut shape, gluing the two B circles. Thus we have manipulated a flat surface into a torus, and indeed we can perform the same bending and gluing algorithm to create almost any surface (without punctures, and without intersecting itself). Indeed, the only instruction we needed was the ordering of the letters around the edge of the sheet, ABAB. Clearly not all such "words" will give rise to nice surfaces (e.g. there is no clear way to glue a pentagon labelled ABBAC); those words which do work are called Wicks forms.Such words can be much easier to deal with than the complex geometry of surfaces, and Bacher and Vdovina (2002) have been able to count all Wicks forms of a given length - this in turn gives us some valuable information on how to triangulate (that is, divide into triangles) a given surface, as each Wicks form corresponds directly to a particular triangulation.Moreover, Wicks forms of a given length are in one-to-one correspondence with subgroups of a given size of the modular group M, by means of technology found in Brenner and Lyndon (1983). This means that understanding the behaviour of Wicks forms can give direct insight into the properties of M. It is our aim to bring together methods from topology, geometry, and combinatorics in ways which have never before been done in order to classify these subgroups further, and learn more about the types of subgroups M has. Hopefully these methods can also be used to study the subgroups of other important mathematical objects, such as Hecke groups or the special linear group, the latter of which is of fundamental importance in Euclidean geometry, linear algebra, and representation theory.We will investigate the use of graphical structures such as Bass-Serre theory and Bruhat-Tits buildings to try and apply these methods to the above groups, along with using the Wicks forms algorithms to explore the connections between the subgroups of M and so-called coset diagrams, similar to the gluing diagrams previously described.Indeed, the algorithmic nature of these methods have resulted in citations from computer science journals, and the geometric nature garners attention from knot theorists and geometric group theorists, so there is plenty of evidence for interest in this research.

模块组（由M表示）是一个具有特殊几何特性的特殊数学对象，我们的目的是使用它们来对M的所谓亚组进行分类，可以将有关该组本身的信息揭示信息的结构进行分类。可以将模块化群体视为围绕特定规则的二维行为和众所周知的行为。 20世纪初期的Cayley。然而，最近，Vdovina（1995）具有创新的使用，即使用称为Wicks形式的组合物体来对一些模块化组的某些属性发明新的启示。Wicks形式的根源在数学的完全不同的分支中根源：拓扑结构。要构建一个灯芯形式，我们考虑如何将“平坦”多边形（例如纸的床单）变成更有趣的形状表面，例如球体或圆环（甜甜圈）。我们利用严格的弯曲和胶水通知。以矩形的纸张，然后在其边缘围绕字母abab时顺时针循环，以使相反的边缘标记为相同的字母。我们的目标是将所有边对与相同的标签粘合在一起。如果首先将A边缘粘合，则获得一个圆柱体，其圆形末端都标记为B。然后，我们将圆柱体缠绕成甜甜圈形，将两个B圆胶粘成。我们已经将平坦的表面操纵到圆环中，实际上，我们可以执行相同的弯曲和胶合算法来创建几乎所有表面（没有穿刺，而不会与自身相交）。的确，我们唯一需要的指示是在纸的边缘Abab的边缘订购字母。显然，并非所有这些“单词”都会引起良好的表面（例如，没有明确的方法可以粘合标有Abbac的五角大楼）；与表面的复杂几何形状相比，这些工作的单词被称为灯芯形式。通过Brenner和Lyndon（1983）中发现的技术，在与给定大小M的给定大小的亚组的一对一对应中。这意味着了解WICKS形式的行为可以直接深入了解M的特性。我们的目的是以前所未有的方式将方法汇集在一起，以便进一步对这些亚组进行分类，并了解更多子组类型M的方法。希望这些方法也可以用于研究其他重要数学对象的亚组，例如Hecke群体或特殊线性组，后者在欧几里得几何形状，线性代数和表示理论中至关重要，我们将使用图形结构，并将其与诸如Bass-serre理论构建相同的图形结构进行操作，以尝试这些方法，并将其与Bruhat-Tits Trubsers一起使用，以实现这些方法。探索M和所谓的固定图表之间的连接算法，类似于先前描述的粘合图。这些方法的算法性质导致了计算机科学期刊的引用，而几何性质也引起了从结论理论家和几何组的关注，因此对此有很多兴趣，因此对此有很多兴趣。