Classifying subgroups of the modular group using Wicks forms

模群（用M表示）是一个特殊的数学对象，它具有特殊的几何性质，我们的目的是用这些特征来分类M的所谓子群，这些结构可以揭示群本身的信息。模群可以被认为是一个机器，根据特定的规则移动二维网格上的点。这种几何行为的大部分是众所周知的，研究可以追溯到20世纪初的庞加莱、德恩和凯利。然而，最近，Vdovina（1995）创新了组合对象的使用，称为Wicks形式，以揭示模块化组的一些属性。威克斯的形式源于数学的一个完全不同的分支：拓扑学。为了构造威克斯形状，我们考虑如何将“平面”多边形（如纸张）变成更有趣形状的表面，如球体或环面（甜甜圈）。我们使用了弯曲和粘合的严格概念。举个例子，拿一张长方形的纸，在它的边缘顺时针写上字母ABAB，这样相对的边缘就标有相同的字母。我们的目标是将所有具有相同标签的边粘合在一起。如果我们先粘合A边，我们得到一个圆柱体，其圆形的两端都标记为B。然后，我们将圆柱体包裹成一个甜甜圈形状，粘合两个B圆。因此，我们已经将一个平面操纵成一个环面，实际上，我们可以执行相同的弯曲和粘合算法来创建几乎任何表面（没有穿孔，也没有交叉）。事实上，我们唯一需要的指令就是纸边上字母的顺序：ABAB。显然，并不是所有这样的“单词”都会产生漂亮的表面（例如，没有明确的方法来粘合标记为ABBAC的五边形）；那些做功的词叫做威克斯形。这样的词比复杂的几何曲面要容易处理得多，Bacher和Vdovina（2002）已经能够计算出给定长度的所有Wicks形式——这反过来又给了我们一些有价值的信息，告诉我们如何对给定的曲面进行三角剖分（即分成三角形），因为每个Wicks形式都直接对应于一个特定的三角剖分。此外，通过Brenner和Lyndon（1983）中的技术，给定长度的Wicks形式与模群M中给定大小的子群是一一对应的。这意味着理解Wicks形式的行为可以直接洞察M的属性。我们的目标是将拓扑，几何和组合学的方法以前所未有的方式结合在一起，以便进一步对这些子群进行分类，并更多地了解M具有的子群类型。希望这些方法也可以用于研究其他重要数学对象的子群，如Hecke群或特殊线性群，后者在欧几里得几何、线性代数和表示理论中具有重要的基础意义。我们将研究使用图形结构，如Bass-Serre理论和Bruhat-Tits建筑，尝试将这些方法应用于上述组，同时使用Wicks形式算法来探索M的子组和所谓的协集图之间的连接，类似于前面描述的粘合图。事实上，这些方法的算法性质已经被计算机科学期刊引用，而几何性质引起了结理论家和几何群理论家的注意，因此有很多证据表明对这项研究感兴趣。

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.