Classifying subgroups of the modular group using Wicks forms

模群（M）是一个特殊的数学对象，它具有特殊的几何性质，我们的目标是利用这些性质来分类M的所谓子群，这些子群的结构可以揭示关于群本身的信息。模群可以被认为是一个机器，它可以根据特定的规则在二维网格上移动点。这种几何行为的大部分是众所周知的，研究可以追溯到世纪初的庞加莱、德恩和凯莱。然而，最近，Vdovich（1995）创新地使用了称为Wicks形式的组合对象，以揭示模群的一些性质。Wicks形式的根源在于数学的一个完全不同的分支：拓扑学。为了构造Wicks形式，我们考虑如何将“平面”多边形（例如纸张）变成更有趣形状的表面，如球体或圆环（甜甜圈）。我们利用弯曲和胶合的严格概念。举个例子，一张长方形的纸，在它的边缘顺时针写上字母ABAB，这样相对的边缘就用同一个字母标记。我们的目标是用相同的标签将所有成对的边粘在一起。如果我们先把A的边粘在一起，我们就得到一个圆柱体，它的圆端都标为B。然后，我们将圆柱体缠绕成甜甜圈形状，将两个B圆粘合在一起。因此，我们已经将一个平面处理成一个环面，实际上我们可以执行相同的弯曲和胶合算法来创建几乎任何曲面（没有穿孔，也没有相交）。事实上，我们需要的唯一指令是字母在纸边缘的顺序，ABAB。显然，并不是所有这些“词”都会产生漂亮的表面（例如，没有明确的方法来粘一个标有ABBAC的五边形）;那些起作用的词被称为Wicks形式。2这样的词比复杂的曲面几何形状容易处理得多，Bacher和Vdovich（2002）已经能够计算给定长度的所有Wicks形式-这反过来又为我们提供了一些关于如何三角测量的有价值的信息此外，给定长度的Wicks形式与模群M的给定大小的子群一一对应，这是通过Brenner和Lyndon（1983）中发现的技术实现的。这意味着理解Wicks形式的行为可以直接洞察M的性质。这是我们的目标，汇集方法从拓扑学，几何学和组合学的方式从来没有做过，以进一步分类这些子群，并了解更多的类型的子群M。希望这些方法也可以用来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，线性代数和表示论中具有根本的重要性。我们将研究使用图形结构，如Bass-Serre理论和Bruhat-Tits建筑物，尝试将这些方法应用于上述群，沿着使用Wicks形式算法来探索M的子群和所谓的陪集图之间的联系，类似于前面描述的胶合图。事实上，这些方法的算法性质已经引起了计算机科学期刊的引用，几何性质引起了纽结理论家和几何群理论家的注意，所以有大量的证据表明人们对这项研究很感兴趣。

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.