Classifying subgroups of the modular group using Wicks forms

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.