Infinitesimal Lie rings: gradings & obstructions

Mathematical objects are often not as difficult as they seem. In order to show that a given object A is easy, simple, or small in some precise sense, we approximate it with other another kind of object B that is better understood. In the context of this project, we approximate a group G with a ring L(G), and we use the techniques that are available for rings to show that G is not so bad after all. This method helped solve some of the most interesting problems in group theory. It contributed to the solution of the restricted Burnside conjecture, of the co-class conjectures of Leedham-Green and Newman, and of the Frobenius conjecture. If we want to go in the other direction, and show that a group G is difficult, complicated, or large in a precise sense, it sometimes suffices to show that the ring L(G), which approximates G, behaves badly. This was the case with a geometric conjecture of Milnor. In fact: the conjecture was disproven by showing that L(G) cannot always be written down using small matrices. In this project, we want to detect when the ring L(G) of a group G is well-behaved. And if L(G) is not well-behaved, we want to know how just how bad its behaviour can be. An answer to these questions will help us come to a better understanding of what groups are like, and it will allow us to solve many more interesting problems about groups.

The world is full of symmetries. We find them in a star-fish, in a honeycomb, and even in a board of chess. In the world of mathematics, we study these symmetries in a very precise way. It turns out that all the symmetries of any given object form an interesting mathematical idea known as a "group." And the properties of this group can tell us something important about the original object (the star-fish, or honeycomb, or chess board, or a Rubick's cube, or ). Let us explain this definition of a group by means of a simple example. We consider a circle and we divide it into twelve equal segments (like a clock without numbers). If we first pick up the circle, then rotate it by 60 degrees, and then put it back down, then we end up with the same object that we started out with. Similarly, if we first pick up the circle, then reflect it horizontally, and then put it back down, then we again end up with the same object. These two operations are examples of symmetries. But our segmented circle has many more symmetries. For example: the "trivial" symmetry (in which we pick up the circle and then immediately put it back down again). Note that we can also "combine" symmetries. If we first perform the reflection-symmetry and then perform the rotation-symmetry, we end up with yet another symmetry of the circle. The second important property of symmetries is that they can be "undone". For example: the symmetry that reflects the circle horizontally undoes itself. And the symmetry that rotates the circle by 60 degrees is undone by the symmetry that rotates the circle by 60 degrees in the other direction. But this combination of symmetries depends on the order in which we perform them! For example: If we first perform the reflection-symmetry and then the rotation-symmetry, then we obtain a symmetry of the circle that is different from the symmetry that is obtained by first rotating and then reflecting. This abstract definition of "groups" can help us study problems in physics, geometry, algebra, topology, analysis, and so on. But it is also possible to study problems about groups themselves by looking at symmetries of groups! Such symmetries of groups are called "automorphisms". In this project, we study these "symmetries of symmetries." We have generalized this classical theorem (known as the positive solution of the Frobenius conjecture) to a very general one about finite groups with a fixed-point-free automorphism satisfying a polynomial identity.