Research in Algebra
Matrix groups Matrix groups of classical type include, for example, the general linear, symplectic and orthogonal groups, and they play an important role in many different branches of mathematics. A group of this type consists of n-by-n matrices defined over some ring, R, and its structure is determined by both n and R. For a given R, the groups tend to be more complicated structure when n is "low". In contrast for "high" n, the groups usually have much simpler properties, the most important of which are independent of n. In the 1950's it was discovered that many important topological results (arising from homological algebra) followed from properties of the classical matrix groups. This has led to the development of an area called Algebraic K-Theory. As a result a great deal of attention has been devoted to, for example, the normal subgroup and automorphism group structure of the classical matrix groups.This work has been extended to other types of matrix group, for example, the Chevalley groups.
Lower dimensional groups however are also important .The study of such groups began in the 19th century when mathematicians were searching for functions (of a complex variable) which would generalize the classical trigonometric and elliptic functions. The search for such functions, invariant under a given group of linear fractional transformations, focussed attention on certain 2-by-2 matrix groups. By far the most important of these is the modular group, SL(2,Z). This group is the best known example of a larger class called the Fuchsian groups. Such groups are studied via their action, as isometries (or "distance preserving") maps, on a metric space called hyperbolic 2-space. The study of the Fuchsian groups led to the introduction of an imporant area called combinatorial group theory.
The study of groups of this type, via their action on some space, has been extended in a number of directions. Of particular importance in, for example, manifold theory are the so-called Bianchi groups. They are (low dimensional) groups which act as isometries on hyperbolic 3-space and generalize the Fuchsian groups in a natural way. On the other hand some of these groups act on a graph type structure called a tree (as "adjacency preserving" maps) and their study has led to the development of a geometric area called the theory of Bruhat-Tits buildings.
There are important connections between the theory of matrix groups and number theory which arise via the arithmetic groups. Roughly speaking such groups are matrix groups defined over one of the "basic" rings of number theory, for example the integers, Z. Much of the study of these groups has concentrated on the so-called congruence subgroups which are defined arithmetically.
The study of classical matrix groups can be extended to the algebraic groups and this is where currently much of the research activity is concentrated. Here discrete groups like the Fuchsian and Bianchi groups generalize to the so-called lattices. Again number theory plays an important role. ReadingThe following are among the most important texts in this area.
- A. J. Hahn and O. T. O'Meara, The Classical Groups and K-Theory, Springer 1989.
- M. S. Raghunathan, Discrete subgroups of Lie groups, Springer 1972.
- J-P. Serre, Trees, Springer 1980.
- J. Elstrodt, F. Grunewald and J. Mennicke, Groups acting on Hyperbolic Space, Springer 1998.