Groups: representations and presentations

Yale University, New Haven CT

Investigators

Abstract

Groups are basic objects in mathematics - expressing symmetries of various objects. The groups themselves can be described by generators and relations - so called "presentations of groups". Given a group, one may want to understand its "representation" i.e. the way to present the group as a group of matrices over the field of complex numbers. The proposed research will focus on quantitative questions of both aspects: How e±ciently can a group be presented? How many genera- tors and relations are needed and what is their length? The motivation for these questions comes from computational group theory: an efficient presentation is important for efficient computation. The quantitative aspects of representation theory deal with questions of the form: What is the rate of growth of the number of representations as a function of the dimension? It is an elaboration of the theory of subgroup growth which deals with permutational representations. Usually presentations and representations are very much disjoint in their studies. One of the interesting features of the proposed research is a connection between the two. If successful, it can shed new light on the representation theory of the automorphism group of the free group and of the mapping class groups.