Groups: representations and presentations

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