Groups: representations and presentations

群是数学中的基本对象--表示各种对象的对称性。群体本身可以用生成者和关系来描述--即所谓的“群体呈现”。给定一个群，人们可能想要理解它的“表示”，即将群表示为复数域上的一组矩阵的方法。本文的研究将集中在这两个方面的定量问题上：如何表示群？需要多少类属和关系，它们的长度是多少？这些问题的动机来自计算群论：有效的表示对于有效的计算是重要的。表示理论的量化方面处理形式的问题：表示的数目作为维度的函数的增长率是多少？它是关于置换表示的子群增长理论的阐述。在它们的研究中，表示和表示通常是非常脱节的。这项拟议的研究的一个有趣的特点是两者之间的联系。如果成功，它可以为自由群的自同构群和映射类群的自同构群的表示理论提供新的启示。

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.