Representation zeta functions of groups and a conjecture of Larsen-Lubotzky

这项建议的目的是将来自英国和以色列的一支经验丰富的数学家研究团队聚集在一起，在证明渐近群论领域的一个重要猜想方面取得实质性进展。参与研究的研究人员将在2010年夏天在伦敦皇家霍洛威大学和内盖夫大学(以色列)会面两次，以巩固他们最近取得的突破，这些突破已经在一个相当特殊的情况下证明了这一猜想。数学、物理或化学对象--如图形、分子或晶体--的对称性形成了一个称为群的代数结构。对有限群的研究导致了20世纪数学最显著的成就之一：所有有限单群的分类。一个重要的工具是通过群的线性表示来研究群，即通过群的像作为矩阵群来研究群。渐近群论是关于群的某些算术不变量的渐近性质的，其目的是理解有限群和无限群。在这个相对年轻的群论领域，一个经典的方向是研究单词增长，这一研究因格罗莫夫的开创性工作而闻名。在另一个方向，子群增长理论，人们通过研究无限群的有限指标子群的分布来研究无限群。“群的Zeta函数”--某些产生复数函数的无穷级数--被用来编码相关生长序列的算术。近些年来，渐近群论的研究人员已经开始研究群的表示的分布，利用发展起来的技术，例如，在词和子群增长的背景下。在表示增长中，人们研究群所提供的任何给定次数的不可约复表示的数目的渐近性和算术。再一次，Zeta功能在这一领域近年来取得的第一个重大成果中发挥了关键作用。李群中的格是一类重要的无限群，由于多种原因而受到人们的关注。这些群通常是非对易的，但它们推广了整数的交换环，而交换环是经典数论的中心研究对象。Larsen和Lubotzky的一个重要猜想指出，渐近地，高阶半单李群中算术格的表示数只取决于环境群，而不取决于所选择的特定格。就所涉及的Zeta函数而言，这意味着这种格子的Zeta函数的收敛横坐标实际上是环境李群的不变量。我们证明这一猜想的方法建立在来自渐近群论和同源学科的相当不同的技术的综合之上。

The aim of this proposal is to bring together an experienced research team of mathematicians from the UK and Israel to make substantial progress towards proving an important conjecture in the area of asymptotic group theory. The researchers involved will meet twice during the summer 2010, at Royal Holloway University of London and at the University of the Negev (Israel), to build upon their recent breakthrough which already led to a proof of the conjecture in a rather special case. The symmetries of a mathematical, physical or chemical object - such as a graph, a molecule or a crystal - form an algebraic structure called a group. The study of finite groups led to one of the most striking achievements of 20th century mathematics: the classification of all finite simple groups. An important tool is to investigate groups by means of their linear representations, i.e. by their images as matrix groups. Asymptotic group theory, which is aimed at understanding finite and infinite groups alike, is concerned with the asymptotic properties of certain arithmetic invariants of groups. A classical direction in this comparatively young area of group theory is the study of word growth, made famous by groundbreaking work of Gromov. In another direction, the theory of subgroup growth, one studies infinite groups by investigating the distribution of their finite index subgroups. `Zeta functions of groups' - certain infinite series which give rise to complex functions - are used for encoding the arithmetic of associated growth sequences. They have proved powerful tools in developing the theory.More recently, researchers in asymptotic group theory have begun to study the distributions of representations of groups, utilizing the techniques developed, for instance, in the context of word and subgroup growth. In representation growth one studies the asymptotics and the arithmetic of the numbers of irreducible complex representations of any given degree afforded by a group. Again, zeta functions have played a key role in establishing the first significant results in this area during recent years. An important class of infinite groups, which have received attention for manifold reasons, consists of lattices in Lie groups. These groups are typically non-commutative, but generalize the commutative rings of integers which are the central objects of study in classical number theory. An important conjecture of Larsen and Lubotzky states that, asymptotically, the numbers of representations of an arithmetic lattice in a higher rank semisimple Lie group only depend on the ambient group and not on the particular lattice chosen. In terms of the zeta functions involved this means that the abscissa of convergence of the zeta function of such a lattice is, in fact, an invariant of the ambient Lie group. Our approach towards proving this conjecture builds upon a synthesis of quite distinct techniques from asymptotic group theory and cognate disciplines.