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

该提案的目的是汇集来自英国和以色列的经验丰富的数学家研究团队，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。参与的研究人员将在2010年夏天在伦敦的皇家霍洛威大学和内盖夫大学（以色列）举行两次会议，以建立他们最近的突破，这已经导致了一个相当特殊的情况下证明了猜想。数学、物理或化学对象的对称性--如图形、分子或晶体--形成了一种称为群的代数结构。对有限群的研究导致了20世纪世纪数学中最引人注目的成就之一：所有有限单群的分类。一个重要的工具是通过它们的线性表示来研究群，即通过它们作为矩阵群的图像。渐近群论，旨在理解有限和无限群一样，关注的是某些算术不变量的群的渐近性质。在这个相对年轻的群论领域中，一个经典的方向是研究词的增长，这是由格罗莫夫的开创性工作而闻名的。在另一个方向，子群增长理论，研究无限群通过调查他们的有限指数子群的分布。“群的Zeta函数”-产生复杂函数的某些无穷级数-用于对相关增长序列的算术进行编码。最近，渐近群论的研究者们开始研究群的表示的分布，他们利用了在词和子群增长的背景下发展起来的技术。在表示增长研究的渐近性和算术的数量不可约复表示的任何给定程度提供了一个组。同样，zeta函数在近年来建立该领域的第一个重要成果方面发挥了关键作用。一类重要的无限群，由于多种原因而受到关注，由李群中的格组成。这些群是典型的非交换群，但它们是交换整数环的推广，而交换整数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，在一个高阶半单李群中，算术格的表示数渐近地只依赖于周围群，而不依赖于所选择的特定格。就所涉及的zeta函数而言，这意味着这样一个格的zeta函数的收敛横坐标实际上是环境李群的不变量。我们对证明这一猜想的方法建立在一个相当不同的技术，从渐近群理论和同源学科的合成。

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.