Sieve Methods in Group Theory

最近几年在扩展图和性质“tau”方面取得了巨大的进步；这些发展导致了“仿射筛法”的出现，它表明，对于作用于n维欧氏空间中整数格点集的许多群，每个轨道都有无穷多个向量，它们的项几乎都是素数，也就是说，每个项都是有限数目素数的乘积。这是经典数论结果的非对易版本。我们计划将这一方向进一步发展，并将类似的方法应用于发展研究纯群论问题的“群筛法”。这种方法似乎适合于解决经典群论方法无法解决的问题。将这种方法应用于几何和拓扑学中的各种重要群，例如映射类群，也有望给出一些几何应用。因此，广义数论方法有望有一些重要的几何应用。群作为对称性作用于某些数学对象是数学研究的核心。大多数数学和物理问题都被建模为作用于某一集合的群体。这项建议涉及某些几何和组合对象上的群，并研究这些作用的性质。本提案中讨论的主题涉及几个研究领域之间的联系，并说明了数学的统一性及其与计算机科学的联系。

The last few years showed a dramatic progress on expander graphs and property 'tau'; these developments led to the 'affine sieve method' showing that for many groups acting on the set of integer lattice points in n-dimensional Euclidean space, each orbit has infinitely many vectors whose entries are all almost primes, i.e., every entry is a product of a bounded number of primes. This is a non-commutative version of classical number theoretic results. We plan to take this direction some steps further and to apply similar methods for developing 'group sieve method' for the study of pure group theoretical problems. This method seems to be suitable for solving problems which are out of reach by the classical group theoretic methods. Applying this method to various groups of important in geometry and topology- e.g., the mapping class group, it is expected to give also some geometric applications. So all together generalized number theoretic methods are expected to have some significant geometric applications.Groups acting on certain mathematical objects as symmetry is in the heart of mathematical research. Most mathematical and physical questions are modeled as group acting on a certain set. This proposal deals with groups on certain geometric and combinatorial objects and is to study properties of these actions. The topics discussed in this proposal involve connections between several areas of research and illustrate the unity of mathematics and its connection with computer science.