Sieve Methods in Group Theory

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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.