Lior Rosenzweig: Random walks inside linear algebraic groups

Let G be a linear algebraic group (e.g. SL(n)), and let A be a finitely generated subgroup of it. In the talk I will describe how the sieve method, a combinatorial method used traditionaly to count primes, can be adopted to prove 'genericity' of certain properties inside A. If time allows, I will show how using this method can prove that a generic element in an arithmetic group ( SL(n,Z) for example) has as irreducible characteristic polynomial as possible over the rationals. A special role is played by property tau, and the recent results that many finitely generated groups have this property.