Sieving, Property Τ , and Random 3-manifolds

These are notes for a talk I gave in the Group theory seminar during the Fall of 2007 at the University of Wisconsin. The goal is to introduce the reader to the method of sieving from analytic number theory in the context of arithmetic groups. We do this by discussing properties of Random 3-Manifolds which were studied by Dunfeld and Thurston [2]. The primary source for this talk is Kowalski’s paper [3] and his book [4]. For an introduction to sieving I highly recommend the book of Cojocaru and Murty [1] 1. Outline The goal is to try to convey how sieving on a group can be done and how it is useful. We motivate the talk witht he results of Dunfeld and Thurston. We will see that to actually achieven the “sieve result” properties like Property (τ) will become important. We will start by describing what we mean by “random 3-manifold”. We will study the the first homology of these groups with coefficients in various rings. To study these groups we will quote some lemmas from [2]. Once we have some mechanism for studying these groups we will attempt to count how often the groups have a given size. We will see that the conditions we have look a lot like conditions that one uses in classical sieves from analytic number theory. We conclude by introducing Kowalski’s sieve results for arithmetic groups and comparing these results to classical sieve results. 2. Random 3-Manifolds One way to go about obtaining a 3-manifold is to take two surfaces of genus g and glue them together in a prescribed way. This will be how we obtain 3-manifolds. We just need to have an appropriate notion of ‘random’. For g ≥ 1 we let Σg be the genus g surface. Additionally, let G be the mapping class group of Σg and S a set of generators for G (with S = S−1 for technical reasons). We define a random walk on G by X0 = 1, Xk+1 = Xkξk+1 for k ≥ 0 and the sequence (ξk)k≥0 is a sequence of random variables of S with Pr (ξk = s) = 1 |S| for s ∈ S. We generate random 3-manifolds by taking two g-handle bodies, denoted Hg, and gluing the boundaries, ∂Hg = Σg using Mk. This gives a sequence of random 3-manifolds (Mk)k≥0. 3. Homology We want to study the probabilities that H1(Mk, R) is zero for R = Q,Z,F`. The following is one of the main results of [2]. Date: December 31, 2007. 1