The Large Sieve, Property (t ) and the Homology of Dunfield-thurston Random 3-manifolds

In their paper [DT1], N. Dunfield and W. Thurston define a notion of “random 3-manifold” and study some properties of those manifolds with respect (among other things) to the existence of finite covers with certain covering groups, especially with regard to their homological properties (in particular, which ones have positive first Betti number). In this note, we show that some applications of the large sieve to random walks on groups with Property (T ) (or Property (τ)) may be used to refine some of their results. To state our results, let g > 1 be an integer. Let G denote the mapping class group of a closed surface Σg of genus g, and let S be a fixed finite set of generators, such that S = S−1 (i.e., a symmetric generating set). For g = 1, assume that 1 ∈ S (this is to avoid periodicity issues with the random walk; it can also be assumed for simplicity if g > 2, but there it is not necessary). Associated to this is a simple random walk (Xk) on G defined by X0 = 1, Xk+1 = Xkξk+1 for k > 0, where (ξk) is a sequence of independent S-valued random variables with uniform distribution