On the Complexity of Dunfield-thurston Random 3-manifolds

This short note is a follow-up to (the second part of) Section 7.6 of my book [3]. There, earlier results of N. Dunfield and W. Thurston [1] on a certain type of random 3-manifolds (compact, connected) were improved and were made quantitative, using large-sieve inequalities for random walks on discrete groups. Roughly speaking, these results expressed the fact that “typical” 3-manifolds could be expected to have finite, but large, first homology with integer coefficients H1(M,Z). Here, we will first obtain a stronger form of the basic result of [3], by a simple refinement of the underlying arithmetic argument. Then, we will obtain a more satisfactory understanding of the nature of the result by relating the asymptotic parameter defining Dunfield-Thurston manifolds (the length of the underlying random walk) with more classical and intrinsic invariants of the manifolds themselves. This is of interest because otherwise it is by no means clear how, exactly, the results can be interpreted as probable (heuristic) properties of a fixed 3-manifold. We first recall the definition of the Dunfield-Thurston manifolds, and the result obtained in [3, Prop. 7.19]. Their construction is based on a classical topological description of compact 3manifolds, due to Heegaard. First, fix an integer g > 2. Let Γg denote the mapping class group of a closed surface Σg of genus g, and let S be a fixed finite set of generators of Γg, such that S = S−1 (i.e., a symmetric generating set). Then consider a random walk (Xk) on Γg 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