Hyperbolicity and the number of components for random link ランダム絡み目の最頻成分数と双曲性

In recent Low-dimensional topology, to study 3-manifolds and/or knot via random methods could be a hot topic now. As a pioneering work, in [1], Dunfield and W.Thurston introduced a random model of 3-manifolds by using random walks on the mapping class group of a surface, and a theory of random 3-manifolds has started. Actually they considered random Heegaard splittings by gluing a pair of handlebodies by the result of a random walk in the mapping class group. In view of this, the second author introduced and studied two models of random links in [4]. We here report two recent results based on joint works Ichihara-Yoshida [3] and Ichihara-Ma [2]. One model of random link introduced in [4] is defined as the braid closures of the randomly chosen braids via random walks on the braid groups. Suppose that a random walk on the braid group Bn of n-strings induces a uniform distribution on the symmetric group Sn on n letters via the natural projection Bn → Sn (n ≥ 3). Then, the second author showed in [4, Theorem 1.1] that, for the random link coming from a random walk of k-step on Bn (n ≥ 3), the expected value of the number of components converges to 1 + 1 2 + 1 3 + · · ·+ 1 n when k diverges to ∞. Then it is natural to ask what is the most expected number of components for such a random link. We first answer to this question as follows in [3].