Research on various geometric structures on 3-manifolds

1.Morimoto关于3-流形中复合结点隧道数的猜想设t(K)为3-流形中结点K的隧道数。假设对于m个小结点K_1，…，K_n，对于#^n_<i=1> K_i，隧道数的超可加性不成立，则证明了{1，【三键】，n}存在一个子集i，使得#_<i∈1>/K_i允许一个原始子午线。对于3流形M中的结点K，我们定义了一个称为K的隧道数增长率的数值不变量，并证明了以下几点。假设K的Heegaard属大于m的Heegaard属，则K隧道数的增长率小于1.3。Gersten提出了以下问题：“每个自动群要么是(1)一个有限群，(2)包含秩为2的自由阿贝尔群，要么(3)一个词双曲群。”我们证明了对于一类自动群（称为n星群），这个问题是肯定解决的。当我们完全确定了哪个空间是塞弗特纤维化的，赫格特梯度就消失了。

1.Morimoto's Conjecture on the tunnel numbers of composite knots in 3-manifoldsLet t(K) be the tunnel number of a knot K in a 3-manifold. Suppose for m-small knots K_1,…,K_n, the super additivity of tunnel number does not hold for #^n_<i=1> K_i, Then we proved that there exists a subset I of {1,【triple bond】,n} such that #_<i∈1>/K_i admits a primitive meridian.2.The growth rate of tunnel number of knotsFor a knot K in a 3-manifold M, we defined a numerical invariant called the growth rate of the tunnel numbers of K, and proved the following.Suppose that the Heegaard genus of K is greater than the Heegaard genus of M. Then the growth rate of the tunnel numbers of K is less than 1.3.Gersten's Problem for automatic groupGersten posed the following problem "Each automatic group is eithr (1)a finite group, (2)contains free abelian group of rank 2,or (3)a word hyperbolic group." We showed that for a class of automatic group (called n-starred groups) this problem is solved affirmatively.4.Heegaard gradients Seifert fibered spacesWe completely determined for which Seifert fibered space, the Heegaard gradient vanish.