Research on various geometric structures on 3-manifolds

1.关于3-流形中复合纽结的隧道数的Morimoto猜想设t（K）为3-流形中纽结K的隧道数。设对m-小纽结K_1，.，K_n，隧道数的超可加性对K_i不成立，则证明了存在{1，[三键]，n}的子集I，使得K_i有一个本原纽结. 2.纽结隧道数的增长率对3-流形M中的纽结K，定义了一个数值不变量K的隧道数增长率，并证明了：假设K的Heegaard亏格大于M的Heegaard亏格。Gersten提出了如下问题：“每个自动群是（1）有限群，（2）包含秩为2的自由交换群，或（3）字双曲群。“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。4. Heegaard梯度Seifert纤维空间我们完全确定了对于哪种Seifert纤维空间，Heegaard梯度为零。

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.