Geometric structures of 3-manifolds and various related structures

在这个研究项目中，我们获得了以下结果。我们定义了一个数字变量，称为隧道数的增长率，3块中的knots。对于m-小结，我们得到了下面的结果，Suppose K是一个m-小结。A3-manifold M. Let g = g（X）-g（M）， and b_i（i =1，...，g）是K的桥指数，尊重基因g（X） - i Heegaard表面的M.然后K的增长率被min_i =_<1，...，n>{1-i/（b_i）}.2. Heegaard的外部节点的碎片。让K_1和K_2在3块块中运行，并相应地使用K_1和K_2的T_1和T_2隧道系统。我们为隧道系统t_1 ∪ T_2中的K_1#K_2提供了一种必要和适当的条件。对于每个自然数字n，存在一个结K这样的平等性g（nK） = gt（K）持有，在哪里nK拒绝n份K份的连接总和。这一实例对Morimoto's Concerne's Concerne's tunnel number的超附加现象的例子。我们展示了在3个球体中的任何链接L，这里有一个Seifert表面S为L，它从一个成功的Plumbing Flat annuli的磁盘中获得，所有附加区域都包含在磁盘中。我们研究了Gersten的问题：每个自动群体都有（1）一个有限的群体， （2）包含一个自由的阿拉伯群体的排名2。或（3）一个词超自然组，我们展示了n星自动组持有此断言。2-桥接链接组的增长函数我们对2-桥接链接组的增长函数进行了计算机实验，并对增长函数结构的现有概念。

In this research project, we obtained the following results.1. We defined a numerical invariant, called growth rate of tunnel numbers, of knots in 3-manifolds. For m-small knots, we obtained the following.Suppose K is a m-small knot in. a 3-manifold M. Let g = g(X)-g(M), and b_i (i =1,..., g) be the bridge index of K with respect to genus g(X) - i Heegaard surface of M. Then the growth rate of K is given by min_i=_<1,..., n>{1-i/(b_i)}.2. Heegaard splittings of exteriors of knots.・ Let K_1, K_2 be knots in 3-manifolds, and T_1,T_2 tunnel systems of K_1, K_2 respectively. We gave a necessary and sufficient condition for the tunnel system t_1 ∪ T_2 of K_1#K_2 giving a stabilized Heegaard splitting.・ For each natural number n, there exists a knot K such that the equality g(nK) = gt(K) holds, where nK denotes the connected sum of n copies of K. This implies the existence of counterexample to Morimoto's Conjecture concerning super additive phenomina of tunnel number of knots.3. We showed that for any link L in the 3-sphere, there is a Seifert surface S for L such that S is obtained from a disk by successively plumbing flat annuli, where all of the attaching regions are contained in the disk.4. We made research on Gersten's Problem : each automatic group is either (1) a finite group, (2) contains a free abelian group of rank 2. or (3) a word hyperbolic group.We showed that for the n-starred automatic groups this assertion holds.5. Growth function of 2-bridge link groupsWe made computar experiments on the growth functions of 2-bridge link groups, and posed conjectures on the structure of the growth functions.