Geometric structures of 3-manifolds and various related structures

In this research project，we obtained the following results.1。We defined a numerical invariant，called growth rate of tunnel numbers，of knots in3-manifolds。For m-small knots，we obtained the following.Suppose K is a m-small knot in.A3-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=_&lt；1，...，n&gt；{1-i/(B_I)}.2。Heegaard splittings of exteriors of knots.·Let K_1，K_2be knots in3-manifolds，and T_1，T_2tunnel systems of K_1，K_2respectively。We gave a necessary and sufficient condition for the tunnel system t_1áT_2 of K_1#K_2giving 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 rank2.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.

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.