The complex hyperbolic structures on the configuration spaces of points on the sphere and surface subgroup of mapping class groups

(1)非双曲型自动群的结构（与Y. Nakagawa， M. Tamura合著）设G为有限呈现群。如果G包含Z + Z子群，那么众所周知G不可能是双曲的。一个很自然的问题是“Z + Z是一个有限呈现的群体的唯一障碍吗？”换句话说，“如果G不包含任何Z + Z子群，它是单词双曲的吗？”Baumslag-Solitar小组是这个问题的反例。因此，最好将我们的注意力限制在一些好的群体上。这里我们关注的是自动组。注意，Baumslag-Solitar组不是自动的。我们的问题显示在开放问题列表中，并归因于Gersten。我们称这个问题为“格斯滕问题”。回想一下，所有自动群的类包含所有双曲群、所有虚阿贝尔群和所有几何有限双曲群的类。几何上有限的双曲群在某种意义上类似于双曲群，但它可能包含Z + Z子群。因此，自动组类是考虑前面提到的问题的一个很好的目标。我们定义了“n条轨道长度为n”的概念，给出了Z + Z子群存在的线索，并证明了Z + Z子群存在于所有条件温和的非双曲自动群中。(2)单孔环面特征的变化（与S.P. Tan合著）近十年来，许多人对穿孔环面群的拟紫红空间进行了深入的研究，并解决了一些关于穿孔环面群的重大猜想。但是，对于一般的“单孔”案例，我们知道的并不多。在这项研究中，我们制作了计算机软件来研究单孔环面的特征变化，并发现了许多有趣的现象

(1) Structures of non-hyperbolic automatic groups(Joint work with Y. Nakagawa, M. Tamura)Let G be a finitely presented group. If G contains a Z + Z subgroup, then it is well known that G cannot be word hyperbolic. A natural question is that "is Z + Z the only obstruction for a finitely presented group to be word hyperbolic?" In other words, "if G does not contain any Z + Z subgroups, is it word hyperbolic?" Baumslag-Solitar groups are counter examples to this question. Thus it would be better to restrict our attention to some good class of groups. Here we focus on automatic groups. Note that Baumslag-Solitar groups are not automatic. Our problem is indicated in the list of open problems and attributed to Gersten. We call this problem "Gersten's problem".Recall that the class of all automatic groups contains the class of all hyperbolic groups, all virtually abelian groups and all geometrically finite hyperbolic groups. A geometrically finite hyperbolic group is, in some sense, similar to hyperbolic groups, but it might contain a Z + Z subgroup. Thus the class of automatic groups is a nice target to consider the question mentioned before.We define the notion of "n-tracks of length n", which suggests a clue of the existence of Z + Z subgroup and shows its existence in every non-hyperbolic automatic groups with mild conditions.(2) The character variety of one-holed torus(Joint work with S.P. Tan)The quasifuchsian space of punctured torus groups is deeply studied by many people and some of the major conjectures on them are solved in the last decade. But, for general "one-holed" cases, not much is known. In this study, we produced computer software to investigate the character variety of one holed torus and were able to find many interasting phenomena