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

(1)非双曲自动群的结构（与Y. Nakagawa，M. Tamura）设G是一个群G.如果G包含一个Z + Z子群，则G不可能是字双曲的。一个很自然的问题是：“Z + Z是唯一的障碍，一个共同提出的集团是字双曲？换句话说，如果G不包含任何Z + Z子群，它是字双曲的吗？“鲍姆斯拉格-索利塔团体是这个问题的反例。因此，最好把我们的注意力限制在某些好的群体类别上。在这里，我们专注于自动组。请注意，Baumslag-Solitar组不是自动的。我们的问题在开放问题列表中列出，并归因于Gersten。我们称这个问题为“Gersten问题”。回想一下，所有自动群的类包含所有双曲群、所有虚交换群和所有几何有限双曲群的类。一个几何有限的双曲群在某种意义上类似于双曲群，但它可能包含一个Z + Z子群。我们定义了“n-长n的迹”的概念，它给出了Z + Z子群存在的线索，并在较弱的条件下证明了Z + Z子群在所有非双曲自动群中的存在性。(2)穿孔环面群的拟fuchsian空间在近十年来得到了广泛的研究，并解决了一些重要的问题。但是，对于一般的“一个洞”的情况下，没有太多的了解。在本研究中，我们制作了计算机软件来研究单孔环面的性质变化，并能够发现许多有趣的现象

(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