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

(1)非双曲自动机群的结构(与Y.Nakagawa，M.Tamura联合工作)设G是有限表示群。如果G包含Z+Z子群，那么众所周知，G不可能是字双曲的。一个自然的问题是“Z+Z是有限表示群成为词双曲的唯一障碍吗？”换句话说，“如果G不包含任何Z+Z子群，它是双曲型的吗？”鲍姆斯拉格-索利塔群就是这个问题的反例。因此，将我们的注意力限制在一些好的群体上会更好。在这里，我们重点介绍自动分组。请注意，鲍姆斯拉格-Solitar组不是自动的。我们的问题出现在未解决的问题列表中，并归因于格斯滕。我们称这个问题为Gersten问题，所有自动机群的类包含所有双曲群、所有虚拟阿贝尔群和所有几何有限双曲群的类。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Z+Z子群。定义了长度为n的n-迹的概念，给出了Z+Z子群存在的一个线索，并证明了Z+Z子群的存在性。(2)单孔环面的特征变化(与S.P.Tan联合工作)许多人深入研究了穿孔环面群的拟富氏空间，解决了过去十年中关于它们的一些主要猜想。但是，对于一般的“一洞”案例来说，我们所知的并不多。在这项研究中，我们制作了计算机软件来研究一个开孔环面的特征变化，并能够发现许多中间现象

(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