Actions on cube complexes and homomorphisms to families of groups

[摘要]获奖：DMS 1507067，首席研究员：Daniel GrovesA群是对几何对象的对称性进行编码的代数对象。几何和代数之间的这种联系意味着群的研究提供了一种代数语言和一套研究几何问题的技术。近年来，三维几何中的一些最重要的问题（如关于覆盖三维流形的空间结构的虚哈肯猜想）已经通过将群论和这些空间的二维子流形的几何性质相结合来解决。与这一行工作相关的一些名字是Agol， Wise, Kahn, Markovic, Haglund, Hsu， Bergeron和Manning。本项目前半部分的主要重点是扩大现有理论的应用范围，并将这些新工具应用于几何和群论中的更多问题。在这个项目的后半部分，将开发技术来研究关于族群之间的地图问题。同样，来自三维几何的问题将提供关键的动机。由于Agol和其他许多人的工作，这种设置中的对象现在已经很好地理解了。然而，关于这些对象之间的映射存在许多基本问题，本提案将解决其中的一些问题，并为研究这些问题建立一个通用工具包。在项目的前半部分，我们研究了紧作用于CAT(0)立方配合物上的双曲群。在细胞稳定剂是有限的情况下，Agol的著名定理意味着这样的群是“实际上特殊的”（Haglund和Wise引入的一个重要概念），这意味着这样的群具有许多强性质，如线性和强剩余有限性。在一维复合体（树）的情况下，如果单元稳定器是无限的，但是是拟凸的，并且实际上是特殊的，那么Wise的拟凸层次定理意味着双曲群也是实际上是特殊的。本课题前半部分的主要目标是证明这两个定理的一个同时推广：如果一个双曲群G紧作用于一个具有拟凸和虚特殊元稳定器的CAT(0)立方复上，则G是虚特殊的。这对于作用于立方体配合物的相对双曲和双曲群有许多应用。在本提案的后半部分，我们将开发一个相当通用的框架，用于研究从任意有限生成群到自然群族的映射集。这些族的例子将被研究包括：Kleinian群，任意三流形群，相对双曲和非圆柱双曲群，等等。

AbstractAward: DMS 1507067, Principal Investigator: Daniel GrovesA group is an algebraic object which encodes the symmetries of a geometric object. This connection between geometry and algebra implies that the study of groups provides an algebraic language and a set of techniques for studying problems throughout geometry. In recent years, some of the most important problems in three-dimensional geometry (such as the Virtual Haken Conjecture on the structure of spaces that cover three-dimensional manifolds) have been solved by a combination of group theory and geometric properties of two-dimensional submanifolds of these spaces. Some of the names associated to this line of work are Agol, Wise, Kahn, Markovic, Haglund, Hsu, Bergeron, and Manning.A major focus of the first half of this project is to broaden the range of applications of the available theory, and to apply these new tools to many more problems in geometry and group theory. In the second half of this project, techniques will be developed to study problems about maps between families of groups. Again, problems from three-dimensional geometry will give key motivation. Due to the work of Agol and many others, the objects in this setting are now quite well understood. However, there are many fundamental questions about the maps between these objects, and this proposal will tackle some of these problems, as well as building a general toolkit for studying questions such as these. In the first half of the project, we study hyperbolic groups acting cocompactly on CAT(0) cube complexes. In the case that the cell stabilizers are finite, Agol's famous theorem implies that such a group is "virtually special" (an important concept introduced by Haglund and Wise) which implies many strong properties, such as linearity and strong residual finiteness properties for such a group. In the case of a one-dimensional complex (a tree), if the cell stabilizers are infinite but quasi-convex and virtually special, then Wise's Quasiconvex Hierarchy Theorem implies that the hyperbolic group is again virtually special. The main goal of the first half of this project is to prove a simultaneous generalization of these two theorems: If a hyperbolic group G acts cocompactly on a CAT(0) cube complex with quasi- convex and virtually special cell stabilizers, then G is virtually special. This will have many applications to relatively hyperbolic and hyperbolic groups acting on cube complexes. In the second half of this proposal, we will develop a quite general framework for studying the set of a maps from an arbitrary finitely generated group into natural families of groups. Examples of such families which will be studied include: Kleinian groups, arbitrary three-manifold groups, relatively hyperbolic and acylindrically hyperbolic groups, and others.