Groups and Group Actions in Low-Dimensional Topology

我们的项目将研究单调性在低维几何和拓扑中的作用，在一些特定的问题。首先，我们将研究（拓扑自然）群在1维流形上的作用，特别是在直线和圆上的作用，以及在非豪斯多夫1-流形上的作用。这种作用与三维面理和层理理论密切相关。我们的工具将包括理论的叶理和叠层，最近的结果在规范理论，计算工具，并建设新的拓扑对象与这些行动。我们还将研究问题的存在性和分类等行动时，一个通过有限的指数子群，在类比虚拟哈肯猜想。与此相关，我们将研究一个随机3-流形群在直线上是否有忠实作用的问题。最后，我们将通过研究具有2维横向叶理对的四维流形，把这些技巧的应用范围从3维扩展到4维。通过改变这些叶理的解析性质，我们希望对规范理论中某些神秘的几何不等式有不同的看法。我们将利用关于作用在具有良好解析性质的平面上的群的有界上同调性质的新旧结果来做到这一点。单调性是分析、几何和拓扑学中的一个重要原则。最著名的例子基本上是一维的;单调性让人们根据其元素的序结构来定义这样的对象作为真实的数。更一般地说，这种序结构提供了从几何问题到代数语言的桥梁，并允许人们使用计算机进行实验和构造证书。单调性变得越来越难以applyas维上升，因此，它是最强大的，当使用在某些动力系统的背景下，减少了研究的流形两个互补的问题较小的尺寸：研究的轨道系统，和研究的参数空间的轨道。

Our project will study the role of monotonicity in low dimensional geometry and topology, in the context of a number of specific problems. Firstly, we will study actions of (topologically natural) groups on 1-dimensional manifolds, especially actions on the line and the circle, but also on non-Hausdorff 1-manifolds. Such actions are closely related to the theory of foliations and laminations in dimension 3. Our tools will include the theory of foliations and laminations, recent results in gauge theory, computational tools, and constructions of new topological objects associated to such actions. We will also study the question of existenceand classification of such actions when one passes to finite indexsubgroups, in analogy with the Virtual Haken conjecture. Related tothis, we will investigate the problem of whether a random 3-manifoldgroup has a faithful action on the line. Finally, we will expand therange of these techniques from dimension 3 to dimension 4 by studying4-manifolds with pairs of transverse 2-dimensional foliations. By varying the analytic quality of these foliations, we hope to arriveat a different point of view on some of the mysterious geometricinequalities arising from gauge theory. We will do this by exploitingold and new results about bounded cohomological properties of groupsacting on the plane with good analytic qualities.Monotonicity is an important principle in analysis, geometry and topology.The best known examples are essentially 1-dimensional in origin;monotonicity lets one define such objects as the real numbers in termsof the order structure of its elements. More generally, such orderstructures provide a bridge from geometric problems to algebraic language,and permit one to perform experiments and construct certificates withthe use of computers. Monotonicity becomes increasingly hard to applyas dimension goes up; consequently it is most powerful when used inthe context of certain dynamical systems which reduce the study of themanifold to two complementary problems of smaller dimension: the studyof the orbits of the system, and the study of the parameter space ofthe orbits.