Groups and Group Actions in Low-Dimensional Topology

我们的项目将在一些具体问题的背景下研究单调性在低维几何和拓扑中的作用。首先，我们将研究（拓扑自然）群在一维流形上的作用，特别是在直线和圆上的作用，以及在非hausdorff 1-流形上的作用。这种作用与三维的叶理和层理理论密切相关。我们的工具将包括叶状和层状理论，规范理论的最新成果，计算工具，以及与这些行为相关的新拓扑对象的构建。我们还将与虚哈肯猜想类比，研究当传递到有限索引子群时这些作用的存在性和分类问题。与此相关，我们将研究一个随机的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对横向二维叶状的4-流形，将这些技术的范围从3维扩展到4维。通过改变这些叶化的解析性质，我们希望对规范论中产生的一些神秘的几何不等式得出不同的观点。我们将利用关于平面上具有良好解析性质群的有界上同调性质的旧的和新的结果来做到这一点。单调性是分析、几何和拓扑学中的重要原理。最著名的例子本质上是一维的；单调性允许根据其元素的顺序结构定义实数等对象。更一般地说，这种秩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建证书。单调性随着维数的增加而变得越来越难以应用；因此，当将流形的研究简化为两个较小维度的互补问题：系统轨道的研究和轨道参数空间的研究时，它在某些动力系统的背景下是最强大的。

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.