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(拓扑自然的)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。这些作用与三维空间中的层理和层理理论密切相关。我们的工具将包括层理和层理理论、规范理论的最新结果、计算工具，以及与这些作用相关的新拓扑对象的构造。类似于虚Haken猜想，我们还将研究当一个人转到有限指数子群时，这类作用的存在和分类问题。与此相关，我们将研究随机3-流形群在直线上是否有忠实行动的问题。最后，我们将通过研究具有两对横向2维叶状结构的4维流形，将这些技巧的范围从3维扩展到4维。通过改变这些叶的分析性质，我们希望对规范理论中出现的一些神秘的几何不等式得出不同的观点。我们将通过利用作用在具有良好分析性质的平面上的群的有界上同调性质的新旧结果来实现这一点。单调性是分析、几何和拓扑学中的一个重要原理。最著名的例子本质上是一维的起源；单调性允许人们用其元素的序结构来定义这样的对象为实数。更广泛地说，这种有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。随着维度的增加，单调性变得越来越难应用；因此，当它被用于某些动力系统的背景下时，它是最强大的，这些动力系统将流形的研究减少到两个更小维度的互补问题：系统轨道的研究和轨道参数空间的研究。

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.