Low dimensional topology, ordered groups and actions on 1-manifolds

组是从研究对称性和刚性运动发展而来的数学对象，尽管在现代研究中，有许多地方组作为一种复杂的方式出现，对某些类型的数据进行编码。因此，自然而然的挑战之一就是学习如何从对其代数性质的研究中提取由一组编码的数据。解决这一挑战的一种方法是证明你感兴趣的组与另一个更好理解的组的子组同构(相同)。我在我的研究中使用了这种方法，其中共同的主题是试图将给定的群实现为低维空间(即，作为低维对象的变形的集合)的保序同胚群的子群，例如实线或圆。更广泛地说，有时将群实现为从任意有序集到自身的保序函数的集合是有用的。这个程序的目的是研究一个群可以被实现为实线、圆或有序集的变形的子群的不同方法，并从这些结构中提取关于给定群的代数信息。尤其重要的是，当一个群通过一种被称为“基本群”的结构从3维空间的研究中产生时。在这种情况下，目标变成了从基本群中提取关于底层空间的拓扑的信息；在这个方向上，有两个特别值得注意的可疑联系。首先，一个空间的基本群是否同构于实直线的一组保序变形，被怀疑与叶层有关(即如何用较低维空间“紧填充”一个空间)。这种联系在少数几个案例中已经被理解了。它还推测与Heegaard-Floer同调有关，Heegaard-Floer同调是一种强大的新同调理论，在过去的十年里一直被用来解决三维空间研究中的许多长期悬而未决的问题。总而言之，这一系列可疑的联系被称为“L空间猜想”。这个计划的主要短期目标之一是开发和应用代数工具来解决L空间猜想的某些情况。这一方向的进展直接促进了低维拓扑界正在进行的努力，即将现代工具和技术(如Heegaard-Floer同调)与经典不变量和来自代数拓扑的拓扑构造联系起来。

Groups are mathematical objects developed from a study of symmetries and rigid motions, though in modern research there are many places where groups spring up as a sophisticated way of encoding certain kinds of data. It is natural that one of the challenges, then, is to learn how to extract the data encoded by a group from a study of its algebraic properties. One method of tackling this challenge is to show that your group of interest is isomorphic to (the same as) a subgroup of another group that is better understood. I employ this method in my research, where the common theme is to try to realize a given group as a subgroup of the group of order-preserving homeomorphisms of a low-dimensional space (that is, as a collection of deformations of a low-dimensional object) such as the real line or the circle. More generally, it is sometimes useful to realize the group as a collection of order-preserving functions from an arbitrary ordered set to itself. The goal of this program is then to study the different ways that a group can be realized as subgroups of deformations of the real line, the circle or an ordered set, and to extract algebraic information about a given group from these structures. Of particular importance is the case when a group arises from the study of a 3-dimensional space via a construction known as the "fundamental group". In this case, the goal becomes to extract from the fundamental group information about the topology of the underlying space; and in this direction there are two suspected connections of particular note. First, whether or not the fundamental group of a space is isomorphic to a group of order-preserving deformations of the real line is suspected to be connected with foliations (that is, how to "tightly pack" a space with lower--dimensional spaces). This connection is already understood in a select few cases. It is also conjecturally connected to Heegaard--Floer homology, a powerful new homology theory that has been used over the past decade to resolve many long-standing open questions in the study of 3-dimensional spaces. Together, this package of suspected connections has become known as "the L-space conjecture". One of the primary short-term goals of this program is the development and application of algebraic tools to tackle certain cases of the L-space conjecture. Advances in this direction contribute directly to the ongoing effort in the low-dimensional topology community to relate modern tools and techniques (such as Heegaard-Floer homology) to classical invariants and topological constructions from algebraic topology.