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

群是从对称性和刚性运动的研究中发展起来的数学对象，尽管在现代研究中，有许多地方出现了群，作为编码某些类型数据的复杂方式。很自然，挑战之一就是学习如何从对群的代数性质的研究中提取出由群编码的数据。 解决这一挑战的一种方法是证明你感兴趣的群体与另一个更好理解的群体的子群同构（相同）。我在我的研究中采用了这种方法，其中的共同主题是试图将给定的群实现为低维空间的保序同胚群的子群（即，作为低维对象的变形集合），例如真实的线或圆。更一般地，有时将群实现为从任意有序集到其自身的保序函数的集合是有用的。这个程序的目标是研究不同的方式，一个组可以实现为子群的变形的真实的线，圆或有序集，并提取代数信息的一个给定的组从这些结构。特别重要的是，当一个群通过一个被称为“基本群”的构造从三维空间的研究中产生时。在这种情况下，我们的目标是从基本群中提取关于底层空间拓扑的信息;在这个方向上，有两个值得特别注意的可疑连接。首先，空间的基本群是否同构于真实的直线的保序变形群，被怀疑与叶理有关（也就是说，如何用低维空间"紧密包装"空间）。这种联系已经在少数几个案例中得到了理解。它也是连接到heegaard-Floer同源性，一个强大的新的同源性理论，已被用于在过去十年来解决许多长期悬而未决的问题，在研究三维空间。总之，这一系列可疑的联系被称为“L空间猜想”。该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。在这个方向上的进展直接有助于在低维拓扑社区正在进行的努力，将现代工具和技术（如Heegaard-Floer同源）与代数拓扑的经典不变量和拓扑结构联系起来。

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.