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

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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.