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 existence and classification of such actions when one passes to finite index subgroups, in analogy with the Virtual Haken conjecture. Related to this, we will investigate the problem of whether a random 3-manifold group has a faithful action on the line. Finally, we will expand the range of these techniques from dimension 3 to dimension 4 by studying 4-manifolds with pairs of transverse 2-dimensional foliations. By varying the analytic quality of these foliations, we hope to arrive at a different point of view on some of the mysterious geometric inequalities arising from gauge theory. We will do this by exploiting old and new results about bounded cohomological properties of groups acting 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 terms of the order structure of its elements. More generally, such order structures provide a bridge from geometric problems to algebraic language, and permit one to perform experiments and construct certificates with the use of computers. Monotonicity becomes increasingly hard to apply as dimension goes up; consequently it is most powerful when used in the context of certain dynamical systems which reduce the study of the manifold to two complementary problems of smaller dimension: the study of the orbits of the system, and the study of the parameter space of the orbits.
