"Laminations, foliations and flows in 3-manifolds"
Essential laminations and Reebless foliations are a basic and fundamental tool in the study of 3-manifolds. Their use yields deep results concerning 3-manifold topology and they are also related to the geometrization conjecture for 3-manifolds. One goal is to analyse the existence question for laminations in 3-manifolds, particulary for hyperbolic 3-manifolds. The PI has recently shown there are infinitely many hyperbolic 3-manifolds which do not admit essential laminations. A natural question is how common are essential laminations. The project will investigate Dehn surgery on torus bundles over the circle and general surface bundles and also special types of laminations/foliations. The project will also consider more general structures, such as loosesse laminations and which properties they have. Another part of the project is to understand the geometric behavior of foliations and transverse pseudo-Anosov flows in hyperbolic 3-manifolds - which is a very large class of manifolds. The focus will be on the large scale geometric behavior in the universal cover, which is fundamental for such manifolds. An important question is whether such flows are quasigeodesic - this is true in certain cases by previous work of the PI and Lee Mosher. One goal is to use the quasigeodesic property for the flows to derive information about the asymptotic behavior of the foliation transverse to the flow. This is specially promising in the case of general finite depth foliations. Another goal is to connect these properties with the universal circle of the foliation - establishing a link between the global structure of the foliation and the global geometry of the universal cover. Another project is to study properties of limit sets of leaves of foliations in hyperbolic manifolds.
The project aims to better understand 3-manifolds. 3-manifolds are widely used in the sciences: for example knots and their properties are very relevant to DNA research. 3-manifolds are also used to get mappings of the brain and in other visualization techniques. Understanding 3-manifolds can lead to progres in these other areas. The project focuses on laminations and foliations, which are an essential and currently extremely dynamic area in low dimensional topology. The proposed projects will have an effect on some main research directions in the area. One important goal is to attract students and postdoctoral researchers in this exciting area.
