Laminations, Foliations and Flows in 3-Manifolds

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Abstract."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 topologyand they are also related to the geometrization conjecturefor 3-manifolds. One goal is to analyse the existencequestion for laminations in 3-manifolds, particulary forhyperbolic 3-manifolds. The PI has recently shown thereare 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 circleand general surface bundles and also special types oflaminations/foliations. The project will also considermore general structures, such as loosesse laminations andwhich properties they have. Another part of the project isto understand the geometric behavior of foliations andtransverse 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 speciallypromising in the case of general finite depth foliations.Another goal is to connect these properties with the universalcircle of the foliation - establishing a link between the global structure of the foliation and the global geometry of theuniversal cover. Another project is to study properties oflimit 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 exampleknots and their properties are very relevant to DNAresearch. 3-manifolds are also used to get mappings of thebrain and in other visualization techniques. Understanding3-manifolds can lead to progres in these other areas.The project focuses on laminations and foliations, whichare an essential and currently extremely dynamic area inlow 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.