Georgia Institute of Technology College of Sciences

A contact 3-manifold is planar if it is compatible with an open book whose page genus is 0. Wendl's theorem that any symplectic filling of a planar contact 3-manifold comes as a Lefschetz fibration filling a given planar open book translates questions about symplectic fillings of planar contact 3-manifolds into group theoretical questions about Dehn twist factorizations in the planar mapping class groups. I will outline work with Matt Day studying the nilpotent levels and truncations of the planar mapping class groups. We have calculated the ranks of the first three levels in the lower central series. I will explain these calculations and will outline a procedure to determine whether a mapping class element has a positive factorization. This work is joint with Matt Day.