Puncture–forgetting maps have been studied in various contexts, including Teichmüller spaces and mapping class groups. In this talk, we present several approaches to forgetting punctures in the setting of measured foliations, leading to upper semi-continuous maps between spaces of measured foliations.
A key ingredient of the proof is the introduction of complexes of pre-homotopic multicurves, which we show are hyperbolic CAT(0) cube complexes. We then analyze the action of point-pushing mapping classes on these complexes. This framework is motivated by applications to Teichmüller geodesics and the dynamics of post-critically finite rational maps. This is joint work with Jeremy Kahn.