The study of translation surfaces has captivated both dynamicists and algebraic geometers; at this point the deep links between the two points of view are beginning to be understood. Translation surfaces are organized into families called strata, and a study of the topology and geometric group theory of strata is a subject that is still very much in its infancy. I will describe my work with Aaron Calderon that gives a first glimpse at what the subject might have in store. Speaker(s): Nick Salter (Columbia)