Graphs, surfaces and handlebodies

In Teichmüller space Riemann surfaces are marked by a homeomorphism from a model surface. We consider the notion of a complex handlebody, which can be marked by either a homeomorphism or a homotopy equivalence from a model handlebody. In all of these situations the space T of marked objects has a partial compactification Tbar with a “boundary complex” that encodes the structure of Tbar - T. These boundary complexes are, respectively, the curve complex of a surface, the disc complex of a handlebody and the simplicial closure of Outer space. In joint work with R. Ramadas, R. Silversmith and R Winarski we fill in the details of this picture.