Combinatorially defined complexes appear throughout geometric group theory. The most famous example is the curve complex, and it is unreasonably effective for studying mapping class groups. In terms of geometry the curve complex is Gromov hyperbolic and it is the departure point for exploring the large-scale geometry of mapping class groups. However, the curve complex is also important in terms of cohomology: by work of Harer, the reduced homology of the curve complex is the so-called dualising module of the mapping class group.

In joint work with Dan Petersen we show that a space called the complex of non-simple disc systems plays the same role in the setting of handlebody groups. The key idea in both Harer’s work and ours is that these complexes appear dual to stratifications of certain subspaces of Teichmüller space. I shall give an overview of the tools which are used in both of these proofs. Time permitting I’ll show how these stratifications can also be used to think about tropicalisation and the links to Out(F_n) and Outer space.