Last week I was at Oberwolfach for a meeting on geometric group theory. My friend and collaborator Koji Fujiwara gave a very nice talk about constructing actions of groups on quasitrees (i.e. spaces quasiisometric to trees). The construction is inspired by the famous subsurface projection construction, due to MasurMinsky, which was a key step in their proof that the complex of curves (a natural simplicial complex on which the mapping class group acts cocompactly) is hyperbolic. Koji’s talk was very stimulating, and shook up my thinking about a few related matters; the purpose of this blog post is therefore for me to put some of my thoughts in order: to describe the MasurMinsky construction, to point out a connection to certain geometric phenomena like winding numbers of curves on surfaces, and to note that a variation on their construction gives rise directly to certain natural chiral invariants of surface automorphisms (and their generalizations) which should be relevant to 4manifold topologists.

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