As an experiment, I plan to spend the next five weeks documenting my current research on this blog. This research comprises several related projects, but most are concerned in one way or another with the general program of studying the geometry of a space by probing it with surfaces. Since I am nominally a topologist, these surfaces are real -manifolds, and I am usually interested in working in the homotopy category (or some rational “quotient” of it). I am especially concerned with surfaces with boundary, and even (occasionally) with corners.
Since it is good to have a “big question” lurking somewhere in the background (for the purposes of motivation and advertising, if nothing else), I should admit from the start that I am interested in Gromov’s well-known question about surface subgroups, which asks:
Question (Gromov): Does every one-ended word-hyperbolic group contain a closed hyperbolic surface subgroup?
I don’t have strong feelings about whether the answer to this question is “yes” or “no”, but I do think the question can be sharpened usefully in many ways, and it is my intention to do so. Gromov’s question is certainly inspired by questions such as Waldhausen’s conjecture and the virtual fibration conjecture in -manifold topology, but it is hard to imagine that a proof of one of these conjectures would shed much light on Gromov’s question in general. At least one essential tool in -manifold topology — namely Dehn’s lemma — has no meaningful analogue in geometric group theory, and I think it is important to try to imagine different methods of constructing surface groups from “first principles”.
Another long-term project that informs much of my current research is the problem of understanding stable commutator length in free groups. The interested reader can learn something about this from my monograph (which can be downloaded from this page). I hope to explain why this is a fundamental and interesting problem, with rich structure and many potential applications.