When I started in graduate school, I was very interested in 3-manifolds, especially Thurston’s geometrization conjecture. Somehow in dimension 3, there is a marvelous marriage of flexibility and rigidity: generic 3-manifolds are flexible enough to admit hyperbolic structures — i.e. Riemannian metrics of constant curvature -1, modeled on hyperbolic space. But these structures are so rigid that they are determined up to isometry (!) entirely by the fundamental group of the manifold, and provide a bridge from topology to the rigid world of number fields and arithmetic. 3-manifolds, especially the hyperbolic ones, display an astonishing range of interesting phenomena, so that even though the individual manifolds are discrete and rigid, they come in infinite families parameterized by Dehn surgery. When Perelman proved Thurston’s conjecture, I gradually moved away from 3-manifold topology into some neighboring fields such as dynamics and geometric group theory; subjectively this move felt to me like a transition from a baroque world of highly intricate, finely tuned and beautiful objects to more rough and disordered domains in which the rule was chaos and disorder, and where one had to restrict attention and focus to find the kinds of structured objects that one can say something about mathematically. In these new domains my familiarity with 3-manifold topology was always extremely useful to me, but almost always as a source of inspiration or analogy or example, rather than that some specific theorem about 3-manifolds could be used to say something about groups in general, or dynamical systems in general, or whatever. Many important recent developments in geometric group theory are generalizations of geometric ideas which were first identified or studied in the world of 3-manifolds; but there was not much connection at the deepest level, at least as far as I could see.
This impression was dramatically shaken by Agol’s proof of the virtual Haken conjecture and virtual fibration conjectures in 3-manifold topology by an argument which depends for one of its key ingredients on the theory of non-positively curved cube complexes — a subject in geometric and combinatorial group theory which, while inspired by key examples in low-dimensions (especially surfaces in the hands of Scott, and graphs in the hands of Stallings), is definitely a high-dimensional theory with no obvious relations to manifolds at all. Even so, the transfer of information in this case is still from the “broad” world of group theory to the “special” world of 3-manifolds. It shows that 3-manifold topology is even richer than hitherto suspected, but it does not contradict the idea that the beautiful edifice of 3-manifold topology is an exceptional corner in the vast unstructured world of geometry.
I have just posted a paper to the arXiv, coauthored with Henry Wilton, and building on prior work I did with Alden Walker, that aims to challenge this idea. Let me quote the first couple of paragraphs of the introduction:
Geometric group theory was born in low-dimensional topology, in the collective visions of Klein, Poincaré and Dehn. Stallings used key ideas from 3-manifold topology (Dehn’s lemma, the sphere theorem) to prove theorems about free groups, and as a model for how to think about groups geometrically in general. The pillars of modern geometric group theory — (relatively) hyperbolic groups and hyperbolic Dehn filling, NPC cube complexes and their relations to LERF, the theory of JSJ splittings of groups and the structure of limit groups — all have their origins in the geometric and topological theory of 2- and 3-manifolds.
Despite these substantial and deep connections, the role of 3-manifolds in the larger world of group theory has been mainly to serve as a source of examples — of specific groups, and of rich and important phenomena and structure. Surfaces (especially Riemann surfaces) arise naturally throughout all of mathematics (and throughout science more generally), and are as ubiquitous as the complex numbers. But the conventional view is surely that 3-manifolds per se do not spontaneously arise in other areas of geometry (or mathematics more broadly) amongst the generic objects of study. We challenge this conventional view: 3-manifolds are everywhere.
Read the rest of this entry »