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.

The generic objects that we discuss in the paper are random groups, in the sense of Gromov. In fact, there are two models of random groups that one usually encounters in geometric group theory. First, fix a finite number k (at least 2) of generators x_1, x_2,\cdots, x_k, and a length n; and then throw in \ell random relations r_1,r_2,\cdots,r_\ell all reduced words of length n in the generators and their inverses, chosen randomly and independently from amongst all possible words of that length. The two models are distinguished by how the number of relators (i.e. \ell) depends on the length n. In the few relators model, one takes \ell to be a fixed (positive!) constant. In the density model, one fixes a constant D between 0 and 1, and lets \ell = (2k-1)^{nD}. The point is that there are approximately (2k-1)^n possible reduced words of length n to add as relators (each successive letter of a random word could be any generator or its inverse except for the inverse of the previous letter) and we are choosing to throw in a fixed multiplicative density of these words.

Suppose we are interested in some property of a group; for instance, that it should be infinite, or torsion-free, or abelian, or whatever. For each fixed n, we get a probability law on groups, and we can ask what the probability is that our random group (with relators of length n) has the desired property. Then one takes n to infinity and looks at the way in which the probability behaves; usually we are interested in properties for which the probability goes to 1 as n goes to infinity. We say then that a random group has the desired property with overwhelming probability.

Gromov showed that there is a natural phase transition in the behavior of random groups; at any fixed density D bigger than 1/2, a random group is either trivial or isomorphic to \mathbb{Z}/2\mathbb{Z}, with overwhelming probability. Conversely, at any fixed density less than 1/2, a random group is infinite, torsion-free, hyperbolic, and 2-dimensional. Since the group is 2-dimensional and hyperbolic, the boundary is 1-dimensional. Dahmani-Guirardel-Przytycki show that the boundary is a Menger sponge with overwhelming probability — i.e. the universal compact 1-dimensional topological space that every other 1-dimensional compact topological space embeds into it (one should say “metrizable” to be really rigorous here).

So in one sense, we know what the “generic” objects look like amongst finitely generated groups. But in another sense, the answer is unsatisfying — these groups are unfamiliar, and not obviously related to the sorts of groups that we understand well, like free groups, surface groups, matrix groups, and so on. So it becomes important to try to understand the structure of subgroups of random groups; do they contain subgroups that are familiar, which we can use as key structural elements to understand the big group? and is this subgroup structure rich enough that we can hope to find similar structure in all hyperbolic groups?

In order to make progress, we must first be clear about what sorts of subgroups we are looking for. We are interested in our groups not only as algebraic objects, but as geometric objects (with respect to some choice of word metric), and it is important to look for subgroups whose intrinsic and extrinsic geometry are uniformly comparable, so that the geometry of the subgroup (which we understand) tells us something about the geometry of the ambient group (which we want to understand). Since the random group G is hyperbolic, this means looking for subgroups H which are quasiconvex. Such groups are themselves necessarily hyperbolic, and the boundary of a quasiconvex subgroup H embeds in the boundary of G. Since the boundary of G is (topologically) 1-dimensional, the same is true of H, so we are led to the natural question: what hyperbolic groups have 1-dimensional boundary?

The answer to this question is essentially known, by work of Kapovich-Kleiner. First of all, a hyperbolic group with disconnected boundary splits over a finite group, by Stallings theorem on ends. Second of all, a hyperbolic group with connected boundary with local cut points is either virtually a surface group or splits over a cyclic group, by Bowditch. So we are led to essentially four cases:

  1. a Cantor set; in this case, H is (virtually) free. All nonelementary hyperbolic groups contain free subgroups, by Klein’s ping-pong argument; so random groups certainly contain such subgroups;
  2. a circle; in this case, H is (virtually) a surface group. It is a famous open problem of Gromov whether all one-ended hyperbolic groups contain surface subgroups. A positive answer is known in a few cases: Kahn-Markovic showed that closed hyperbolic 3-manifold groups contain surface subgroups (a key ingredient in Agol’s theorem). About a year ago, Alden Walker and I showed that random groups contain surface subgroups, and these subgroups are quasiconvex;
  3. a Sierpinski carpet; in this case, conjecturally H is (virtually) the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. This conjecture more-or-less reduces (by a doubling argument) to the Cannon conjecture — that a hyperbolic group has boundary homeomorphic to the 2-sphere if and only if it is virtually the fundamental group of a closed hyperbolic 3-manifold; or
  4. a Menger sponge; this is the boundary of the random group itself!

In view of this classification, François Dahmani asked me (after hearing the proof of my theorem with Alden) whether random groups could contain subgroups isomorphic to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. This is precisely the main theorem that Henry and I prove in the paper; explicitly:

3-Manifolds Everywhere Theorem: A random group, either in the few relators model or in the density model at any density less than 1/2, contains many quasiconvex subgroups isomorphic to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary.

The proof is direct — we basically show that one can directly construct a map from such a 3-manifold group into a random group (given by a random presentation) in such a way that it is very likely to be quasiconvex and injective. The argument borrows very heavily from many parts of my earlier paper with Alden, although the construction step is much more complicated.

It is possible to say something in general terms about the combinatorial construction. Our random presentation can be realized in geometric terms by building a 2-dimensional complex K, whose 1-skeleton X is a wedge of k circles (one for each generator), to which we attach \ell disks along loops corresponding to the relators. Let r be one such relator; it is a long (cyclic) reduced word in the generators and their inverses. We can think of this word as being written along the edges of a circle L subdivided into intervals, with one letter in each interval. Imagine taking this circle and gluing it up to itself, matching sets of edges with the same label, so that the result is a labeled graph Z. If we then attach a disk along the boundary of the circle, we get a 2-complex M(Z), and this 2-complex immerses in K. If we are careful, we can arrange for M(Z) to have the homotopy type of a 3-manifold with boundary; and if the manifold is acylindrical and freely indecomposable with infinite fundamental group, it is the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary.

Gluing up L to produce the “spine” Z so that M(Z) is homotopic to a 3-manifold is thus the bulk of the work. The spine Z will be a 4-valent graph, and the circle L will map to Z with degree 3 (i.e. every edge of Z has 3 preimages). At each vertex of Z, 6 edges in L run over the vertex in all possible ways from one incident edge of Z to another. The figure below shows three local models; the correct local model is the third one:

local_model

The key to the construction is to glue up collections of segments in L in triples, leaving a gap of three unglued segments of some fixed length \lambda which are the three edges of a theta graph (we call them “football bubbles”). Almost all the mass of L can be glued up this way, so we produce a reservoir of bubbles in a predictable distribution, and a remainder with relatively small mass. There are some operations that can then be performed on the remainder, gluing it up into the desired form, at the cost of adjusting the reservoir somewhat. Then the great mass of the reservoir is glued up into small disjoint collections whose local combinatorics can be completely specified; one particularly pretty move glues up four football bubbles (with suitably labeled edges) by draping them along the edges of a cube, each bubble aligned with one of the diagonal axes:

cube_move

This idea of first performing a random matching which is “almost” right, which can then be adjusted at the cost of perturbing the distribution of an almost equidistributed “sea” of predictable pieces of bounded size, so that the rest of the matching decouples into a massive number of matching problems of uniformly bounded size that can be solved once and for all — is one that has come up in several places recently, including in the papers of Kahn-Markovic and my paper with Alden mentioned above, but also in Peter Keevash’s construction of General Steiner Systems and Designs (a paper  I learned of from Gil Kalai’s blog). This is an idea with remarkable power and potential, beyond the already impressive (but well-known) power of “random constructions”. And it shows that highly constrained and beautiful combinatorial and geometric objects — designs as well as 3-manifolds — can be built out of generic pieces.

I am not sure what the moral of the story is; perhaps in every corner of the geometric desert, beautiful flowers bloom.