At the end of his monumental preprint addressing the Virtually Fibred Conjecture for Haken 3-manifolds , Wise makes a remarkably bold conjecture. (Nathan Dunfield blogged about Wise’s work here.) The purpose of this post is to highlight that conjecture and explain what it means. It’s such a remarkable conjecture that it’s difficult to believe it’s true, but it’s also a win-win in the sense that either a positive or a negative answer would be a huge advance in geometric group theory.
Wise’s Conjecture (Conjecture 20.5 of ): Let be a word-hyperbolic group which is also the fundamental group of a compact, non-positively curved cube complex . Then has a finite-sheeted covering space which is special.
Most of the rest of this post will be an attempt to explain what ‘special’ means, but let me first whet your appetite by giving some consequences.
Consequences of a positive answer
1. The Virtually Haken Conjecture (and, indeed, the Virtually Fibred Conjecture, LERF etc) for the fundamental group of any closed hyperbolic 3-manifold . The point is that Kahn and Markovic , in proving the Surface Subgroup Conjecture, actually construct enough surface subgroups to show that is the fundamental group of a compact, non-positively curved cube complex. The details of this were worked out by Bergeron and Wise , using foundational work of Sageev .
2. A random finitely presented group (in the density model, for the experts) at density less than 1/6 is virtually special (and hence residually finite, indeed -linear, and have all sorts of other nice properties). This would be a truly remarkable discovery. I think most experts believe that a random group is not residually finite (although no one knows how to prove this). These groups are known to be word-hyperbolic, and Ollivier and Wise proved that they are also fundamental groups of non-positively curved cube complexes .
Consequences of a negative answer
1. There exists a non-residually finite word-hyperbolic group. Indeed, Haglund and Wise  proved that, under the hypotheses of the conjecture, is virtually special if and only if every quasi-convex subgroup of is separable (ie closed in the profinite topology). But Agol, Groves and Manning proved that if every word-hyperbolic group is residually finite then every quasi-convex subgroup of every word-hyperbolic group is separable .
This is certainly enough to convince me that resolving Wise’s Conjecture is of the utmost importance. On the other hand, notice that a positive answer is probably no easier than the work contained in Wise’s preprint, together with the other papers that go into the main results of !
In the remainder of this post, I want to introduce Haglund and Wise’s special cube complexes, introduced in . These are well worth understanding if you are interested in 3-manifold topology, as Wise’s programme aims to derive nice theorems about hyperbolic 3-manifolds by proving that they are homotopy equivalent to (virtually) special cube complexes. Indeed, the main theorem of  implies that any closed hyperbolic 3-manifold that contains an embedded geometrically finite surface is indeed homotopically equivalent to a virtually special cube complex.
Much of Wise’s programme is rather technical, but special cube complexes are a beautiful, simple and appealing idea, and I think anyone can understand the definitions and basic properties quite quickly.
Special Cube Complexes
The context here is that of cube complexes, ie cell complexes in which each cell is a cube and the attaching maps are combinatorial isomorphisms. In geometric group theory we also like to impose a condition on the geometry of a cube complex , namely that should admit a locally CAT(0) (ie non-positively curved) metric. One of the attractions of cube complexes is that this condition can be phrased purely combinatorially. Note that the link of a vertex in a cube complex naturally has the structure of a simplicial complex.
Gromov’s Link Condition: A cube complex admits a non-positively curved metric if and only if the link of each vertex is flag. Recall that a simplicial cube complex is not flag if there is a subcomplex isomorphic to the boundary of an -simplex (for ) but is not the boundary of an -simplex in .
Examples: Salvetti complexes. Let be any graph. We build a cube complex as follows:
- has one 0-cell;
- has one (oriented) 1-cell for each vertex of ;
- has a square 2-cell with boundary reading whenever and are joined by an edge in ;
- for , the -skeleton is defined inductively—simply glue in an -cube wherever you see the boundary of an -cube.
The fundamental group of the Salvetti complex is the right-angled Artin group . It’s an easy exercise to check that is non-positively curved.
Another nice feature of cube complexes is that they have natural codimension-one subcomplexes, called hyperplanes. If an -cube in is identified with , then the hyperplanes of are just the intersection of with the coordinate hyperplanes of . We then glue together hyperplanes in adjacent cubes whenever they meet, to get the hyperplanes of of , which naturally immerse into . Pulling back the cubes in which the cells of land gives an interval bundle over , which also naturally immerses into . Using this language, we can write down a short list of pathologies for hyperplanes in cube complexes.
1. A hyperplane is one-sided if is not a product bundle. Otherwise it is two-sided.
2. A hyperplane is self-intersecting if is not an injection.
3. A hyperplane is self-osculating if does not inject the boundary of .
4. A pair of hyperplanes is inter-osculating if they both intersect and osculate; that is, the map is not an embedding and the map maps a point of the boundary of to the same place as a point of the boundary of .
These are illustrated in the above picture. A self-intersection is in the top left, a self-osculation in the top right and an inter-osculation below.
Definition (Haglund–Wise, ): The cube complex is special if none of the above pathologies occur.
Remark: I’m lying slightly. In fact, the definition of a special complex is slightly less restrictive. But, up to passing to finite covers, this definition coincides with their actual definition. Similarly, some of the statements below are slightly over-simplified.
Definition: The hyperplane graph of a cube complex , , is the graph with vertex-set equal to the hyperplanes of , and with two vertices joined by an edge if and only if the corresponding hyperplanes intersect.
Now, here’s the remarkable observation that Haglund and Wise made.
If every hyperplane of is two-sided, then there is a natural map
Indeed, there is only one place to send each vertex of . Each 1-cell of goes to the unique 1-cell in which corresponds to the unique hyperplane that crosses, and the two-sided-ness assumption ensures that we can choose orientations consistently. Tracing through the definitions, one sees that for every higher-dimensional cube of , there is always a higher-dimensional cube of to send it to.
Pathologies 2-4 above correspond exactly to the failure of the map to be a local isometry. Indeed, if it is not, then it does not induce an isometric embedding on the link of some vertex; if two 0-cells of are identified then we have a self-intersection or a self-osculation; if two -cells that were not joined by an edge in are joined by an edge in the image, then we have an inter-osculation.
This proves one direction of Haglund and Wise’s main theorem.
Theorem (Haglund–Wise ): A non-positively curved cube complex is special if and only if there is a local isometry for some graph.
To prove the other direction, you simply need to notice that is special, and that covering spaces and locally convex subcomplexes of special cube complexes are also special. Lifting this local isometry to universal covers, we get a genuine isometry
Corollary: A cube complex is special if and only if is a subgroup of a right-angled Artin group.
This is really remarkable: from quite simple combinatorial conditions, we get an embedding of our group into something as concrete as a right-angled Artin group. From this you can deduce all sorts of nice properties. For instance, Agol showed that right-angled Artin groups, and hence their subgroups, are Residually Finite Rational Solvable (RFRS), and also that if is RFRS then is virtually fibred .
There are many more nice observations that one can say about special cube complexes, but for the sake of brevity I’ll finish off here. But I should just highlight one further theorem, which I alluded to above.
Theorem (Haglund–Wise ): If is a word-hyperbolic group and the fundamental group of a compact, non-positively curved cube complex , then is virtually special if and only if every quasi-convex subgroup of is separable.
The proof is, in essence, very attractive. On the one hand, if is virtually special then we have a nice embedding of (a finite-index subgroup of) into a right-angled Artin group, and it follows that has nice separability properties. On the other hand, if has nice separability properties then one can lift away pathologies 1-4 in a finite-sheeted covering space.
So Wise’s Conjecture is equivalent to the claim that the quasiconvex subgroups of the group are separable; in particular, has a lot of finite-index subgroups. This is one reason why the conjecture, on the face of it, seems so implausible—a priori, one wouldn’t expect such a complex to have any finite-sheeted covering spaces. (That said, if your primary interest is 3-manifolds, then you may also assume that is residually finite, as it is linear by Geometrization.) But implausible or not, I think this conjecture is already a major open problem.
 Agol, Groves and Manning, Residual finiteness, QCERF, and fillings of hyperbolic groups, Geom. Topol. 13 (2009), no. 2, 1043–1073, arXiv:0802.0709v2 .
 Bergeron and Wise, A boundary criterion for cubulation, arXiv:0908.3609v3 .
 Haglund and Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551–1620 .
 Kahn and Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, arXiv:0910.5501v5 .
 Ollivier and Wise, Cubulating random groups at density less than 1/6, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4701–4733 .
 Sageev. Codimension-1 subgroups and splittings of groups. J. Algebra 189 (1997), no. 2, 377–389 .
 Wise, The structure of groups with a quasiconvex hierarchy, available online at http://comet.lehman.cuny.edu/behrstock/cbms/program.html .