Ian gave his second and third talks this afternoon, completing his (quite detailed) sketch of the proof of the Virtual Haken Theorem. Recall that after work of Kahn-Markovic, Wise, Haglund-Wise and Bergeron-Wise, the proof reduces to showing the following:

Theorem (Agol): Let G be a hyperbolic group acting properly discontinuously and cocompactly on a CAT(0) cube complex X. Then there is a finite index subgroup G’ so that X/G’ is special; in other words, G is virtually special.

Agol works with a characterization of virtually special groups, due to Wise, which is more closely tied to the notion of hierarchies.

Definition: A hyperbolic group G is QVH if it is obtained inductively by the following procedures:

1. the trivial group is QVH;
2. If G splits as an amalgam $G = A *_B C$ where A and C are QVH, and B is quasiconvex in G, then G is QVH;
3. similarly for an HNN extension $G = A*_B$; and
4. If H is QVH and is contained in G with finite index, then G is QVH.

Wise shows the following:

Theorem (Wise): A group is hyperbolic and acts cocompactly on a CAT(0) cube complex with special quotient if and only if it is QVH.

Thus the virtually special groups are the analogue in the world of hyperbolic groups of virtually Haken hyperbolic 3-manifolds in topology; we can say they are the fundamental groups of hyperbolic NPC cube complexes that admit a “quasiconvex virtual hierarchy”. So Ian’s argument works by exhibiting a finite cover of X/G with such a hierarchy.

The groups along which we would like to split in this hierarchy are (finite index in) the fundamental groups of the hyperplanes in X/G. Finding the cover amounts to separating these subgroups in finite covers. We don’t know how to do this directly, but the Weak Separation Theorem of Agol-Groves-Manning (discussed yesterday) shows that these subgroups can be “separated” in infinite covers (in a certain sense). The separation refers to the fact that a finite index subgroup of the hyperplane groups lifts to the cover, so that the hyperplanes in the infinite cover will be compact, 2-sided and embedded. In algebraic language, the Weak Separation Theorem guarantees the existence of a normal subgroup G’ of G so that if we write $\mathcal{G} = G/G'$ and $\mathcal{X} =X/G'$ then $\mathcal{X}$ has locally 2-sided embedded compact acylindrical hyperplanes (the acylindricity implies that the fundamental groups of the hyperplanes are malnormal). We can split $\mathcal{X}$ along these hyperplanes to produce a kind of hierarchy, so if $\mathcal{X}$ were compact we would be done. The idea is to take the (infinitely many) pieces that are created by the cutting, separate them into finitely many classes, and glue them together in such a way as to create a finite covering of X with a hierarchy of its own.

Ian calls the pieces into which $\mathcal{X}$ is decomposed by its hyperplanes cubical polyhedra, although they are not really polyhedra, but rather cubical subcomplexes of the cubical barycentric subdivision of $\mathcal{X}$. The combinatorics of the system of (compact!) hyperplanes in the (noncompact!) $\mathcal{X}$ is encoded by the so-called crossing graph $\Gamma(\mathcal{X})$. This graph has as vertex set equal to $\mathcal{W}$, the set of hyperplanes of $\mathcal{X}$ (there is a slightly unfortunate point that Ian uses the terminology “hyperplane” for what Wise calls “walls” when they are embedded; the letter W is supposed to represent the word “walls”; anyway, Ian used the terms hyperplane and walls synonymously in his talk, so if I sometimes accidentally use one word instead of the other, that’s the reason). A pair of hyperplanes (i.e. vertices) share an edge in $\Gamma$ in two cases, if either:

1. the hyperplanes $W_1, W_2$ intersect; or
2. conjugates of their fundamental groups have an infinite intersection (i.e. they are not acylindrical as a pair)

The group $\mathcal{G}$ acts on $\Gamma$, and moreover the maximum degree of $\Gamma$ is finite, because hyperplanes are compact and fall into finitely many $\mathcal{G}$-orbits, and $\mathcal{X}$ is locally compact (the hyperbolicity of G guarantees that there are no arbitrarily long essential cylinders running between distinct hyperplanes, so hyperplanes which are sufficiently far away from each other will not contribute an edge as in case 2 above) . Let k be the maximum degree of the vertices of $\Gamma$.

Suppose we could color the graph with finitely many colors in such a way that adjacent vertices have different colors, and the coloring is invariant under some finite index subgroup $\mathcal{G}'$ of $\mathcal{G}$. Then the quotient $\mathcal{X}/\mathcal{G}'$ would be compact with a quasiconvex hierarchy, and we would be done. Another way of saying that the coloring should be invariant under a finite index subgroup of $\mathcal{G}$ is to say that we have a finite set of colorings of $\Gamma$ which are permuted by the action of $\mathcal{G}$. Let $C_{k+1}(\Gamma)$ denote the set of all colorings of $\Gamma$ by the numbers $1,2,\cdots,k+1$ such that adjacent vertices get different colors. The set $C_{k+1}(\Gamma)$ can be topologized with the topology of convergence of colorings on finite subgraphs, making it into a compact (totally disconnected) space, and the group $\mathcal{G}$ acts on this space by homeomorphisms. Translating the statement above into this language, if we could find an invariant finite set on this space, we would be done. Instead, Ian finds an invariant probability measure. This is a completely general statement, and applies to all locally finite graphs with cocompact group actions.

Theorem: Let $\Gamma$ be a graph with bounded valence k, and G a group acting cocompactly on $\Gamma$. Then there is a G-invariant probability measure on the space of (k+1)-colorings of $\Gamma$.

Ian gave a very elegant proof of this theorem; after working it out, Lewis Bowen informed him that the theorem is a consequence of known work of Kechris-Solecki-Todorocevic on Borel colorings of Borel graphs (the kind that arise in the theory of measure equivalence of group actions). But Ian’s proof is so elegant that I can’t resist reproducing it here.

Pick orbit classes $e_1,\cdots,e_m$ of the edges of $\Gamma$. We call an assignment of colors to the vertices which does not necessarily assign different colors to adjacent vertices a labeling. The weight of a labeling is the number of $e_i$ whose vertices get the same color. Weight extends linearly to the space of probability measures on labelings; a G-invariant probability measure $\nu$ on labelings of weight 0 will be a G-invariant probability measure on colorings.

Now, a random labeling with n colors will have weight m/n. There is a function from labelings with n colors to labelings with (n-1) colors whenever $(n-1)>k$ that takes each vertex labeled n to the smallest number which is not a label on an adjacent vertex. Extend this operation by linearity to probability measures on labelings. This operation is G-equivariant and does not increase weight. So start with a random G-invariant probability measure on labelings with n colors (call it $\mu_n$) and reduce the number of colors one by one to $(k+1)$. This gives a measure $\nu_n$ on G-invariant labelings with $(k+1)$ colors whose weight is at most m/n (the weight of the random labeling). Take a weak limit of the $\nu_n$ as $n \to \infty$; this is a G-invariant probability measure $\nu_\infty$ on $(k+1)$-labelings whose weight is 0; that is, it is a probability measure on $(k+1)$-colorings, as desired. qed

OK, we now have a $\mathcal{G}$-invariant probability measure on colorings of $\Gamma(\mathcal{X})$. The colors 1 to (k+1) correspond to the order in which to cut along the hyperplanes in a hierarchy, so we can think of this probability measure as a kind of “superposition” of hierarchies of $\mathcal{X}$, which want to be pulled back from some hierarchy on a finite cover of $\mathcal{X}/\mathcal{G}$. When we cut along hyperplanes and then try to glue them back up, we need to remember the labels on all the cut open facets meeting the hyperplanes we are gluing. So it is important not just to remember the colors on vertices, but also all the colors on adjacent vertices we have cut up earlier, and the colors on vertices adjacent to them that we have cut up earlier, and so on.

Given a graph and a coloring of the vertices by numbers from 1 to (k+1), each vertex determines a “descending link”, which is the union of all simplicial paths emanating from that vertex along which the numbers decrease. The supercolor of a vertex is the structure of its descending link as a colored graph. Since the valence is bounded, there are only finitely many supercolors. Supercoloring determines an equivalence relation finer than coloring, and therefore an equivalence relation $\sim$ on $\mathcal{W} \times C_{k+1}(\Gamma)$ where $(v,c) \sim (w,d)$ if there is some g in $\mathcal{G}$ so that gv=w and the supercolor of v with respect to c is equal to the supercolor of w with respect to $d\circ g^{-1}$ (remember that $\mathcal{W}$ denotes the set of hyperplanes). Similarly we can define equivalence relations on $\mathcal{F} \times C_{k+1}(\Gamma)$ where $\mathcal{F}$ denotes the collection of faces of $\mathcal{X}$ in the barycentric subdivision dual to edges, by saying that $(F,c) \sim (E,d)$ if $(v,c)\sim(w,d)$ where $v$ is the hyperplane containing $F$, and $w$ is the hyperplane containing $E$. Finally we can define an equivalence relation on $\mathcal{P} \times C_{k+1}(\Gamma)$ where $\mathcal{P}$ denotes cubical polyhedra, by $(P,c) \sim (Q,d)$ if $(F,c) \sim (E,d)$ for corresponding faces F,E of P,Q.

A non-negative $\mathcal{G}$-invariant real valued function

$\omega: \mathcal{P}\times C_{k+1}(\Gamma) \to \mathbb{R}_{\ge 0}$

satisfies the gluing equations if for every face $F$ in the boundary of polyhedra $P_1,P_2$, and for every equivalence class $[F,d]$ we have $\sum \omega [P_1,c] = \sum \omega [P_2,c]$ where the sum is taken over equivalence classes $[P_i,c]$ restricting on F to equivalence classes $[F,c]$ which are equivalent to $[F,d]$ (i.e. the supercolors agree on the given face). Note that this is a finite sum, since there are only finitely many supercolors and orbits of faces or cubical polyhedra.

The point of the measure $\nu_\infty$ is that it defines a solution to the gluing equations, by setting $\omega[P,c]$ to be equal to the $\nu_\infty$ measure of the set of colors inducing the given supercolor on P. Since there are only finitely many gluing equations, and they have integer coefficients, the existence of one nontrivial solution implies the existence of a nontrivial integer solution; i.e. we can find an $\omega$ taking integer values.

I think it’s time for me to take a break now, so I’m posting this with the intention of coming back to add more details about how to use this integer solution to the gluing equations to get a hierarchy. Let me just say cryptically that this solution lets one glue up a finite collection of pieces which immerses into our (partially glued up) hierarchy in such a way that at the next step what needs to be glued are a finite collection of pieces which cover the same compact hyperplane with equal total degrees. It is at this point that Wise’s Malnormal Special Quotient Theorem lets one find finite covers in which the pieces can be matched in pairs and glued up along the hyperplane in question. More later (I hope). For the moment, here’s Ian explaining a detail to some doofus.

OK, after a break and another full day of lectures, I’m suitably rested, and ready to (briefly!) describe the endgame.

We imagine that we have already glued everything up to get a quasiconvex hierarchy, and then we inductively split along hyperplanes in the order of their colors $1,2,\cdots,(k+1)$. The result will be a collection of cubical polyhedra whose boundaries are decorated with what is known as a boundary pattern, which keeps track of where the cuts where made.

If V is a finite cube complex obtained by (perhaps partially) cutting open along a quasiconvex hierarchy, we will denote its boundary pattern by $\lbrace \partial_1(V),\cdots,\partial_j(V)\rbrace$. It is glued back to a hierarchy by first gluing up $\partial_j(V)$; this gives a new finite cube complex V’ with a new boundary pattern $\lbrace \partial_1(V'),\cdots,\partial_{j-1}(V'\rbrace$ which are the image of the boundary pattern in V.

So Ian’s inductive gluing procedure starts with $\mathcal{V}_{k+1}$ with boundary pattern $\lbrace \partial_1(\mathcal{V}_{k+1}),\cdots,\partial_{k+1}(\mathcal{V}_{k+1})\rbrace$ which consists simply of a collection of cubical polyhedra. The number of colored polyhedra of each type is the corresponding coefficient of our (integer) solution to the gluing equations, and the facets in the given boundary pattern are those with the corresponding color.

There is no obstruction to gluing up the $\partial_{k+1}$ facets in pairs, since this is exactly what the gluing equation guarantees we can do. This gives rise to $\mathcal{V}_k$ with boundary pattern $\lbrace \partial_1(\mathcal{V}_k),\cdots,\partial_k(\mathcal{V}_k)\rbrace$. Now the components of $\partial_k$ are more complicated, and it is not immediately clear how to glue them up. It will turn out that the components of $\partial_k$ all cover certain boundary components (with respect to a particular boundary pattern) of a particular compact cube complex $\mathcal{Y}_k$ in such a way that the sum of the degrees of the covers on either side of the component agrees. We would therefore like to take a finite cover of $\mathcal{V}_k$ in which the components of $\partial_k$ map to corresponding components of the boundary of $\mathcal{Y}_k$ in a way which can be matched up and then glued. In Ian’s paper he describes a method to find such a finite cover inductively, using a method in an appendix of Agol-Groves-Manning; however he pointed out in his talk on Wednesday that the cover can be found in one step by the MSQT. Anyway, I won’t say any more about it here.

OK, in this way we glue up $\partial_k$ in a finite cover to get $\mathcal{V}_{k-1}$, and then the components of $\partial_{k-1}(\mathcal{V}_{k-1})$ immerse into $\mathcal{Y}_{k-1}$ covering the same object from two sides with the same degree, so we can pass to a further cover (by the MSQT) where they can be paired, and glued up to get $\mathcal{V}_{k-2}$, and so on. Eventually we have glued up everything in a finite cover, obtained a hierarchy, applied Wise’s theorem that QVH is equivalent to virtually special, and then it’s time to break out the champagne.

So what’s $\mathcal{Y}_k$? In fact there is a $\mathcal{Y}_j$ for each j; it is a disjoint union of hyperplanes of the original complex split open along the hyperplanes they intersect of smaller color, quotiented out by the action of the stabilizer in $\mathcal{G}$ of the associated equivalence class. This complex has the property that each equivalence class $[F,c]$ of face for which the color of F in the coloring c is j has a unique representative in the complex $\mathcal{Y}_j$. It is this fact, together with the fact that the set of polyhedra in $\mathcal{V}_j$ satisfies the gluing equations (because inductively it is a cover of a partially glued union of polyhedra which as a set satisfy the gluing equations) which implies that the map is an immersion, and then the gluing equations say that the degrees on either side have the same sum, and now the previous paragraph makes sense (ahem!).

Well, that’s it for my summary. I will post a few photos of the blackboard taken by Patrick Massot and Alden Walker when I get a chance (one such photo by Patrick is above). If you want more details, then I believe Ian intends to post his preprint before too long, so keep watching the skies arXiv.

Update (April 13): Ian’s preprint is now available on the arXiv here.

Photos by Patrick Massot:

Photos by Alden Walker: