Cubulating groups

 In this post, we discuss ways of producing nice actions of groups on CAT$(0)$-cube complexes. We will follow two papers: one by Bergeron--Wise, which cubulates many hyperbolic manifolds, and one by Hruska--Wise, titled "Finiteness properties of cubulated groups", which gives a beautiful unified treatment of various concepts related to cubulation. Both are easy to read and some of the best pieces of writing to bear Wise's name. 

First let us recall what these cube complexes are. We will say why we want to produce these actions after.

An $n$--cube is a copy of $[-1,1]^n$. Restricting $i$ of its coordinates to $\pm 1$ yields a \emph{subcube} which can be identified with an $(n-i)$--cube in various ways.
A cube complex is obtained by gluing cubes along subcubes. (The actual gluing maps are modeled on isometries but are determined completely by the combinatorial data.)

A flag complex is a simplicial complex with the property that $n+1$ vertices span an $n$--simplex if and only if they are pairwise adjacent.

The link of a $0$--cube $c \in C^0$ is the complex associated to the ``$\epsilon$--sphere'' about $c$ in $C$. More precisely, $\text{link}(c)$ has an $n$--simplex for each corner of $(n+1)$--cube at $c$, where such corners are glued together precisely according to the way their associated cubes are glued together. A cube complex $C$ is nonpositively curved if $\text{link}(c)$ is a flag complex for each $c \in C^0$.

Finally, $C$ is $\CAT(0)$ if $C$ is simply connected and nonpositively curved. It is a fact that when $C$ is simply connected and satisfies the local nonpositive curvature condition, then $C$ has a $\CAT(0)$ metric where each $n$--cube is isometric to the standard Euclidean $n$--cube.

A group is said to be ($\mathrm{CAT}(0)$-)cubulated if it acts properly and cocompactly on a (finite-dimensional) $\mathrm{CAT}(0)$-cube complex.

The earliest motivation for defining nonpositively curved cube complexes is simply that a proper/cocompact action on a $\CAT(0)$ cube complex yields a proper/cocompact action on a $\CAT(0)$ metric space.
Interestingly enough, this remains a fascinating albeit simple application, especially in view of the methodology promoted in this paper, which often surprisingly provides a $\CAT(0)$ metric from codimension--$1$ subgroups. Some further properties one can get are:

• If $G$ acts essentially on a $\CAT(0)$ cube complex then $G$ cannot have Property~(T) by Niblo--Roller. Moreover, it is implicit in their argument that if $G$ acts metrically properly then $G$ is a-T-menable.
• If $G$ has bounded torsion and acts properly on a finite dimensional $\CAT(0)$ cube complex then $G$ satisfies a strong form of the Tits Alternative by Sageev--Wise.
• If $G$ acts properly on a $\CAT(0)$ cube complex $C$, then the asymptotic dimension of $G$ is bounded above by the dimension of $C$ by work of Wright.

Most notably, $\CAT(0)$ cube complexes play a key role in the resolution of the virtual Haken and virtual fibring conjectures. Agol proved that a cubulated hyperbolic group must be virtually special, and for these there is a lot of structure that can be exploited.

Definition: The class of word-hyperbolic groups with a malnormal quasiconvex hierarchy is the smallest class of groups, closed under isomorphism, that contains the trivial group, and such that, if $G = A *_BC $ and $A, B$ each have a malnormal quasiconvex hierarchy, or $G = A*_C$ and $A$ has a malnormal quasiconvex hierarchy, and $C$ is malnormal and quasiconvex in $G$, then $G$ also has a malnormal quasiconvex hierarchy.

Theorem 1 (Haglund, Hsu and Wise): Let G be a word-hyperbolic group. The following are equivalent: 

1. $G$ is virtually special; 
2. $G$ has a subgroup $G_0$ of finite index with a malnormal quasiconvex hierarchy;
3. $G$ has a subgroup $G_1$ of finite index with a quasiconvex hierarchy

So how do we show a group is cubulated? The first major result in this direction is due to Sageev and most cubulation results seem to be variations of his method. The main observation is in a cube, there are associated midcubes coming from setting one of the coordinates to 0, and one can extend this to a hyperplane in a cube complex by joining up these midcubes. If a group acts by cubical isometries on a cube complex, the subgroup stabilising a given hyperplane should be 'codimension 1' in some sense. If one can find appropriate codimension 1 subgroups, Sageev's construction builds the cube complex around these subgroups.  Let me know elaborate on what codimension 1 means.

Codimension 1 subgroups

Let $G$ be a finitely generated group with Cayley graph $\Gamma=\Gamma(G,S)$. A subgroup $H\subset G$ is codimension--$1$ if the coset graph $\bar\Gamma = H\backslash \Gamma$ is a \emph{multi-ended} graph in the following sense: $\bar\Gamma - \Lambda$ has $2$ or more infinite components for some compact subgraph $\Lambda \subset \bar\Gamma$. Note that this definition is independent of the finite generating set $S$.

For example, it is well known that any copy of $\Z^n$ in $\Z^{n+1}$ is codimension--$1$. A frequently encountered example arises when $G$ is isomorphic to a nontrivial amalgamated product $A *_C B$ or an HNN extension $A *_C$, in which case $C$ is a codimension--$1$ subgroup by a result of Scott. Another suggestive example is any infinite cyclic subgroup of a closed surface group.

Every finitely generated infinite index subgroup of a free group is codimension--$1$. However the same cannot be said for a closed surface group. The reader is urged to consider a covering space with a compact core having exactly one boundary circle; the corresponding subgroup is not codimension--$1$, but it does have a property we now briefly turn to.

Observe that if $H$ is codimension--$1$, then $H$ is a \emph{divisive} subgroup of $G$ in the sense that for some $d>0$ the complement $\Gamma-\mathcal{N}_d(H)$ has more than one component $K$ that is deep, meaning that $K$ does not lie in $\mathcal{N}_c(H)$ for any $c>0$.

Returning to the case when $G$ splits, we note that $A$, $B$, and $C$ are all divisive subgroups in a nontrivial amalgam $A *_C B$ and likewise $A$ and $C$ are divisive in $A *_C$.

A divisive subgroup is codimension--$1$ precisely when there exists $d>0$ such that $\Gamma-\mathcal{N}_d(H)$ has more than one $H$--orbit of deep components. The notions of codimension--$1$ and divisive subgroups are not equivalent but can be confusing, so we take the time to describe the difference. Let $n$ denote the number of deep components in $\Gamma - \mathcal{N}_d(H)$. In the special case when $1 < n < \infty$, the divisive subgroup $H$ has a finite index subgroup $H'$ that is a codimension--$1$ subgroup of $G$. Indeed $H$ acts on the collection of deep components and we can let $H'$ be the kernel of the resulting permutation homomorphism.

There is a more sensitive notion called the number of ends of the group pair $(G,H)$, which takes a value between $0$ and $\infty$. A group is codimension--$1$ precisely when this end invariant takes the value $e(G,H) \ge 2$. The property that $H$ is divisive in $G$ is equivalent to saying that the pair $(G,H)$ has more than one ``filtered end'', indicated in the literature by $\tilde{e}(G,H) \ge 2$.

We will use codimension 1 subgroups to produce many different ways of cutting up the group into two halves, analogous to how hyperplanes separate Euclidean space into two halves. The next section makes this idea formal with wallspaces, but the reader is also welcome to skip it since the intuition is just that a wallspace encodes the different ways of cutting a space up.

Wallspaces

Let $X$ be a nonempty set. A wall of $X$ is a pair of subsets $\{U,V\}$ called (closed) halfspaces such that $X = U\cup V$. We shall not assume that $U \cap V = \emptyset$. The open halfspaces associated to the wall $\{U,V\}$ are the sets $U-(U\cap V)$ and $V-(U\cap V)$.

Points $x,y$ of $X$ are separated by a wall $W$ if they lie in distinct open halfspaces of $W$. The notation $\#(x,y)$ denotes the number of walls separating $x$ and $y$. We say that a point $x$ and a wall $\{U,V\}$ betwixt each other if $x \in U \cap V$.

Distinct walls $W=\{U,V\}$ and $W' = \{U',V'\}$ are transverse if all four of the following intersections are nonempty:
\[
   U \cap U', \quad U \cap V', \quad V \cap U', \quad V\cap V'.
\]

The central example of a wallspace arises from a $\CAT(0)$ cube complex $C$ by letting $X=C^0$ and letting $\mathcal{W}$ consist of the partitions of $C^0$ induced by hyperplanes. Note that the walls are genuine partitions here. A similar example is a wallspace on $C$ whose walls are pairs of closed halfspaces associated to hyperplanes of $C$.

A wallspace is a pair of sets $(X,\mathcal{W})$ where each element $W \in \mathcal{W}$
is a pair of indexed subsets $\{H_{-W}, H_{+W}\}$ of $X$ called the closed halfspaces of $W$. Note that the subsets are indexed by the elements of $\{-,+\} \times \mathcal{W}$.

Moreover we require that the following properties are satisfied:

• $H_{-W} \cup H_{+W} = X$ for all $W \in \mathcal{W}$, 
• For all $x,y \in X$, there are finitely many $W \in \mathcal{W}$ such that $x \in (H_{-W}) - (H_{+W})$ and $y \in (H_{+W}) - (H_{-W})$.
• For each $x \in X$, there are finitely many $W \in \mathcal{W}$ such that $x \in H_{-W} \cap H_{+W}$. 
• If $\{H_{-W},H_{+W}\} = \{H_{-W'},H_{+W'}\}$ and $H_{-W} \cap H_{+W} = \emptyset= H_{-W'} \cap H_{+W'}$ then $W=W'$.

Note that the last item excludes the duplicate walls that are genuine partitions.

However, for walls that are not partitions, it is now sensible for $W,W'$ to be distinct walls with the same closed halfspaces. For, though $H_{-W},H_{-W'}$ and $H_{+W},H_{+W'}$ have the same underlying sets, their indices differ.

The dual cube complex

We now define the $\CAT(0)$ cube complex $C$ \emph{dual} to a wallspace $(X,\mathcal{W})$. An orientation $\sigma(W)$ of a wall $W =\{U,V\}$ is a choice of one of the two ordered pairs: $(U,V)$ or $(V,U)$. We use the notation $\sigma(W) =\bigl(\overleftarrow{\sigma}(W),\overrightarrow{\sigma}(W)\bigr)$.

An orientation $\sigma$ of the wallspace $\mathcal{W}$ is a choice of orientation $\sigma(W)$ for each wall $W \in \mathcal{W}$. We emphasize that duplicated walls need not be oriented in the same way by $\sigma$.

A $0$--cube $c^o$ of $C$ is an orientation of the wallspace that satisfies the following conditions:

• $\overleftarrow{c^o}(W) \cap \overleftarrow{c^o}(W') \ne \emptyset$ for all $W,W'\in\mathcal{W}$.
• For each $x \in X$, we have $x \in \overleftarrow{c^o}(W)$ for all but finitely many $W\in\mathcal{W}$.

Two $0$--cubes are connected  by a $1$--cube $c^1$ if there is a unique wall $W$ to which they assign opposite orientations. In this case, $c^1$ is \emph{dual} to the wall $W$. For $n\ge 2$, we add an $n$--cube whenever its $(n-1)$--skeleton is present.

The advantage of this construction is that it is very explicit in terms of the wallspace, so one can show many properties by hand (careful bookkeeping). For example,

Theorem 2: The dual cube complex $C$ is connected, simply connected, and non-positively curved. Hence $C$ is a $\rm{CAT}(0)$ cube complex.

One can similarly give criteria for $C$ to be finite dimensional. In order to deduce something about the group however, one wants the action to be nice as well

Desirable properties

Sageev showed that when $G$ is hyperbolic and the codimension 1 subgroups are quasi-convex, then the action on $C$ is cocompact. There exist various criteria for the action to be proper. The following is due to Bergeron--Wise:

Theorem 3: Let \( G \) be word-hyperbolic. Suppose that for each pair of distinct points \((u, v) \in (\partial G)^2\) there exists a quasi-convex codimension-1 subgroup \( H \) such that \( u \) and \( v \) lie in distinct components of \( \partial G - \partial H \). Then there is a finite collection \( H_1, \ldots, H_k \) of quasi-convex codimension-1 subgroups such that \( G \) acts properly and cocompactly on the resulting dual CAT(0) cube complex.

This gives one the added flexibility of just adding in as many codimension 1 subgroups as possible and then the theorem will take care of finding a finite collection. Finiteness comes from the cocompactness of the action of $G$ on the space of triples of distinct points on the boundary; this isn't an easy result but was known from the 20th century.

Using Theorem 3 one can cubulate many hyperbolic manifolds:

Theorem 4: Let \( M \) be a closed hyperbolic 3-manifold, and regard \( \pi_1 M \) as acting on \( \mathbb{H}^3 \cong \widetilde{M}^3 \). For each great circle \( C \subset \partial \mathbb{H}^3 \) there is a sequence of immersed quasi-Fuchsian surfaces \( F_i \to M \) such that \( \partial \widetilde{F}_i \) pointwise converges to \( C \).

Corollary 5: Let \( M \) be a closed hyperbolic 3-manifold. For each pair of distinct points \( p, q \in \partial \widetilde{M} \) there is an immersed quasi-Fuchsian surface \( F \to M \) with a lift of universal cover \( \widetilde{F} \subset \widetilde{M} \) such that \( \partial \widetilde{F} \) separates \( p, q \) in \( \partial \widetilde{M} \).

Proof: Let \( C \) be the great circle that is the perpendicular bisector of a geodesic from \( p \) to \( q \). By Theorem 4, there exists \( F_i \to M \) a sequence of surfaces whose universal covers have boundaries that limit to \( C \).

For sufficiently large \( i \), the \( \epsilon \)-neighborhood of \( C \) in \( \partial \widetilde{M} \) has the property that \( \partial \widetilde{F}_i \) separates its two bounding circles (just like \( C \)). Thus \( \partial \widetilde{F}_i \) separates \( p, q \) in \( \partial \widetilde{M} \). $\blacksquare$

Theorem 6: Let \( G \) be a uniform arithmetic hyperbolic lattice with a codimension-1 quasi-convex subgroup \( H \). Then \( G \) acts properly and cocompactly on a CAT(0) cube complex.

Proof: Let \( U, V \) be the components of \( \partial G \) separated by \( \partial H \). For any two distinct points \( p, q \in \partial G \) there is an element \( c \) of the commensurator of \( G \) such that \( cp \in U \) and \( cq \in V \).  
Note that \( H_c = c^{-1}Hc \cap G \) is of finite index in \( c^{-1}Hc \), and is thus itself a codimension-1 quasi-convex subgroup. Observe that \( \partial H_c = c^{-1}\partial H \) separates \( p, q \) since \( p \in c^{-1}U \) and \( q \in c^{-1}V \). We have thus satisfied the criterion of Theorem 3. 

Agol's result then kicks in to show all these groups are virtually special. Many of these results have extensions to the relatively hyperbolic setting, but there things get (even) more technical as one has to be very careful when controlling the behaviour of the parabolics. We leave it to the motivated reader to look up what these are in the references. Nonetheless, there are pleasing results which show that reasonable things are still true. For instance, Reyes and Groves--Manning independently showed that if a group $G$ which is hyperbolic relative to virtually special subgroups can be cubulated such that the cubulation is 'compatible' with the cubulations of the parabolics, then $G$ is again virtually special. There are also instances when one can 'truncate' the cusps in the relatively hyperbolic case to make the group act cocompactly. For details, we refer the reader to Hruska--Wise.