5CAT(0) spaces and groups
IV Topics in Geometric Group Theory
5.8 Special cube complexes
Let
X
be a non-positively curved cube complex. We will write down explicit
geometric/combinatorial conditions on
X
so that
π
1
X
embeds into
A
N
for some
N.
Hyperplanes and their pathologies
If
C
∼
=
[
−
1
,
1]
n
, then a midcube
M ⊆ C
is the intersection of
C
with
{x
i
= 0
}
for some i.
Now if
X
is a non-positively curved cube complex, and
M
1
, M
2
are midcubes
of cubes in
X
, we say
M
1
∼ M
2
if they have a common face, and extend this to
an equivalence relation. The equivalence classes are immersed hyperplanes. We
usually visualize these as the union of all the midcubes in the equivalence class.
Note that in general, these immersed hyperplanes can have self-intersections,
hence the word “immersed”. Thus, an immersed hyperplane can be thought of
as a locally isometric map H # X, where H is a cube complex.
In general, these immersed hyperplanes can have several “pathologies”. The
first is self-intersections, as we have already met. The next problem is that of
orientation, or sidedness. For example, we can have the following cube complex
with an immersed hyperplane:
This is bad, for the reason that if we think of this as a (
−
1
,
1)-bundle over
H, then it is non-orientable, and in particular, non-trivial.
In general, there could be self intersections. So we let
N
H
be the pullback
interval bundle over
H
. That is,
N
H
is obtained by gluing together
{M ×
(
−
1
,
1)
|
M is a cube in H}
. Then we say
H
is two-sided if this bundle is trivial, and
one-sided otherwise.
Sometimes, we might not have self-intersections, but something like this:
This is a direct self-osculation. We can also have indirect osculations that look
like this:
Note that it makes sense to distinguish between direct and indirect osculations
only if our hyperplane is two-sided.
Finally, we have inter-osculations, which look roughly like this:
Haglund and Wise defined special cube complexes to be cube complexes that
are free of pathologies.
Definition
(Special cube complex)
.
A cube complex is special if its hyperplanes
do not exhibit any of the following four pathologies:
– One-sidedness
– Self-intersection
– Direct self-osculation
– Inter-osculation
Example. A cube is a special cube complex.
Example.
Traditionally, the way to exhibit a surface as a cube complex is to
first tile it by right-angled polygons, so that every vertex has degree 4, and then
the dual exhibits the surface as a cube complex. The advantage of this approach
is that the hyperplanes are exactly the edges in the original tiling!
From this, one checks that we in fact have a special curve complex.
This is one example, but it is quite nice to have infinitely many. It is true
that covers of special things are special. So this already gives us infinitely many
special cube complexes. But we want others.
Example.
If
X
=
S
N
is a Salvetti complex, then it is a special cube complex,
and it is not terribly difficult to check.
The key theorem is the following:
Theorem
(Haglund–Wise)
.
If
X
is a compact special cube complex, then there
exists a graph N and a local isometry of cube complexes
ϕ
X
: X # S
N
.
Corollary. π
1
X → A
N
.
Proof of corollary.
If
g ∈ π
1
X
, then
g
is uniquely represented by a local geodesic
γ
:
I → X
. Then
ϕ ◦ γ
is a local geodesic in
S
N
. Since homotopy classes
of loops are represented by unique local geodesics, this implies
ϕ
X
◦ γ
is not
null-homotopic. So the map (ϕ
X
)
∗
is injective.
So if we know some nice group-theoretic facts about right-angled Artin groups,
then we can use them to understand π
1
X. For example,
Corollary.
If
X
is a special cube complex, then
π
1
X
is linear, residually finite,
Hopfian, etc.
We shall try to give an indication of how we can prove the Haglund–Wise
theorem. We first make the following definition.
Definition
(Virtually special group)
.
A group Γ is virtually special if there
exists a finite index subgroup Γ
0
≤
Γ such that Γ
0
∼
=
π
1
X
, where
X
is a compact
special cube complex.
Sketch proof of Haglund–Wise.
We have to first come up with an
N
. We set
the vertices of
N
to be the hyperplanes of
X
, and we join two vertices iff the
hyperplanes cross in
X
. This gives
S
N
. We choose a transverse orientation on
each hyperplane of X.
Now we define ϕ
X
: X # S
N
cell by cell.
– Vertices: There is only one vertex in S
N
.
–
Edges: Let
e
be an edge of
X
. Then
e
crosses a unique hyperplane
H
.
Then
H
is a vertex of
N
. This corresponds to a generator in
A
N
, hence a
corresponding edge
e
(
H
) of
S
N
. Send
e
to
e
(
H
). The choice of transverse
orientation tells us which way round to do it
– Squares: given hyperplanes
f
2
e
2
f
1
e
1
H
0
H
Note that we already mapped
e
1
, e
2
to
e
(
H
), and
f
1
, f
2
to
e
(
H
0
). Since
H
and
H
0
cross in
X
, we know
e
(
H
) and
e
(
H
0
) bound a square in
S
N
. Send
this square in X to that square in S
N
.
–
There is nothing to do for the higher-dimensional cubes, since by definition
of S
N
, they have all the higher-dimensional cubes we can hope for.
We haven’t used a lot of the nice properties of special cube complexes. They
are needed to show that the map is a local isometric embedding. What we do
is to use the hypothesis to show that the induced map on links is an isometric
embedding, which implies ϕ
X
is a local isometry.
This really applies to a really wide selection of objects.
Example. The following groups are virtually special groups:
– π
1
M for M almost any 3-manifold.
– Random groups
This is pretty amazing. A “random group” is linear!