On CAT(0) cube complexes
Graham A. Niblo and Michah Sageev
University of Southampton
5th July 2006
Graham A. Niblo
A worked example - PGL2 (Z)
We will study the linear group G = PGL2 (Z) following the
treatment in Buildings by Ken Brown.
“The book should be
accessible to any mathematics
graduate student who knows
very little group theory and
topology.”
Editorial Review Amazon
Graham A. Niblo
The definition
The group PGL2 (Z) is the quotient of GL2 (Z) (the group of
2 × 2 integer matrices with determinant ±1) by
! the central
"
−1 0
subgroup of order 2 generated by the matrix
.
0 −1
1
2
It is obtained
!
" from GL2 (Z) by identifying each matrix
a b
A=
with −A.
c d
We will write#the element
of PGl2 (Z) represented by the
$
a b
matrix A as
.
c d
Graham A. Niblo
The Mobius action
PGL2 (Z) acts by Mobius transformations on the upper half
plane model of the hyperbolic plane so that
%
#
$
az+b
if ad − bc = 1,
a b
(z) = cz+d
az+b
c d
if ad − bc = −1.
cz+d
Graham A. Niblo
Three reflections
Consider the three matrices each of order 2:
Matrix
#
$
0 1
S1 =
1 0
#
$
−1 1
S2 =
0 1
#
$
−1 0
S3 =
0 1
Isometry
s1 : z #→ 1/z
s2 : z #→ 1 − z
s3 = z #→ −z
Fixed set
{z | z = 1}
{z | Re(z) = 1/2}
{z | Re(z) = 0}
Graham A. Niblo
The reflections
l2
l3
Q
l1
P
The Mobius action of P GL 2 (Z) on the hyp erb olic plane
Graham A. Niblo
Generators for PGL2 (Z)
Theorem
PGL2 (Z) = %S1 , S2 , S3 &
Proof
The first step is to interpret multiplication of a matrix A by the
matrices Si as performing elementary row operations.
Multiplying A on the left or right by S1 has the effect of
switching the rows (respectively columns) of A.
Multiplying on the left or right by S3 has the effect of
multiplying a row (respectively column) by −1,
Multiplying by a product of S2 and S3 allows us to add one
row to another (respectively one column to another).
Graham A. Niblo
Carrying out these operations we can reduce any matrix A
to upper triangular form, and since the matrix has
determinant ±1 the diagonal entries will also be ±1.
Carrying out further elementary row operations we can
reduce the matrix to the identity.
Hence we can find elements M1 , M2 in the subgroup of
PGL2 (Z) generated by the matrices S1 , S2 , S3 such that
M1 AM2 = I, and it follows that every matrix in PGL2 (Z) lies
in this subgroup.
Graham A. Niblo
Picturing the action
Now we can examine the action of PGL2 (Z) by studying the
group of isometries generated by the three reflections s1 , s2 , s3 .
1
The mirror√lines !1 , !2 intersect at the point
P = (1 + 3i)/2 at an angle of 2π/6, so the two reflections
s1 , s2 generate the finite dihedral group D6 .
2
The reflection lines !1 , !3 intersect at right angles at the
point Q = i and therefore generate the finite dihedral group
D4 .
3
The lines !2 , !3 only share the ideal point ∞ and therefore
the isometries s2 , s3 generate a parabolic subgroup
isomorphic to the infinite dihedral group D∞ .
4
The three lines !1 , !2 , !3 cut the plane into six regions, one
of which, labelled C in the following diagram, is bounded by
all three lines. This region forms a fundamental domain for
the action of PGL2 (Z).
Graham A. Niblo
The Coxeter complex
The tiling of the plane by images of C is well known and
appears in a paper by Fricke which appeared in 1890. It was
reproduced in Brown’s book in 1988.
l3
Q
Graham A. Niblo
l2
C
l1
P
The cubing associated to the Coxeter complex
Recall that maximal cubes in the cube complex associated to a
wall system correspond bijectively to maximal families of
pairwise crossing walls. It is relatively easy to see that there are
two orbits under the action of such families:
1
!1 crossing !3 at Q,
2
!1 , !2 and s2 (!1 ) crossing at P.
It follows that there are two orbits of maximal cubes, one
square and one 3-cube.
Each square corresponds to a translate of the point Q and each
3-cube to a translate of the point P.
We can now easily draw the corresponding cube complex.
Graham A. Niblo
The cubing associated to PGL2 (Z)
The group acts with two orbits of hyperplanes.
Graham A. Niblo
The Cayley graph and Sageev cubings
Graham A. Niblo
Two almost invariant sets
"
The almost invariant
sets A and B are
stabilised by the
orientation preserving
subgroup (of index 2)
in the corresponding
hyperplane stabilisers.
!
Graham A. Niblo
The cubing associated to the almost invariant set A
Note that this is not the
Bass Serre tree of a
splitting since the
group acts with
involutions on the
edges.
0
4HETREEDUALTOTHECONFIGURATIONOFLINES
Graham A. Niblo
The Bass-Serre tree associated to the set A
Taking the barycentric
subdivision repairs this
and we obtain a
splitting of PGL2 (Z) as
the amalgamated free
product of the
stabilisers of P and Q
over their intersection.
1
0
4HETREEDUALTOTHECONFIGURATIONOFLINES
PGL2 (Z) = D4 ∗ D6 = %S3 , S1 & ∗ %S1 , S2 &.
Z2
Graham A. Niblo
"S1 #
The cubing associated to the almost invariant set B
Note that while this
cubing is not a tree it is
quasi-isometric to the
Bass-Serre tree. The
question is how we
derive the splitting from
this picture.
Graham A. Niblo
There is a map from the B-cubing to the A-tree which takes the
centre of each 3-cube to the corresponding vertex of the tree. If
we pull back the midpoints of edges we find a tree-like pattern
of walls (NOT hyperplanes) in the cube complex:
1
0
Graham A. Niblo
Coxeter groups
Any Coxeter group acts on its Coxeter complex with one (free)
orbit of chambers, allowing us to identify the group with the set
of chambers.
Each reflection (conjugate of a generator) has a codimension-1
fixed set called a wall and the walls are separating.
Each half space corresponds to an almost invariant set for the
wall stabiliser (the centraliser of the corresponding reflection).
Theorem
Niblo, Reeves and Caprace Any finitely generated Coxeter
group acts properly discontinuously on a finite dimensional
CAT(0) cube complex which is locally finite. If G contains no
special subgroups isomorphic to the 3, 3, 3 or 4, 4, 2 triangle
groups the action is also co-compact.
Graham A. Niblo
Example
The cubing associated to the 3, 3, 3 triangle group is R3 with its
usual cubing.
The Coxeter complex can be seen as a slice of this picture,
orthogonal to the line x = y = z.
The group PGL2 (Z) is a hyperbolic Coxeter group and we have
already constructed its cubing.
Graham A. Niblo
Hyperplanes
1
2
Each hyperplane H is itself a cube
complex and is contained in a totally
geodesic subspace of the form H × [0, 1].
It is therefore a CAT(0) cube complex and
has hyperplanes of its own.
3
Given a hyperplane K of H the subspace
K × [0, 1] hits each edge of a cube which
meets it orthogonally at its midpoint.
4
It follows that the hyperplanes in H are
precisely the intersection of H with the
other hyperplanes of X .
Graham A. Niblo
;=
H
K
It follows that a hyperplane is finite if and only if it is crossed by
only finitely many hyperplanes.
Lemma
Let X be a CAT(0) cube complex and G be a group G acting on
X with a single orbit of hyperplanes. Let H be the stabiliser of a
hyperplane H and let SingG (H) be the set
{g ∈ G | gH crosses H}. Then SingG (H) is of the form HFH for
some subset F ⊂ G.
Proof.
If f H crosses H then fh1 H crosses h2−1 H for any h1 , h2 ∈ H, so
SingG (H) is invariant under left and right multiplication by H.
Hence SingG (H) = HFH for some subset F ⊂ G.
Graham A. Niblo
The finite hyperplane theorem
Theorem
Let G be a finitely generated group, and H be a finitely
generated subgroup with e(G, H) ≥ 2. Let X be the Sageev
cubing and H the hyperplane. Then SingG (H) is a finite union
of double cosets HFH and the hyperplane H is finite if and only
if SingG (H) lies in the subgroup
CommG (H) = {g ∈ G | H g ∩ H has finite index in H and H g }.
Graham A. Niblo
Proof of the finite hyperplane theorem - part 1
(<
''
("
Here we use the finite generation of H to arrange for the
coboundary of HB to be connected.
Graham A. Niblo
Proof of the finite hyperplane theorem - part 2
Recall that hyperplanes crossing H correspond bijectively with
cosets gH in HFH.
Since HFH ⊂ CommG (H) each double coset HfH decomposes
as finitely many left cosets of H:
HfH = (ff −1 )HfH = f (H f H) = fSf (H ∩ H f )H = fSf H.
Graham A. Niblo
Applications of the finite hyperplane theorem
If G is finitely generated and has a finitely generated subgroup
H with e(G, H) and satisfying any of the following conditions
then the hyperplanes in the Sageev cubing are finite:
1
H is finite,
2
H is normal in G,
3
The commensurator of H in G is G.
Graham A. Niblo
The Plan
When G acts on a CAT(0) cube complex with finite hyperplanes
then the complex is sufficiently tree-like for us to obtain an
action on a tree.
Graham A. Niblo
Stallings’ theorem
Theorem
Let G be a finitely generated group. Then G
splits over a finite subgroup if and only if
e(G) ≥ 2.
Graham A. Niblo
Outline proof of Stallings’ theorem
1
e(G) > 1 ⇒ G has an almost invariant set B with finite
stabiliser.
2
Therefore G acts on a CAT(0) cube complex with finite
hyperplane stabiliser.
3
The hyperplanes are compact and the cube complex is
contractible.
4
So G acts on a simply connected 2-complex X which
contains a finite essential track.
5
So X contains a G invariant pattern of minimal tracks.
6
So G acts on a tree.
Graham A. Niblo
Steps 1 and 2 were outlined by Michah earlier on.
Step 3 The cubing is contractible because it is CAT(0) so
geodesics are unique and vary continuously with their
endpoints. Hyperplanes are finite because
CommG (H) = G.
Step 4 asserts that G acts on a simply connected
2-complex. For this we take a suitably refined triangulation
of the 2-skeleton of the cubing.
Graham A. Niblo
Dunwoody patterns
Definition
A pattern in the 2-complex consists of a closed subset P in the
complement of the 0-skeleton of X such that P meets each
closed 1-simplex γ in a finite union of points all lying in the
interior of γ, and each closed 2-simplex in a finite union of
disjoint closed line segments, each joining two points in the
boundary edges.
Graham A. Niblo
Dunwoody patterns
Definition
A pattern in the 2-complex consists of a closed subset P in the
complement of the 0-skeleton of X such that P meets each
closed 1-simplex γ in a finite union of points all lying in the
interior of γ, and each closed 2-simplex in a finite union of
disjoint closed line segments, each joining two points in the
boundary edges.
Note that patterns are
locally separating.
Graham A. Niblo
Notes on patterns
1
This definition does not quite agree with that given by
Dunwoody since we allow the line segments to join two
points in the same edge of a 2-simplex.
2
A connected pattern is called a track
Graham A. Niblo
Theorem
If X is a simply connected 2-complex then any track is a
separating set.
Graham A. Niblo
Proof
Given a track τ and an edge e define w(e) = |τ ∩ e|. Reducing
mod 2, we obtain a Z2 -valued 1-cochain z.
4HEWEIGHTFUNCTIONWANDCOCHAINZ
Graham A. Niblo
Proof
Given a track τ and an edge e define w(e) = |τ ∩ e|. Reducing
mod 2, we obtain a Z2 -valued 1-cochain z.
4HEWEIGHTFUNCTIONWANDCOCHAINZ
ZDF
DZS
δz(σ) = 0 for each 2-simplex σ.
The corresponding class in
H 1 (X , Z2 ) is zero if and only if the
pattern separates X
If π1 (X ) = {0} then H 1 (X , Z2 ) = 0
so every track separates.
Graham A. Niblo
Least weight tracks
Now assume that X is a simply connected, triangulated
2-complex (so every track separates)
Definition
A pattern P (or track) is said to be finite if it intersects only
finitely many 1-simplices, and in this case we assign it a weight,
/P/ = |P ∩ X (1) . A finite pattern is said to be essential if at
least two of its complementary components are unbounded. It
is said to be least weight if it is essential and has least weight
among all essential tracks.
The Big Idea: “Least weight tracks do not intersect”
Graham A. Niblo
Example
. . . unfortunately least weight tracks can (and often do)
intersect.
Graham A. Niblo
Cut and paste
Lemma
Any least weight pattern on X consists of a single track which
intersects each edge of X in at most one point and each
2-simplex in at most a single arc joining distinct edges of the
simplex.
Graham A. Niblo
Proof
Given a pattern P and an edge e define w(e) = |P ∩ e|. If P
has more than one component they each have weight less than
P so they are all inessential. They cut the space in a tree
pattern, and, by hypothesis two of the complementary
components (corresponding to vertices of this tree) are
unbounded. Choose an edge in the tree separating them This
is a track in the pattern and its two complementary components
are both unbounded. This contradicts the minimality of P.
UNBOUNDEDREGION
E
Graham A. Niblo
Reducing the characteristic function of P mod 2 we obtain a
new function with values 0 and 1, from which we can build a
new pattern Q.
Graham A. Niblo
Reducing the characteristic function of P mod 2 we obtain a
new function with values 0 and 1, from which we can build a
new pattern Q.
If P crossed some edge at least twice
then the norm of Q is strictly less than
||P||, but their characteristic functions
give identical elements of Hb1 (X , Z2 ) by
construction, so Q is also a least weight
pattern, contradicting the minimality of P.
So P is a single track which crosses
each 2-simplex in a single arc and which
separates X into two unbounded
components.
Graham A. Niblo
Theorem
Let σ and τ be least weight tracks in a simply connected
2-complex X which intersect transversely in the interior of the
2-cells of X . Then there are disjoint least weight tracks σ % and
τ % in X such that (σ ∪ τ ) ∩ X (1) = (σ % ∪ τ % ) ∩ X (1) .
Graham A. Niblo
Proof
Let /σ/ = /τ / = n (every least weight pattern is a track of
weight n).
Taking the boundary of a small regular neighbourhood of
the union of the two tracks gives a new pattern P with
weight 2n. There are “opposite corners” which are both
unbounded so at least two of the tracks in P are essential.
Hence they each have weight n and they are least weight.
They are given by the canonical cut and paste sketched
above.
Graham A. Niblo
Least length tracks
Now we "hyperbolise" our complex by uniformizing each
simplex as an ideal hyperbolic triangle so that the midpoint of
each edge is identified with the natural midpoint of each edge.
We can replace each arc of our pattern with the corresponding
geodesic arc in the hyperbolic metric and define the length of
the pattern to be the sum of these lengths.
We say that a pattern is minimal if it is a least weight pattern of
least length among all least weight patterns. (In particular it is a
track.)
4HE-EEKS9AUROUNDINGTRICK
Theorem
Minimal tracks do not cross.
#UTTINGANDPASTINGREDUCESLENGTH
Graham A. Niblo
How do we know minimal tracks exist?
We don’t and they might not.
If the complex has a separating vertex which cuts it into at least
two unbounded components then we obtain least weight tracks
in the neighbourhood of that vertex. By moving them out to
infinity we can make them arbitrarily short AND arbitrarily close
to the separating vertex.
3HORTENINGHOROCYCLICTRACKS
Graham A. Niblo
7ECANNOTARBITRARILYSHORTEN
NONHOROCYCLICTRACKS
Since non-horocyclic
tracks have to have length at least
√
log(1 + 2) they cannot be arbitrarily shortened. We can apply
the Arzela-Ascoli theorem to prove that minimal tracks exist in
this case.
Theorem
The Arzela-Ascoli theorem If C is a compact metric space and
Γ is a separable metric space, then every sequence of
equicontinuous maps fn : Γ −→ C has a subsequence that
converges uniformly on compact subsets to a continuous map
f : Γ −→ C.
Graham A. Niblo
Defining the compact metric space C
Let ! = inf{lengths of least weight tracks}.
If there are no horocyclic tracks ! ≥ log(1 +
√
2).
Choose ' > 0 and consider only least weight tracks of
length < ! + '.
There is a neighbourhood of the ideal vertices which no
such track can enter since it is too far from opposite sides
of triangles.
So we can remove an open neighbourhood of all ideal
vertices and obtain a 2-complex Y which contains all
sufficiently short least weight tracks.
This 2-complex is locally finite since all infinite branching in
the cube complex occurs at the vertices. This follows from
the fact that hyperplanes are finite.
Graham A. Niblo
Now take a sequence of least weight tracks with length
converging to ! and all of length < ! + '.
Since there is one hyperplane orbit and only finitely many
edges on each hyperplane there are only finitely many
edge orbits, so, translating by elements of G we can
assume all the tracks in the sequence intersect some given
edge e.
The ball C of radius ! + ' around e in Y is compact and
contains all the tracks of length < ! + ' which meet the
edge.
There are only finitely many possibilities for the topology of
a least weight track in C. So we can take a subsequence
consisting of tracks which are all homeomorphic to some
graph Γ.
Graham A. Niblo
We now have a sequence of maps
fn : Γ −→ C
defining tracks with lengths converging to !.
Theorem
The Arzela-Ascoli theorem If C is a compact metric space and
Γ is a separable metric space, then every sequence of
equicontinuous maps fn : Γ −→ C has a subsequence that
converges uniformly on compact subsets to a continuous map
f : Γ −→ C.
Since length varies continuously with the tracks the map f
defines a track of length ! as required.
Graham A. Niblo
Generalising Stallings’ Theorem
All we needed to make this argument work is an essential
action of G on a CAT(0) cube complex so that each hyperplane
is finite.
Theorem
If G is a finitely generated group, H is a finitely generated
subgroup with e(G, H) ≥ 2 and G = CommG (H) then G splits
over a subgroup commensurable with H.
Graham A. Niblo
Avoiding the singularity set
Theorem
Suppose G acts essentially on a CAT(0) cube complex X with a
single orbit of hyperplanes GH, finitely generated hyperplane
stabiliser H and suppose there is a finite index subgroup K < G
avoiding the singularity set SingG (H). Then K acts essentially
on a tree and splits over H ∩ K (up to index 2).
Graham A. Niblo
Proof
The action of G on X restricts to an action of K on X . If the K
action has a fixed point p then the orbit Gp is finite so G has a
fixed point. This is a contradiction, so the K action is essential.
Now the hyperplane orbit K H consists of disjoint hyperplanes
since SingG (H) ∩ K = ∅.
We can construct the cube complex dual to this family of half
spaces and it will be a tree. The edge stabiliser will be H ∩ K . If
necessary we take the barycentric subdivision to get a splitting.
Graham A. Niblo
Applications
Corollary
Suppose that G is finitely generated, H < G is finitely
generated, e(G, H) ≥ 2 and H is closed in the profinite topology
on G. Then G virtually splits over H, i.e., it has a finite index
subgroup K which splits over H.
A subgroup H < G is closed in the profinite topology if and only
if it is an intersection of finite index subgroups of G. In other
terminology it is separable.
Graham A. Niblo
Proof.
Let X be the Sageev cubing and H a hyperplane with stabiliser
H. Since H is finitely generated, SingG (H) = HFH for some
finite subset F ⊂ G, and F ∩ H = ∅. Since H is an intersection
of finite index subgroups of G there is a finite index subgroup
K < G such that H < K and K ∩ F = ∅. It follows that
HFH ∩ K = ∅ so we can apply the previous theorem to split K
over H.
Graham A. Niblo
Examples
1
If G is a free group and H is an arbitrary infinite index
finitely generated subgroup then G virtually splits over H.
(Hall’s theorem)
2
If G is the fundamental group of an aspherical surface then
G almost splits over most finitely generated subgroups.
3
If G is the fundamental group of a Seifert fibered
3-manifold containing the fundamental group H of a closed
surface then G virtually splits over H.
Graham A. Niblo
Topological consequences
If M is an aspherical manifold and N is an immersed,
π1 -injective submanifold such that π1 (N) is closed in the
profinite topology on G then M has a finite cover to which the
immersion lifts to an embedding.
This connects with the virtual Haken conjecture in the world of
3-manifolds, that every compact aspherical orientable
3-manifold has a finite cover which contains a π1 -injective
embedded surface.
Graham A. Niblo
Returning to the Rubinstein-Wang example
Recall the example of an immersed π1 injective surface in a
3-manifold such that every translate of the surface crosses
every other.
In this case SingG (H) = G, so G = HFH for some finite set F
disjoint from F .
It follows that G has no finite index subgroup containing H and
avoiding F .
So H is not closed in the profinite topology on G.
Graham A. Niblo
Controlling the singularity part 3
Graham A. Niblo
Theorem
If SingG (H) generates a proper subgroup L of G then G splits
over the intersection of a vertex stabiliser of some vertex
adjacent to H with L.
Proof.
The cosets of L in G correspond with components of the union
of the hyperplanes. If this is not connected there is a vertex in
the cube complex which separates it. We can use this vertex to
build a tree.
(In the language of the proof of Stallings’ theorem we will end
up with a horocyclic track, but since hyperplanes are infinite it is
hard to say what the track stabiliser is.)
Graham A. Niblo
Lifting the splitting of PGL2 (Z)
SL2 (Z) (→ GL2 (Z) → PGL2 (Z)
What about SL3 (Z)?
Graham A. Niblo
Kazhdan’s property (T)
Definition
A (locally compact) group G is said to have Kazhdan’s property
(T) if the trivial representation is isolated in the Fell topology on
the space of unitary representations.
Example
SL3 (Z) has Kazhdan’s property (T).
Theorem
Serre: If G has Kazhdan’s property (T) then every action on a
tree has a global fixed point.
Corollary
SL3 (Z) does not split as a non-trivial amalgamated free product
or an HNN extension.
Graham A. Niblo
Property (T) for geometric group theorists
A countable discrete group G has property (T) if and only if
every affine isometric action on a separable Hilbert space has a
global fixed point.
Serre showed that given a G-action on a tree T there is an
embedding of T into Hilbert space so that the action extends by
affine isometries. If the extension has a fixed point then the
action on the tree has a bounded orbit so is inessential.
Graham A. Niblo
Serre’s embedding
Graham A. Niblo
Embedding a tree in Hilbert space
Let E denote the set of edges of the tree and H = !2 (E).
Amongst the elements of H we have the Dirac functions δe
which take the value 1 on an edge e and 0 for every other
edge.
Choose a vertex u of the tree and for any vertex v and any
edge e define wv0 (e) = 1 if e separates u and v and 0
otherwise.
We now define the element f0 (v ) ∈ H by
f0 (v ) =
&
e∈E
Graham A. Niblo
wv0 (e)δe
So fe (v ) is the sum of those Dirac functions corresponding to
the edges separating u and v . Since this is a finite sum it is !2 .
It is easy to prove that the action of F on the tree descends to a
proper affine isometric action on the Hilbert space.
For s ∈ !2 (E), e ∈ E and g ∈ G define
%
s(e),
if e does not separate v , gv ,
gs(e) :=
1 − s(e), if e does separate v , gv .
Graham A. Niblo
Coordinates and the action
U
Graham A. Niblo
V
GV
Medians
V
Given 3 vertices u, v , w in a
CAT(0) cube complex the
hyperplanes separating them
fall into three families. Each
family separates one vertex
from the other two.
U
W
It follows from this that there is a unique vertex m which lies
between all three of them. More precisely:
Definition
The intervals [u, v ], [v , w], [u, w] have intersection comprising a
single vertex m known as the median of u, v , w.
Graham A. Niblo
Intervals in the plane
V
M
U
Graham A. Niblo
W
Replacing the tree with a CAT(0) cube complex we may
generalise the embedding.
Let H denote the set of hyperplanes and let H denote the
Hilbert space of !2 functions on H.
As with the tree we choose a basepoint u and define
f0 : X (1) −→ H by
f0 (v ) =
&
wv (h)δh
h∈H
where wv (h) is 1 if h separates u and v and 0 otherwise.
Graham A. Niblo
Isometries
If we measure the distance between vertices of the cube
complex as the number of hyperplanes separating them
we obtain the edge metric on the vertices.
If we carry out the same procedure on integer lattice points
in the Hilbert space (i.e., the integer points in the countable
dense subspace spanned by the Dirac functions) we obtain
the !1 -metric,
It follows that the embedding is an isometry between the
edge metric and the !1 metric on the vertices.
Graham A. Niblo
Fixed points from bounded orbits
Theorem
Let G be a group acting with a single, infinite orbit of
hyperplanes on a CAT(0) cube complex X with a bounded orbit.
Then G has a unique fixed point and it is a vertex of X .
If X is finite dimensional then it is a complete CAT(0) cube
complex and any bounded orbit has a unique metric centre.
This centre is invariant under the group action as required. An
infinite dimensional cube complex is not complete so we cannot
apply this argument. There are two approaches we can try to
use to remedy this:
Graham A. Niblo
Use the metric completion
Take the metric completion X of X . This is a complete
CAT(0) space.
The G-action extends to X and so has a bounded orbit
there.
Since X is complete this action has a fixed point p.
Unfortunately we do not it a priori expect p to lie in the
subspace X .
Graham A. Niblo
Use the Hilbert space
Embed the cube complex in Hilbert space and extend the
action.
Since G has a bounded orbit in X it has a bounded orbit in
Hilbert space.
Since Hilbert space is complete G has a fixed point s.
Since G acts transitively on the hyperplanes it acts
transitively on the coordinates of s and the action of any
given element converts some coordinate r to either r or
1 − r . zero of the form r , 1 − r .
Graham A. Niblo
It follows that if any coordinate of the invariant function is
neither 0 nor 1 then they are all nonSuch a vector is not !2 so every coordinate of the fixed
vector is 0 or 1. Hence any fixed point represents a vertex
of the unit cube.
Hence the fixed set is convex and it consists of vertices of
the cube in Hilbert space.
Since the set of vertices is disconnected this shows that G
has a unique fixed point in the Hilbert space and this is a
vertex s.
It remains to show that this vertex comes from a vertex of the
original cube complex.
Graham A. Niblo
Now we put the two constructions together.
Since X embeds in Hilbert space and Hilbert space is complete
we can extend the embedding equivariantly to all of X .
8
8
Graham A. Niblo
l
Any fixed point p in X must map to the fixed vertex s so it
follows that s is the limit of a sequence of points in the image of
X.
S
But each point in the sequence lies in the
image of a closed cube of X and the
distance from a vertex s of the unit cube
to a face not containing it is 1.
Hence almost all points in the sequence lie in closed cubes
containing s. It follows that s lies in a closed cube from X as
required.
Graham A. Niblo
The Haagerup property (a-T-menability)
Definition
A countable discrete group G is said to be a-T-menable if it
admits a metrically proper, affine isometric action on a
separable Hilbert space.
Recall that an action is metrically proper if for any ball B of finite
radius the set {g ∈ G | gB ∩ B 2= ∅} is finite.
Graham A. Niblo
Analytic methods in group theory
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Graham A. Niblo
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Thompson’s group
This is the group of piecewise linear diffeomorphisms of
the interval with dyadic rational slopes and dyadic rational
break points.
It is a torsion free group of infinite cohomological
dimension.
It contains no free subgroups.
It is unknown whether or not it is amenable.
It does act properly (not cocompactly) on a CAT(0) cube
complex (Dan Farley), hence it is a-T-menable.
Graham A. Niblo
The embedding is NOT an isometry in the geodesic
metrics.
W
V
W
U
U
V
The geodesic distance between√the vertices labelled v and w is
2 in the cube complex but only 2 in the Hilbert space.
Graham A. Niblo
Uniform embeddings
Recall that a map f : X → Y between metric spaces
(X , dX ), (Y , dY ) is said to be a uniform embedding if there are
increasing functions S, T : R+ → R+ such that
dY (f (x), f (x % )) ≤ S(dX (x, x % ))
dX (x, x % ) ≤ T (dY (f (x), f (x % )))
Our map f0 is contracting (not necessarily strictly) so we may
choose S to be the identity. Since the cube complex is
geodesic this shows that f0 is a coarse Lipschitz map.
Graham A. Niblo
Controlling the compression
Definition
The compression of f is the function ρf : R −→ R defined by:
ρf (r ) =
inf
d(g,h)≥r
d(f (g), f (h))
The asymptotic compression of f is:
Rf = lim inf
r →∞
log ρf (r )
.
log r
Examples
The asymptotic compression of a constant map is 0, while the
asymptotic compression of an isometry is 1. If ρf (r ) ∼ r 1/n then
Rf ∼ 1/n. In general large asymptotic compression is good.
Graham A. Niblo
The picture for the tree embedding
ve the following picture:
dH (f0 (s), f0 (t)) = /f0 (s) − f0 (t)/
s
Set E % = {e ∈ E | e separates s and t}
&
/f0 (s) − f0 (t)/2 = /
±δe /1 = d(s, t)
e∈E !
u
t
ρf0 (r ) =
inf
d(s,t)≥r
dH (f0 (s), f0 (t)) =
inf
d(s,t)≥r
√
r
√
log ρf0 (r )
log r
R(f0 ) = lim inf
= lim inf
= 1/2
r →∞
r →∞ log r
log r
11
Graham A. Niblo
We can deform the standard embedding, to a family of
embeddings with compression arbitrarily close to 1:
Theorem
Let X be a finite dimensional CAT(0) cube complex. For any
' ∈ [0, 1/2) there is a coarse Lipschitz map of X into a
separable Hilbert space with asymptotic compression 1/2 + '.
This means that on large scale a finite dimensional CAT(0)
cube complexes have the same geometry as a separable
Hilbert space.
Graham A. Niblo
Splittings and Sp(1, n)
Suppose that G is a co-compact lattice in Sp(1, n) where n ≥ 2.
Then G has Kazhdan’s property T. It follows that every action of
G on a CAT(0) cube complex has a global fixed point. In
particular e(G, H) ≤ 1 for every subgroup H < G.
Now G acts properly and co-compactly by isometries on the
quaternionic hyperbolic space of dimension 4n so G is a
Poincaré duality group. According to Kropholler a subgroup
H < G will have e(G, H) ≥ 2 if and only if H is a Poincaré
duality subgroup of dimension 4n − 1.
It follows that there are no π1 -injective immersions of an
aspherical 4n − 1 manifold into a Quaternionic 4n-manifold.
Graham A. Niblo
Controlling the singularity obstruction
Turning this round suppose that there is a π1 -injective
immersion of a quaternionic 4n-manifold M into an aspherical
4n + 1 manifold N.
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Graham A. Niblo
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G = π1 (N), H = π1 (M), so e(G, H) ≥ 2. Then G acts
effectively on a CAT(0) cube complex X so that H is the
stabiliser of a hyperplane.
The hyperplane is itself a CAT(0) cube complex so, as H
has property T , it acts a with a global fixed point on the
hyperplane.
It follows (not entirely trivially) that for any g ∈ Sing(H, G),
the subgroup K = H ∩ H g satisfies e(G, K ) ≥ 2.
So H ∩ H g must be a Poincaré duality subgroup of
dimension 4n.
Hence H ∩ H g has finite index in π1 (M) for any
g ∈ Sing(G, H).
Graham A. Niblo
Immersions and embeddings
This shows that Sing(π1 (M), π1 (N)) lies in the commensurator
of π1 (M), and so π1 (N) splits over a subgroup commensurable
with π1 (M).
Theorem
If G is the fundamental group of a closed aspherical
4n + 1-manifold for n ≥ 2 and G contains a subgroup
isomorphic to a co-compact lattice in Sp(1, n) then G splits over
a subgroup commensurable with the lattice .
At the level of topology this shows that any immersion of a
quaternionic hyperpbolic manifold of dimension 4n ≥ 8 into an
aspherical manifold of dimension 4n + 1 must be close to (a
finite cover of) an embedding.
Graham A. Niblo
Non-embedded immersions as a low dimensional
phenomenon
Compare the situation in low dimensions.
In dimension 1 every infinite index, finitely generated
subgroup is virtually a free factor so e(G, H) ≥ 2.
In dimension 2 most finitely generated subgroups satisfy
e(G, H) ≥ 2.
In dimension 3 it is conjectured that every 3-manifold
contains an immersed incompressible surface, though
many contain no embedded surfaces.
Similarly
SL2 (Z) splits as an amalgamated free product.
G = SLn (Z) has no subgroups H with e(G, H) ≥ 2 for
n ≥ 3.
Graham A. Niblo