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Discrete, Algebraic and Geometric Structures II
Sadayoshi Kojima
January 6, 2014
0 Overview
0.1 Virtual Haken Conjecture and its Neighbor
• Virtual Haken Conjecture
Original and the strongest form (largeness)
• Reduction by the solution of the Geometrization Conjecture
• Linear and integral representability
• Virtual Fiber Conjecture
0.2 Flow of the discussion
• Closed hyperbolic 3-manifold admits sufficiently many closed
essential quasi-Fuchsian immersed surfaces. Hence its π1 ad-
mits sufficiently many codimension 1 quasiconvex subgroups
(Kahn-Markovic [12]).
• Sufficiently many codimention 1 quasiconvex subgroups of a
hyperbolic group G =⇒ A proper and cocompact cubulation
of the group G (Sageev [14], Bergeron and Wise [3]).
• Proper and cocompact cubulation of a hyperbolic groupG =⇒Quasiconvex (malnormal) virtual hierarchy of G (Agol [2]).
• Hyperbolicity + Quasiconvex (malnormal) virtual hierarchy
=⇒ Virtually special (Hugland-Wise [10], Wise [16]).
– Special =⇒ Either virtual abelian or large (Wise [16]).
• Special =⇒ Embeddable in RAAG (Haglund and Wise [10]).
– A subgroup of a RAAG is RFRS and an irreducible 3-
manifold with RFRS π1 is virtually fibered (Agol [1]).
– Any RAAG can be embedded in some RACG (Davis and
Januszkiewicz [7]), and hence, a subgroup of RAAG has a
faithful representation in GL(n,Z).
1 Fundamental Group and Covering Space
1.1 Fundamentals
• Fundamental group ; π1(P, x0) = π1(P )
• Covering space ; π : P → P
• Universal cover ; π : X → P
• Isomorphism of covering spaces
• One to one correspondence between conjugacy classes of sub-
groups of π1(P ) and isomorphism classes of covering spaces
over P .
• Regular cover and covering transformation group ;
1→ π1(P )→ π1(P )→ Gal(P /P )→ 1
• Covering transfermation group and its induced hom ;
ϕ : Gal(P /P )→ Out(P ) = Aut(π1(P ))/Inn(π1(P ))
• There is an orbifold version of the theory.
1.2 Examples
• n-fold cover of the circle S1 ;
1→ Z ×n−→ Z→ Z/nZ→ 1.
• An orbifold is a space locally modeled on a quotient Rn/G
of an euclidean space Rn by a finite group action by a finite
group G. Finite group could be trivial and hence a manifold
is an orbifold.
We assume that the orbifold always has a non-singular univer-
sal cover.
Hence, an orbifold in this note is a quotient of a simply con-
nected space by a proper group action.
• S1 divided by the reflection along x-axis provides topologically
I = [−1, 1], however the terminal points has extra structure.
This is probably the simplest orbifold with the manifold uni-
versal cover. The 2-folded orbifold cover of S1 over the interval
: S1 → I provides an associated exact sequence of the covering,
1→ π1(S1)→ πorb1 (I)→ Z/2Z→ 1,
which is group theoretically
1→ Z→ D∞ ' Z/2Z ∗ Z/2Z→ Z/2Z→ 1.
πorb1 (I) ' D∞ can be seen as a group generated by two reflec-
tions in the universal cover R.
• Exercise : Double branched cover of S3 along the trefoil knot
K is the lens space L(3, 1) ; This implies the 3-fold orbifold
covering : L(3, 1) → O3(K), where O3(K) is the orbifold
whose underlying space S3 singular along K with index 3.
Show that
π1(S3 −K) ' 〈a, b | aba = bab〉πorb
1 (O3(K)) ' π1(S3 −K)/〈a2〉 ' D6,
where a is a meridian, and establish the exact sequence,
1→ Z/3Z→ πorb1 (O3(K))→ Z/2Z→ 1,
for this orbifold covering.
1.3 Eilenberg-Maclane space
• An Eilenberg-Maclane space K(G, n) is a CW complex such
that
πi(K(G, n)) '
G if i = n
1 otherwise.
For any G and n ≥ 1, there exits such a CW complex up to
homotopy, infinite dimensional and not locally finite in general.
• The universal cover of K(G, 1) is thus contractible, and we
have
1→ 1→ π1(K(G, 1))→ G→ 1
• Examples :
– A spaceN will be an Eilenberg-Maclane spaceK(π1(N), 1)
if its universal cover is contractible.
For example, aspherical manifolds are K(π, 1) spaces.
– Eercise : Any two-dimensional complex with trivial π2
is K(π, 1) space.
2 Coxeter Group
2.1 Fundamentals
• Coxeter group is defined by its presentation ;
G = 〈g1, g2, . . . , gn | (gigj)mij〉,
where mii = 1 and 2 ≤ mij ≤ ∞ in general.
• Coxeter Graph : To each Coxeter group, we assign a graph
Γ where,
– Vertices are generators,
– Vertices are joined by an edge labelled by mij if 2 ≤mij <∞.
Conversely, we can reconstruct from a labeled simple graph Γ
a Coxeter group G = CΓ.
2.2 Examples
• D∞ ' Z/2Z ∗ Z/2Z = 〈g1, g2 | g21, g
22〉
• Exercise : Show that there is a geodesic triangle Dp,q,r on
– the 2-dimensional sphere S2 if 1p + 1
q + 1r > 1,
– the 2-dimensional euclidean plane E2 if 1p + 1
q + 1r = 1 and
– the 2-dimensional hyperbolic plane H2 if 1p + 1
q + 1r < 1.
• Theorem : Triangle group ∆(p, q, r) generated by reflections
along edges of a geodesic triangle Dp,q,r is a Coxeter group,
∆(p, q, r) ' 〈g1, g2, g3 | g21, g
22, g
23, (g2g3)
p, (g3g1)q, (g1g2)
r〉.
Outline of the proof :
– Define a Coxeter group G abstractly by
G = 〈a1, a2, a3 | a21, a
22, a
23, (a2a3)
p, (a3a1)q, (g1g2)
r〉.
and define an abstract complexX ′ = G×Dp,q,r/ ∼, where
∼ is defined by looking at the action of G on the geometric
plane appropriately.
– Will see that the natural projection π : X ′ → X is a local
isometry and a covering.
– Then, since X is complete, π will be a covering. Moreover
since X is simply connected, π is a homeomorphism.
• Sending g1, g2, g3 to 1 ∈ Z/2Z, we define a homomorphim
∆(p, q, r)→ Z/2Z and we obtain an orbifold coveringO(p, q, r)
associated to its kernel. This is a double covering orbifold un-
derlying on the S2 with three cone singularities with indexed
by p, q, r and we have an exact sequence.
1→ πorb1 (O(p, q, r))→ ∆(p, q, r)→ Z/2Z→ 1.
• Exercise : Show that
πorb1 (O(p, q, r)) ' 〈g, h | gp, hq, (gh)r〉
2.3 Linear Representation
• Theorem : Any Coxeter group has a faithful linear repre-
sentation in R
• Construction : Let V = 〈α1, α2, . . . , αn〉R be a vector space
generated by α1, α2, . . . , αn over R.
Deifne a bilinear form B : V × V → R by
B(αi, αj) = − cosπ
mij,
and a (reflection like) involution σi : V → V by
σi(v) = v − 2B(αi, v)αi.
Then ϕ : G→ GL(V ) which sends Ri to σi becomes a faithful
representation.
• Corollary : Any right angled Coxeter group has a faithful
linear representation in Z.
2.4 Realizing as a reflection group
• The action of G on V through ϕ preserves B, however B is
quite likely to be not positive definite. Thus, we define the
action of G on the dual space V ∗ using B to obtain (nonde-
generate) reflections. Let
α∨i =B(αi, · )B(αi, αi)
and
σi(f ) = f − 2〈f, αi〉α∨ifor any f ∈ B∗, where 〈 , 〉 is a Kronecker product, Then σifixes the hypersurface Hi = f ∈ V ∗ | 〈f, αi〉 = 0 and
C =
n∩i=1
f ∈ V ∗ | 〈f, αi〉 ≥ 0
becomes a fundamental domain, whose interior is called a
chamber.
The union of the orbit of C is called a Tits cone.
• Example : Let G = 〈g1, g2 | g21, g
22〉 ' D∞, then
B =
(1 −1
−1 1
)With respect to the dual basis α∗1, α∗2 to α1, α2 such that
〈α∗i , αj〉 = δij, we have a matrix representation of the action
of G on V ∗ by
σ1 =
(−1 0
2 1
), σ2 =
(1 2
0 −1
).
Then, the real line t ∈ R → tα∗1 + (1− t)α∗2 is a G-invariant
subset and we can easily find a standard D∞ action on the
real line.
Tits cone in this case is an open cone of the line about the
origin.
• Exercise : Find a reflection representation of a triangle group
∆(p, q, r).
3 Right Angled Artin Group
3.1 Fundamentals
• RAAG is defined by its presentation ;
G = 〈g1, g2, . . . . gn | gigj = gjgi for some i 6= j〉.
• To each RAAG, we assign a graph Γ where
– Vertices are generators,
– Vertices are joined by an edge if they commute.
From a simple graph Γ, we can reconstruct a right angled Artin
group G = AΓ.
Remark : The associated graphs for RAAG and RACG are
not directly related each other, though they have some simi-
larities.
• Examples :
1. Zn, Fn, F2 × F2 are RAAG’s.
2. A particular RAAG, Z, can be embedded of index 2 in
Z/2Z ∗ Z/2Z, which is a RACG.
3. Will see that π1(Σg) can be embedded in some RAAG.
3.2 Salvetti complex
• K(AΓ, 1) space ; Let L be an abstract simplicial complex of
the frag complex of Γ. L has a poset structure by inclusions.
For each J ∈ L, let RJ = xi = 0 | i ∈ V (Γ)− J and set
SΓ = [−1, 1]V (Γ) ∩∪J∈L
RJ/ ∼
where ∼ is induced by the identification of the opposite sides
of [−1, 1]V (Γ).
This is called a Salvetti complex sometime.
• Example : Construct SΓ for some simple Γ !
Say, Z2, F2, F2 ∗ Z, Z ∗ Z2 and F2 × F2, etc.
• Exercise : Show that π1(SΓ) ' AΓ.
• SΓ admits a (Z/2Z)V (Γ) action induced by the reflections about
xi = 0 for i = 1, 2, . . . , |V (Γ)|. We denote this orbifold by
KΓ = SΓ/(Z/2Z)V (Γ)
≈ [0, 1]V (Γ) ∩∪J∈L
RJ
Example : Construct KΓ for the above simple Γ !
• Theorem (Davis and Januszkiewicz [7]) :
1→ π1(SΓ) ' AΓ → πorb1 (KΓ)→ Z/2ZV (Γ) → 1
is exact. In particular, any RAAG can be embedded into a
RACG of finite index.
Proof : Because SΓ → KΓ is an orbifold covering and
πorb1 (KΓ) is a RACG.
• Exercise : Given Γ, describe a Coxeter graph of πorb1 (KΓ).
• Corollary : Any subgroup of a RAAG group admits a faith-
ful representation in GL(n,Z) for some n.
4 Separability
4.1 Residually finiteness
• Definition : G is residually finite if there is a descending
sequence
G = G0 > G1 > G2 > · · ·such that [Gi : Gi+1] <∞ for all i ≥ 0 and
∩∞i=0Gi = 1.
Equivlently, for any a ∈ G, there exists a finite index sub-
group H < G such that a /∈ H .
Remark :
1. Any subgroup of a residually finite group is residually fi-
nite.
2. Any finite elements in a residually finite group can be sep-
arated simultaneously from a subgroup of finite index.
• Lemma : Any finitely generated subgroup of GL(n,C) is
residually finite.
In particular, any CG or RAAG is residually finite.
Outline of a proof : When G is in GL(n,Z) and we
have a specified element g ∈ G, then we can find m ∈ N such
that g ∈ G is mapped to a nontrivial element by GL(n,Z)→GL(n,Z/(m)). Then set the subgroup H to be its kernel.
General case is done just by taking analogous argument using
a ringR generated by entries ofG and its appropriate maximal
ideal O.
4.2 Subgroup separability
• Definition : H < G is separable if H is an intersection of
finite index subgroups of G.
Equivalently, if for any a /∈ H , there is a subgroup K < G
of finite index such that K > H and a /∈ K.
Remark :
1. Since any finite index subgroup H < G is separable in G
by definition, the separability concerns with subgroups of
infinite index.
2. If 1 < G is separable in G, G is residually finite.
3. A separable subgroup may not be represented as an inter-
section of normal subgroups.
4. If H < G′ is separable in G′ and [G : G′] <∞, then H is
separable in G.
• Important Proposition : Suppose G is residually finite.
If a subgroup H < G admits a retraction r : G → H < G,
then H is separable.
Proof : If N = Ker r, then G = NH and the intersections
Ni = Gi ∩ N define a descending sequence of finite index
subgroups to 1 because
[G : NiH ] = [N : Ni] ≤ [G : Gi],
Then H =∩∞i=1NiH .
4.3 Graph of groups
• A graph of groups : Start with a graph Γ = (V, E), not
necessarily simple.
A graph of groups is a system G = (V , E) of groups associated
to Γ :
Assign to each vertex and edge of Γ a group G, and injective
homomorphisms from each edge group to vertex groups ac-
cording to adjacency.
A graph of spaces : This is a space associate to a graph
of groups : Assign to each vertex of Γ with G a K(G, 1) com-
plex, to each eege of Γ with G a K(G, 1)×I , then attach them
according to morphisms.
Their fundamental group : π1(G) will be the fundamen-
tal group of a graph of spaces associated to G.
Classical nontrivial examples : A free product with
amalgamation A ∗C B, an HNN extension A∗C .
Remark : Bass and Serre theory says that these groups are
characterized by the existence of the action on a simplicial tree
without global fixed points.
• Example : Any RAAG is a fundamental group of a graph
of groups.
Proof : Vertices are maximal cliques and we assign Zn where
n is the number of vertices of the clique.
The intersection of two maximal clique must be a connected
clique. Two vertices are joined by an edge if the associated
maximal cliques have a common subclique and we assign Zm
where m is the number of vertices of the intersection.
• Lemma : If G is a fundamental group of a graph of groups
with two independent cycles, then G is large.
Proof : G has a surjection to F2.
• Proposition : If G is a fundamental group of a graph of
groups with edges such that an edge group is separable in G,
then G is either large, virtually π1 of a bundle over the circle
or a Baumslag-Solitar like group (We guess that the last case
does not occur !).
Proof :
– Reduction to the case either G = I ∗H J or I∗H where H
is separable.
– In the amalgamated product case, choose ϕ : G → F
to a finite group F such that [ϕ(I) : ϕ(H)] ≥ 2 and
[ϕ(J) : ϕ(H)] ≥ 2.
Proof : Choose a finite index subgroup K < G so that
G−K contains elements of I and J .
Then find a finite index normal subgroup L of G contained
in K, and let ϕ : G→ G/L = F .
– Let n = |F |, i = |ϕ(I)|, j = |ϕ(J)|, h = |ϕ(H)|. Then
h < i ≤ n, h|i|n and h < j ≤ n, h|j|n.
– Take a covering spaces I , J , H ofK(I, 1), K(J, 1), K(H, 1)
associated to Kerϕ∩I, Kerϕ∩J, Kerϕ∩H respectively.
Then the covering space of K(G, 1) associated to Kerϕ,
consists of n/i copies of I , n/j copies of J and n/h copies
of H . If we collapse I and J into vertices and H to edges,
then the degree of vertices corresponding to I is i/h and
that of vertices corresponding to J is j/h. Since the cov-
ering space in question is connected, it is easy to find two
independent cycles unless i/h = 2 = j/h. In this particu-
lar case, the graph is a cycle.
– The exceptional case reduces to the HNN extension case.
– When [I : H ] ≥ 2 in both direction, similar argument
works.
– When [I : H ] = 1 in both direction, G is a fundamental
group of a bundle over the circle.
– The remaining case is when [I : H ] = 1 for one direction.
A typical example of the remaining case is the classical
Baumslag-Solitar group,
B(1, 2) = 〈a, b | aba−1 = b2〉,where b generates an edge group.
Remark : B(1, 2) cannot be a fundamental group of a
compact manifold since it contains a divisible element
(a−nban)2n
= (a−nb2an)2n−1
= (a−naba−1an)2n−1
= (a−n+1ban−1)2n−1
· · ·= a−1b2a
= b.
Thus B(1, 2) contains a group isomorphic to Z[12]. On the
other hand, Z is not separable in Z[12]. This implies at
least that 〈b〉 < B(1, 2) is not separbale.
4.4 Residually finite rationally solvable
• Definition : G is residually finite Q-solvable (RFRS) if fur-
ther more, we require
1. G B Gi for all i ≥ 1,
2. Gi+1 > Ker(Gi → H1(Gi; Q)).
Remark :
1. Suppose that there a descending sequence having the sec-
ond property above, construct a sequence having the first
property.
2. Show that every subgroup of a RFRS group is RFRS.
• Lemma : If G is nontrivial and RFRS, then G is torsion free.
Proof :
– Every torsion in G = G0 dies in H1(G0 : Q).
– G1 contains all torsion of G0.
– Gi contains all torsion of G for any i > 0.
– So does ∩iGi, but the intersection is trivial.
• Lemma : If G is RFRS and rankH1(Gi) is bounded, then
G is virtually abelian.
Proof : Since rankH1(Gi) is non-decreasing, may assume
that rankH1(Gi) = rankH1(G) for all i.
– H1(G1)/Tor→ H1(G0)/Tor is injective.
– g ∈ U0 = Ker(G0 → H1(G0)/Tor) ⇒ g ∈ G1 by defini-
tion and also g ∈ U2 = Ker(G1 → H1(G1)/Tor · · · .– U0 < U1 < U2 < · · · .– Then U0 = 1 since the intersection is trivial.
– Thus G is abelian.
• Corollary : If G is RFRS and not virtually abelian, then
the virtual Betti number of G is ∞.
Question : Is such G large ?
4.5 Theorems by Agol
• Theorem ([1]) : A right angled Coxeter group is virtually
RFRS.
Proof : G acts on Tits cone X and choose H < G so that
X/H is a manifold.
– Choose an index 2 sequence G = D1 > D2 > · · · .– Set Gi = Di ∩H for i ≥ 1.
– G = G0 > G1 > G2 > · · · ,then [G : G1] <∞ and Gi/Gi+1 ' 0, or Z/2Z.
– Show that g ∈ Gi − Gi+1 is not torsion in H1(Gi) by
getting contradiction!
– Choose a representative ` of g in ⊂ X/Gi.
– The image of ` in X/Di has odd intersections with the
wall Fi because otherwise, ` lifts to X/Di+1 and hence g
is in Gi+1.
– Thus ` intersects Fi odd times.
– This means that Fi is non-separating and ` is not homo-
logically torsion.
• Theorem (Fibering Criteria by Agol [1]) ; Let N be
a closed irreducible 3-manifold. If π1(N) is RFRS, then there
is a finite cover of N which fibers over the circle.
5 Special Group
5.1 NPC cube complex
• A cube complex is a cell complex consisting of metric cubes
pasted by isometries.
There is an orbifold version.
• A connected finite dimensional cube complex with cubic met-
ric is a geodesic space (Bridson) ;
a metric space in which any two points are joined by a shortest
path.
We will assume finite dimensionality throughout the discus-
sion.
• CAT(k) space ; a geodesic space where we can compare a dis-
tance of any two points on a geodesic triangle with the triangle
with same three sides in the space of constant curvature k.
• Theorem (Gromov [8]) : A cube complex is locally CAT(0)
if and only if a link of each vertex is a flag complex.
If a cube complex is simply connected and a link of each vertex
is a flag complex, then it is CAT(0).
• A cube complex is said to be nonpositively curved (NPC) if a
link of each vertex is a flag complex.
• Selvetti complex of a RAAG is NPC.
• Subcomplex of a NPC cube complex may not be NPC.
For example, a cube is NPC but its boundary is not.
• Proposition (Lemma 2.5 in [10]) : Let Q and P be
NPC cube complexes and f : Q(2) → P a cellular map. Then
f admits a unique extension.
Idea of proof : By asphericity, there always be a continuous
extension. Suppose we have a 3-cube in Q. If its bounday
degenerates by f , then the extension is obvious. If not, then
since P is NPC, there must be a 3-cube bounded by the image
of the boundary of the 3-cube by f . The rest is by induction
on dimensions.
5.2 Hyperplane
• Midcube ; codimension one hypercube in the middle
Hyperplane ; a connected maximal extension of a midcube in
a cube complex
Hyperplane is a cube complex again.
• Self intersection of a hyperplane is, by definition, an intersec-
tion in interior of midcubes.
Hyperplane in a NPC cube complex may have self intersection
and its normal line bundle may be nonorientable.
• Theorem (Sageev [13] ?) : Hyperplane W in a simply
connected NPC (= CAT(0)) cube complex X is again CAT(0)
without self intersection, and the normal line bundle is ori-
entable.
Moreover, W is convex in X and separates X into two com-
ponents.
5.3 Local isometry
• Combinatorial definition of local isometry for NPC cube com-
plexes ; A cellular map f : Q → P between NPC cube com-
plexes is local isometry if
1. the induced map f∗ : Lk(v)→ Lk(f (v)) between links for
each vertex v ∈ Q is injective,
2. f∗(Lk(v)) is a full subcomplex of Lk(f (v)) (meaning, two
adjacent vertices in f (Lk(v)) ⊂ Lk(f (v)) must be adjacent
in Lk(v)).
• Nonexample : Any embedding of a connected cube complex
by two 1-cubes into a 1-cube never be a local isometry.
• Why care for only adjacent vertices is enough because any link
of an NPC complex is flag.
• Lemma : A local isometry f : Q → P between NPC cube
complexes Q,P has the following property :
1. f# : π1(Q)→ π1(P ) is injective.
2. A lift f : Q → P in the universal cover is an isometric
embedding.
Proof : Since a cellular homotopy from a loop to the identity
in P can be lifted to a homotopy on Q by the second property
of a local isometry.
5.4 Special NPC cube complex
• Definition : A NPC cube complex P is special if it satisfies
1. Each hyperplane has no self intersection.
2. Each hyperplane is co-orientable.
3. There is no direct osculation (show figure !).
4. There is no inter osculation (show figure !).
• Example : Selvetti complexes are spacial.
• Suppose f : Q → P is a local isometry between NPC cube
complexes.
If P is special, then Q is special.
In particular, any covering of P is special.
• Theorem (Haglund and Wise [10]) : A NPC cube
complex P is special if and only if there is a local isometry
f : P → SΓ for some simple graph Γ.
Remark : If P is not compact, then Γ will be a graph with
infinitely many vertices, and SΓ will be not locally finite. We
will allow to have such objects here.
Proof : If part is easy since Selvetti complexes are special.
Only if part will be verified by four steps ;
1. Construct Γ so that vertices are hyperplanes and two of
them are joined by an edge if they intersect.
Then, Γ has no loop since each hyperplane has no self
intersection (1), and no multiple edge by definition.
2. Construct a cellular map f : P (1) → SΓ so that each edge
intersecting a hyperplaneQ is mapped to a circle in SΓ cor-
responding to Q. This is possible since Q is co-orientable
(2).
Since two midcubes in a 2-cube c are contained in differ-
ent hyperplanes by the property (1), by mapping c to a
corresponding torus in SΓ, twe obtain a natural extension
f : P (2) → SΓ.
Do the same process for k midcubes in a k-cube for k =
3, 4, . . . , dimP , we obtain f : P → SΓ.
3. If f∗ : Lk(v)→ Lk(f(v)) is not injective for some v ∈ P (0),
then two oriented edges starting at v intersect a common
hyperplane in P , which produces a direct osculation. This
contradicts the property (3).
4. If f∗(Lk(v)) is not a full subcomplex of Lk(f(v)) for some
v ∈ P (0), then two oriented edges starting at v intersect
different hyperplanes with a common intersection in P ,
which produces an inter osculation. This contradicts the
property (4).
5.5 Special group
• Definition : A group G is cubulated if G acts on a CAT(0)
cube complex X properly and isometrically.
We may say that G is cubulated on X .
We may also say, G is cocompactly cubulated on X if X/G is
compact.
If G is torsion free, then the action is free and X/G is a NPC
cube complex.
If G contains torsion, then X/G is a NPC cube orbi-complex.
• Definition : A group G is spacial if it is cubulated on a
special CAT(0) cube complex X .
A group G is cocompact spacial if one can choose a special X
on which G acts so that X/G is compact.
• Examples :
– Any RAAG is cocompact special. In particular, Zn and
Fn are cocompact special.
– A closed surface group π1(Σg) is cocompact special because
it can be cubulated as a dual to its pentagonal decompo-
sition.
– 3-manifold groups are good candidates for cocompact spe-
cial groups !
• Corollary to HW theorem : G is special if and only if
G is a subgroup of some RAAG.
Proof : Only if part is a direct corollary of HW theorem.
To see if part, suppose G is a subgroup of a RAAG with asso-
ciated graph Γ. Then the covering of SΓ associated to G is a
special NPC cube complex whose π1 is isomorphic to G.
• Lemma : G is cocompact special if G is a subgroup of finite
index of some finitely generated RAAG.
Proof : Suppose G is a subgroup of finite index in a RAAG
with associated finite graph Γ. Then the covering of SΓ asso-
ciated to G is a compact special NPC cube complex whose π1
is isomorphic to G.
5.6 Canonical completion
• Construction (Section 3 in [11]) : Let Q,P be special
cube complexes such thatQ is compact, and f : Q→ P a local
isometry. The canonical completion C(Q,P ) is a canonically
defined finite covering of P such that there is an embedding
lift f : Q→ C(Q,P ) and a retraction r : C(Q,P )→ Q such
that r f = id. There are many steps :
1. When P is a Selvetti complex ;
– P (1) is a bouquet of circles.
– Canonical construction for 1-skeleton. When the path
with same symbol defines a covering already, we do
nothing. Otherwise, add a return path to each maxi-
mal path with same symbol, and loops to creat covering
with P (1) embedded,
Q(1) ← C(Q(1), P (1))→ P (1).
– Retraction is defined by sending an adding edge to an
expanded return path.
– Any boundary loop of a 2-cube in P lifts to a closed
loop in C(Q(1), P (1)) because since f is a local isometry
and hence eitherQ contains a corresponding cube, orQ
contains no corresponding cube but C(Q(1), P (1)) has
an edge with two loops at vertices.
– f on Q(2) can be lifted since f is a local isometry,
Q(2) ← C(Q(2), P (2))→ P (2).
– Done.
2. For general P ;
– Choose a Selvetti complex S = SΓ into which there is
a local isometry of P .
– Apply the first step to the composition Q → P → Sand we obtain a finite covering C(Q,S)→ S .
– Denote by C(Q,P ) a fiber product of C(Q,P ) → Sover P → S .
– Since the local isometry Q→ P and the inclusion map
Q → C(Q,S) coincide on S, they define an inclusion
Q→ C(Q,P ).
– Canonical retraction is defined to be the composition
C(Q,P )→ C(P,S)→ Q.
• Exercise : Find a canonical completion of an inclusion of the
circle to a holed torus.
• Corollary : Let G be cocompact special on X , and H < G
a subgroup of G which stabilizes a convex subcomplex Y ⊂ X
or a hyperplane W ⊂ X . Then, H is separable in G.
Proof : SinceG is cocompact special,G is a subgroup of some
RAAG and in particular residually finite. Let P = X/G and
Q = Y/H or Q = W/H . When Q = W/H , we subdivide P
so that Q = W/H is a subcomplex. Then we have a canonical
completion C(Q, P ), and the retraction of the space defines
a retraction : π1(C(Q, P )) → H . Then apply Proposition in
the previous section.
• Cocompact case of Theorem 14.10 in [16] : If G is
cocompact special on X , then G is either virtually abelian or
large.
Proof : Let us prove this by induction on the dimension of
P = X/G.
– When dimP = 1, a connected component of P is a bou-
quet of circles,
– Choose a hyperplane Q ⊂ P .
Q is again a compact special cube complex.
Thus we may assume that π1(Q) = H is either large of
virtually abelian.
Subdivide P so that Q is a subcomplex and the inclusion
Q ⊂ P is local isometry.
– In the former case, there is a surjection π1(C(Q,P )) →π1(Q) = H and [π1(P ) : π1(C(Q,P ))] <∞, hence π1(P )
is large.
– In the latter case, G is virtually π1(Q) o Z.
If the HNN extension is a fiber case, then since the isometry
group is finite, it is virtually abelian again.
The HNN extension of Baumslag-Solitar type cannot occur
in this situation.
The rest case provides large G.
6 Hyperbolic Group
6.1 Brief review on hyperbolic geometry
• Poincare disk D = z ∈ C | |z| < 1 ; equipped with the
metric4 (dx2 + dy2)
(1− |z|2)2.
This is one model of the plane hyperbolic geometry which
contrasts with the Euclidean geometry.
• Hyperbolic feature :
1. Geodesics in D ; a part of a circle intersecting ∂D perpen-
dicularly.
2. (Asymptotically) Parallel lines ; geodesics meeting at ∂D.
3. Ultra parallel lines ; geodesics having common orthogonals.
4. Every triangles are uniformly thin ;
5. Geodesics rays with bounded Hausdorff distance are par-
allel ;
6.2 δ-hyperbolic space
• Gromov product ; For any x, y, z in a geodesic space (X, d),
let
(x · y)z =1
2(d(x, z) + d(y, z)− d(x, y)).
Tripod picture provides a better understanding.
It is easy to see that
(x · y)z ≤ d(z, xy),
where xy is a geodesic connecting x and y.
• Lemma : The followings are equivalent up to change of a
constant δ.
1. (Gromov product) ∃δ ≥ 0 such that
(x · z)w ≥ min(x · y)w, (y · z)w − δ
for any x, y, z, w ∈ X .
2. (Thin triangle) ∃δ ≥ 0 such that a geodesic xy is contained
in a δ-neighborhood of xz ∪ yz for any x, y, z ∈ X .
3. (Fine triangle) ∃δ ≥ 0 such that for any geodesic triangle
∆xyz = xy∪yz∪zx, diam(π−1(q)) is bounded by δ where
π is the projection onto an associated tripod and q is any
point on the tripod.
Proof : Let cz ∈ xy, cx ∈ yz, cy ∈ zx be three points in the
preimage of the tripod center.
2 ⇒ 3)
– If ∆xyz is δ-thin, then two of d(cy, cz), d(cz, cx), d(cx, cy)
are < 2δ.
– Choose u, v ∈ π−1(q) and apply 2 to the triangle xcycz.
∗ If u is δ-close to xcz, then d(u, v) < 2δ.
∗ If v is δ-close to xcy, the same estimate holds.
∗ If u, v both are δ-close to cycz, then d(u, v) < 6δ.
3 ⇒ 1)
–
1 ⇒ 2)
–
• Definition : A geodesic space is hyperbolic if one of the
conditions in the above lemma satisfies.
Example : Tree, simply connected negatively curved space,
etc.
• Quasi-isometry : A (not necessarily continuos) map f :
X → Y between metric spaces X and Y is quasi-isometric if
there are constants A ≥ 1, B ≥ 0, C ≥ 0 such that
1
AdX(x, y)−B ≤ dY (f (x), f (y)) ≤ AdX(x, y) +B
and
N(f(X), C) ⊃ Y
Exercise : Quasi-isometry gives rise to an equivalence rela-
tion on metric spaces!
• Stability of Quasi-Geodesic : For any δ ≥ 0, A ≥1, B ≥ 0, there is a constant L = L(δ, A,B) such that if
X is a δ-hyperbolic space, c is a quasi-geodesic (the image of
an (A, B)-quasi-isometric embedding of a geodesic) in X and
[p, q] is a geodesic segment joining the endpoints of c, then the
Hausdorff distance between [p, q] and c is less than L.
Proof is rather technical and we omit it.
• Immediate Corollary : A geodesic spaceX is δ-hyperbolic
if and only if for every A ≥ 1 and B ≥ 0, there is a constant
M = M(δ, A,B) such that every (A,B)-quasi-geodesic trian-
gle in X is M -thin.
• Another Immediate Corollary : Let X, Y be geodesic
spaces and suppose that there is a quasi-isometric embedding
f : Y → X . If X is hyperbolic, then Y is hyperbolic.
Proof :
– Let ∆ be a triangle of three sides s1, s2, s3 in Y , then its
image by f is M -thin.
Thus for any x ∈ s1 there is y ∈ s2 ∪ s3 such that
d(f (x), f (y)) ≤M .
– Then, since f is (A, B)-quasi-isomery,
d(x, y) ≤ Ad(f (x), f (y)) + AB ≤ AM + AB,
and we are done.
• Corollary : Hyperbolicity is preserved by quasi-isometry.
6.3 Hyperbolic group
• Cayley graph : Let G be a group and S be a subset of
elements of G. The Cayley graph Γ(G) of G with respect to
S is a graph where
– V = G, and
– two vertices a, b are joined by an edge if there is g ∈ S
such that ag = b.
If S = ∅, then Γ(G) is a graph without edges.
If S = G, then Γ(G) is a complete graph with |V (G)| vertices.
In particular, it is not locally finite if G is an infinite group.
If we can choose S a finite generating set, then Γ(G) is con-
nected and locally finite.
• Definition : A finitely generated group G is hyperbolic if its
Cayley graph Γ(G) with respect to some finite generating set
with path metric is hyperbolic.
Remark :
1. Hyperbolicity does not depend on the choice of generating
sets.
2. Since G acts on the vertices of Γ(G) transitively, we can
choose the identity element e for w in the definition of
hyperbolicity through Gromov product.
Hence, we sometime omit w in the Gromov product (x ·y)to indicate (x · y)e.
• Examples and nonexamples :
1. Every finite group is hyperbolic by definition.
2. Fn is hyperbolic since we can choose Γ(Fn) by a tree.
3. Groups acting properly and cocomactly on hyperbolic space,
4. Z× Z is not hyperbolic.
5. Baumslag-Solitar group
B(n,m) = 〈a, b | abna−1 = bm〉
is not hyperbolic.
Exercise : Draw Γ(B(1, 2)) and show that it is not hy-
perbolic !
• Rips complex : Given a constant d > 0 and define the Rips
complex Pd(Γ(G)) as a simplicial complex over Γ(G) such that
a set σ of vertices in Γ(G) spans a simplex if diamσ ≤ d.
Example : Z = 〈a〉 and d = 1, 2, 3, . . . .
• Remark
1. Pd(Γ(G)) is locally finite.
2. G acts simplicially on Pd(Γ(G)) and freely, transitively on
vertices.
3. Pd(Γ(G))/G is compact.
4. If G is torsion free, then the action is also free.
• Proposition : If Γ(G) is “δ-fine”, then Pd(Γ(G)) is con-
tractible if d > 2δ + 2.
Proof : The steps are :
– Suffices to prove that every subcomplex K is contractible
in Pd(Γ(G)).
– Assume that K contains e and let y0 be the farthest vertex
in K.
– If d(e, y0) ≤ d/2, we are done since K is contained in a
simplex in Pd(Γ(G)) in this case.
– If d(e, y0) > d/2, then choose y1 on a geodesic ey0 such
that d(y0, y1) = [d/2] and will show that any y ∈ K(0) in
d-neighborhood of y0 is also in d-neighborhood of y1.
More concretely, look at a triangle span by e, y0 y and see
where y1 is in the tripod (see note).
– Then, homotopy sending y0 to y1 extends to a homotopy
on K.
– Thus done by induction.
• Corollary :
1. A hyperbolic group is finitely presented.
2. The number of conjugacy classes of finite subgroups is fi-
nite.
Question : Is a hyperbolic group virtually torsion free ?
3. If a hyperbolic group G is torsion free, then Pd(Γ(G))/G
is a compact K(G, 1) space. In particular, the cohomology
dimension of G is finite.
4. In general, H∗∗ (G : Q) is finite dimensional.
6.4 Boundary
• A sequence (xn) of elements of Γ(G) is said to converge to∞if limi,j→∞(xi · xj) =∞.
If xn →∞, then d(e, xn)→∞ since (xn · xn) = d(e, xn).
S∞ = (xn) ; xn →∞ and introduce a relation (xn) ∼ (yn)
if and only if limn→∞(xn · yn) =∞.
Remark : The relation ∼ is not transitive in general.
• Lemma : If G is hyperbolic, then ∼ is transitive.
Proof : Look at
(xn · zn) ≥ min(xn · yn), (yn · zn) − δ.
• Definition : ∂Γ(G) = S∞/ ∼ and Γ(G) = Γ(G) ∪ ∂Γ(G).
We topologies it later.
• Extension of Gromov product : If x = [(xn)], y = [(yn)]
converges in Γ(G), we define an extension of Gromov product
by
(x · y) = inflim infn
(xn · yn),
where inf is taken over all pairs of sequences xn → x, yn → y.
• Remark : Let G be a group presented by
〈a, b | b2 = 1, ab = ba〉 ' Z× Z/2Z.
Then ∂Γ(G) = −∞, ∞.Choose sequences xn = an, yn = a−n, zn = ban, wn = ba−n
and
vn =
an if n = 2m
ban if n = 2m + 1.
Observe that (wn · vn) is 0 if n is even and 1 otherwise. This
is why we take a limit infimum.
Also observe that (xn · yn) = 0 while (zn · wn) = 1. This is
why we take a infimum over all sequences.
• Lemma : We have the following properties.
1. (x · y) =∞⇔ x, y ∈ ∂Γ(G) and x = y.
2. For any x, y, z ∈ Γ(G), we have
(x · z) ≥ min(x · y), (y · z) − δ
Proof : See articles in Ghys’ proceedings.
• Topology on Γ(G) : Using Gromov product, we define a
basis of two types by
1. Br(x) = y ∈ Γ(G) | d(x, y) < r,
2. Nx,k = y ∈ Γ(G) | (x · y) > k.
6.5 Quasiconvex subgroup
• Definition : A subset S of a geodesic spaceX isK-quasiconvex
if for any geodesic γ in X whose end points lie in S, the K-
neighborhood of S contains γ.
• Lemma : Suppose G is a hyperbolic group. H < G is quasi-
convex if and only if Γ(0)(H) is quasi-isometrically embedded
in Γ(0)(G).
Proof : Since G acts transitively on vertices of Γ(G), it is
sufficient to discuss for geodesics from 1 to some element in G.
Only if part :
– Choose a geodesic path a1a2 . . . an connecting 1 and h ∈ Hin Γ(G). Then by K-quasiconvexity, there are hi ∈ H and
ui of word length ≤ K such that hi = ui−1aiu−1i .
– Remark : These hi’s yield a generating set of H where
its word length ≤ 2K + 1 in G and hence H is finitely
generated.
– Moreover, dH(e, h) with respect to this generating set is
at least n/(2K + 1) and at most n, while dG(e, h) = n.
Hence the embedding is quasi-isometric.
If part :
– Fix generating sets for G and H , and suppose there is a
quasi-isometric embedding φ : Γ(H)→ Γ(G).
– Choose a geodesic c in Γ(H) joining 1 and h ∈ H , and
consider the quasi-geodesic φ c in Γ(G).
– According to the stability of quasi-geodesics, this quasi-
geodesic is L-close to any geodesic joining 1 and h in Γ(G),
where L depends only on the hyperbolicity constant and
quasi-isometric constants for φ. Done !
• Corollary : A quasiconvex subgroup of a hyperbolic group
is hyperbolic.
Proof : Apply previous claims.
• Examples :
– quasi Fuchsian subgroup of a Kleinian group is quasicon-
vex.
– a fiber of a hyperbolic 3-manifold which fibers over the
circle is not quasiconvex, but a group itself is hyperbolic.
6.6 Special hyperbolic group
• Theorem (Corollary 7.4 and Theorem 8.13 in [10]) :
Let G be a hyperbolic group. G is cocompact virtually special
if and only if every quasiconvex subgroup of G is separable.
• Only if part is not quite hard !
– Suppose a subgroup G′ < G of finite index is cocompact
special on X , and let H be a quasiconvex subgroup of G′.
Then there is a compact NPC cube complex P and a local
isometry f : P → X/G′ which maps π1(P ) isomorphically
onto H < G′ (this is due to Haglund [9] and/or Sageev-
Wise [15]).
– Apply the Theorem in the previous section to conclude
separability of H .
– The rest is to see that separability of all quasiconvex sub-
group ofG′ implies separability of all quasiconvex subgroup
of G (Exercise).
• If part is quite technical.
7 Cubulation
7.1 Wallspace
• Definition : A wall W in a set X is a partition of X−W =←−W t
−→W .
A pair ofX with a set of walls (X,W) is said to be a wallspace
if the following two conditions are satisfied :
1. The number of walls separating p, q ∈ X is finite for all
p 6= q,
2. For each point p ∈ X , there are only finitely many walls
which separate the points which can be separated by walls.
• Obvious example :
– A finite set and a system of separation.
– (R,Z) is a wallspace in a obvious sense.
– A CAT(0) cube complex together with a set of hyperplanes
is a wallspace.
– Tiling of the hyperbolic space.
• Cubulation (dual construction) : A dual cube complex
W∗ to (X,W) is defined by
1. A 0-cube v is an orientation on each wall, namely v(W ) is
a choice of←−W or
−→W , with the following conditions,
(1) Two walls should not be oriented away from each other,
in other words, v(W ) ∩ v(W ′) 6= ∅ for all W,W ′.
(2) All but finitely many walls are oriented towards any
x ∈ X . In other wores, x 6∈ v(W ) for only finitely
many W .
2. 1-cube joins two 0-cubes if the difference is exactly one
wall.
3. Two vertices possible to change orientations yields four
edges and we attach a 2-cube to them.
4. n-cube is attached whenever there is n vertices possible to
change orientations (like a flag complex construction).
• Remark : Because of the conditions for a wallspace, there
are only finitely many choices of orientation change we can
make for each vertex of W∗. Hence W∗ is locally finite.
• Examples :
– If we start with a CAT(0) cube complex X with a wall
system W by hyperplanes, then X is isomorphic to W∗.– If the tiling of H2 has only normal crossings, then its cubu-
lation is a dual tiling.
See the pentagonal tiling.
– Think of the tiling of H2 by regular squares with angle π/4,
and choose the vertex ofW∗ where all arrow goes towards
the central square.
this vertex shares four 4-dimensional cube C1, C2, C3, C4
where Ci and Ci+1 share 1-dimensional cube in common
where suffix runs cyclically.
• Theorem (Sageev [13]) : W∗ is CAT(0).
Proof : Check that it is connected, simply-connencted and
NPC.
• Exercise : Find aW∗ for a pentagonal tiling of the hyper-
bolic plane.
7.2 Group action
• Geometric Construction :
1. Start with a closed aspherical manifold P .
2. Choose an essential immersion f : Q→ P , such that each
component of P − f(Q) is contractible.
3. Take the universal cover π : X → P .
4. Each component of π−1(f (Q)) will be a wall and and we
obtain a wallspace (X,W).
5. π1(P ) acts on (X,W).
6. π1(P ) acts on its cubulation W∗.
• A subgroup H < G is codimension 1 if Γ(G)/H has at least
two ends.
• Algebraic Construction :
1. Start with a group G and its Cayley graph Γ(G).
2. A wall systemW would consist of (left) cosets of codimen-
sion 1 subgroups H1, H2, . . . , Hn < G.
3. (Γ(G),W) will be a wall space.
4. G acts on (Γ(G),W) from left.
5. G acts on its cubulation W∗.
• Examples :
1. Z ' H = 〈a〉 < G ' π1(Σg), where H is generated by
a simple closed curve, is codimension 1 and Γ(G)/H has
two ends. H does not define a wallspace since it does not
satisfy the second property of the wallspace.
2. Z ' H = 〈a〉 < 〈a, b | ab = ba〉 = G ' Z2 is codimen-
sion 1 and Γ(G)/H has two ends. H defines a wallspace
W = bnH |n ∈ Z. W∗ is of 1 dimensional and W∗/Gis compact but the action is not proper.
3. Z ' H = 〈a〉 < G ' π1(Σg), where H is generated by a
filling closed curve, is codimension 1 and Γ(G)/H has two
ends. H does define a wallspaceW∗ because of the picture
in the universal cover.
Exercise : The action of G onW∗ is proper and cocom-
pact.
• Theorem (Sageev [14]) : LetG be a hyperbolic group and
H1, H2, . . . , Hn < G a collection of codimension 1 quasiconvex
subgroups which defines a wallspace. Then the action of G on
the dual cube complex of (Γ(G),W) is cocompact.
Remark : Theorem does not guarantee the properness of the
action.
• Theorem (Bergeron and Wise [3]) : Let G be a hy-
perbolic group. Suppose that for each pair of distinct points
(u, v) ∈ ∂G2 there exists a quasiconvex codimension 1 sub-
group H such that u and v lie in distinct components of
∂G−∂H . Then there is a finite collection H1, . . . , Hn of qua-
siconvex codimension 1 subgroups which defines a wallspace
(Γ(G),W) such that G acts properly and cocompactly onW∗.
• The properness criterion they use is that the number of walls
to separate 1 and g goes ∞ it the distance between 1 and g
goes ∞ in Γ(G).
8 Combination
8.1 History
• Haken’s hierarchy (1962) ;
Cut along essential surfaces,
N 3 = N0 ⊃ N1 ⊃ · · · ⊃∐
B3.
• The fundamental group of Haken 3-manifolds ;
1. The trivial group is the initial group, which is the funda-
mental group of a ball.
2. If a connected essential surface S separates Ni so that
Ni+1 = A∪B, then the fundamental group is an amalga-
mated free product π1(Ni) ' π1(A) ∗π1(S) π1(B).
3. If a connected essential surface S does not separate Ni,
then the fundamental group is an HNN extension π1(Ni) 'π1(Ni+1)∗π1(S).
4. Thus, it will be an amalgamated free product of two groups,
or an HNN extension according to the geometric splitting.
• Waldhausen’s work (1968) ;
Homotopy equivalence implies homeomorphic for sufficiently
large 3-manifolds.
• Original virtual Haken conjecture (?) ;
Any compact 3-manifold with infinite π1 has a finite Haken
cover.
• Thurston’s geometrization conjecture (1980) ;
Any compact 3-manifold admix a canonical decomposition by
tori so that the resulting piece is geometric.
• Perelman’s solution (2003)
• Reduction to the hyperbolic case ;
Perelman (2003) and Luecke, Kojima (1987).
8.2 Hyperbolic groups along quasiconvex subgroups
• Let G = (V , E) be a graph of groups, and to each v ∈ V and
e ∈ E , we assign groups Gv, Ge.
Also if v = ι(e) is the initial vertex, assign an monomorphism
fe : Ge → Gv.
• Definition : An annulus of length 2m consists of
1. an edge-path e−me−m+1 . . . e0 . . . em in G,
2. a sequence
ε−m, ν−m, ε−m+1, ν−m+1, . . . , εm−1, νm−1, εm
with εi ∈ Gei and νi ∈ Gι(ei+1), and
νifei(εi)ν−1i = fei+1(εi+1), i = −m, −m + 1, . . . ,m− 1.
See the picture !
• Definitions : An annulus is essential if whenever ei+1 = ei,
then νi 6∈ Im fei+1.
An annulus is ρ-thin if for each i, ||νi|| ≤ ρ.
The girth of an annulus if ||ε0||.An annulus is λ-hyperbolic if max||ε−m||, ||εm|| ≥ λ||ε0||.G satisfies a annuli flare condtion if there are numbers λ >
1, m ≥ 1 such that for all ρ, there is a constant H = H(ρ)
such that any ρ-thin essential annulus of length 2m and girth
at least H is λ-hyperbolic.
• Possible long annulus : Z×F where F is a finite group
is a hyperbolic group with infinite long annuli.
• Impossible long annulus : Z × Z is not hypperbolic,
and in fact, it does not satisfy the annulus flare condition.
• Theorem (Bestvina-Feighn [4]) : Let G be a finite graph
of hyperbolic groups such that each edge groups is quasicon-
vex in adjacent vertex groups. If G satisfies an annuli flare
condition, then π1(G) is hyperbolic.
•Weak Corollary : Let N be a compact 3-manifold with
a connected π1-injective surface S, and N ′ a manifold ob-
tained by cutting N along S. Suppose π1(N′) is hyperbolic,
and π1(S) is quasiconvex in π1(N′). If N is homotopicallly
atoroidal (i.e.,Z× Z 6< π1(N)), then π1(N) is hyperbolic.
• Another weak corollary : Free products and HNN ex-
tensions of hyperbolic groups with virtually cyclic hyperbolic
group are hyperbolic if and only if the resulting groups contain
no Baumslag-Solitar groups.
• Coclusion : Combination gives rise to a hyperbolic group
generically.
8.3 QVH (quasiconvex virtual hierarchy)
• Definition : QVH is the smallest family of hyperbolic
groups generated by the following operations.
1. (The starting group) 1 ∈ QVH.
2. (HNN extension) If A ∈ QVH, C is quasiconvex both in
the images of C → A and G = A∗C is hyperbolic, then
G ∈ QVH3. (Amalgam) If A, B ∈ QVH and C is quasiconvex both
in A, B and G = A ∗C B is hyperbolic, then G ∈ QVH4. (Finite extension) If H ∈ QVH, H < G and [G : H ] <
∞, then G ∈ QVH.
• Examples :
1. All finite groups are in QVH.
2. Fn ∈ QVH for n ≥ 1.
3. All surface groups except π1(T2) ' Z× Z are in QVH.
4. All closed Haken hyperbolic 3-manifold groups are inQVHby Thurston, and all closed hyperbolic 3-manifold groups
will be in QVH by Agol and Wise.
5. A lots of hyperbolic Coxeter groups are in QVH.
• Question : Find a fundamental group of a hyperbolic n-
manifold which is not contained in QVH ?
• Theorem (Wise [16] and Agol-Glove-Manning [2])
: A group G is in QVH if and only if G is a cocompact
virtually special hyperbolic group.
9 Kleinian groups
9.1 Hyperbolic 3-manifolds
• A Fuchsian group Π0 is a discrete torsion free subgroup of
PSL(2,R).
Then H2/Π0 is a hyperbolic surface.
• A Kleinian group Λ is a discrete torsion free subgroup of
PSL(2,C).
Then H3/Λ is a hyperbolic 3-manifold.
A Fuchsian group is regarded to be embedded in PSL(2,R) ⊂PSL(2,C), and in particular it is a Kleinian group.
• Limit set LΛ ;
The set of accumulation points of the action of Λ on the sphere
at infinity.
The limit set of a Fuchsian group is a great circle.
• The convex full CΛ of LΛ in H3 supports the topology of H3/Λ.
In fact, CΛ/Λ injects to H3 as a homotopy equivalence.
• A quasi-Fuchsian group Π is, by definition, a Kleinian group
whose limit set is a quasi circle.
It is known that.
1. Π is isomorphic to a fundamental group of a surface Σ.
2. H3/Π is homeomorphic to Σ × (0, 1) and CΠ is homeo-
morphic to Σ × [0, 1].
3. If Π is contained in a cocompact Kleinian group Λ, then Π
is quasiconvex in Λ.
• Remark : There is a hyperbolic 3-manifold H3/Λ which
fibers over the circle. The fundamental group of the fiber is
not quasiconvex in Λ.
9.2 Surface subgroups of Kleinian groups
• Theorem (Kahn-Markovick [12]) : Let Λ be a cocom-
pact Kleinian group and S1 ⊂ S2∞ a great circle. Then there
is a quasi-Fuchsian subgroup Π < Λ whose limit set LΠ is
arbitrary close to S1.
Proof : Will be done in the last two lectures by Masai-kun.
• Corollary : A cocompact Kleinian group can be cocom-
pactly cubulated.
Proof : Apply Bergeron-Wise [3] and the theorem above.
10 Cubulated hyperbolic groups (Simplified current
version)
10.1 Main Result
• Theorem (Theorem 1.1 in Agol [2]) : A cocompactly
cubulated hyperbolic group is in QVH. In particular, it it
virtually special.
• Corollary : A cocompactly cubulated hyperbolic group is
either virtually abelian or large. In particular, a fundamental
group of a hyperbolic 3-manifold is large.
• Setting for the proof :
G ; a hyperbolic group cocompactly cubulated on X .
P = X/G has only finitely many hyperplanes.
The preimage of hyperplanes define a wall system of X .
There are finitely many orbits of walls W ⊂ X .
The stabilizer GW is quasiconvex in G.
• Two big steps of the proof :
1. Step 1 : Choose an infinite regular cover Q of P so that
each hyperplane in P is covered by compact embedded
hyperplanes in Q.
2. Step 2 : Q has a finite hierarchy, and we construct a
finite cover of P as a quotient of Q so that it admits a
hierarchy modeled on the hierarchy of Q.
10.2 Comments to Step 1
• Need a hard theorem !
Theorem A.1 in Agol-Glove-Manning [2] : G ; a
hyperbolic group, H < G ; a quasiconvex virtually special
subgroup. For any g ∈ G − H , there is a homomorphism
φ : G→ J onto a hyperbolic group J such that
1. φ(g) 6∈ φ(H),
2. φ(H) is finite.
• Applying the above with several technicalities, we may con-
clude
Lemma : ∃ a surjective homo φ : G→ J such that if we let
Q = X/Kerφ, then
1. Q is a NPC cube complex.
2. Q admits a wall system U so that each wall U ∈ U is
compact, co-oriented.
3. NL(W )/(GW ∩ Kerφ) embeds in Q under the natural
covering map, where L > 0 is a constant such that if
d(W,W ′) > R, then |GW ∩GW ′| <∞.
4. X(1)/Kerφ has no loops.
10.3 Model hierarchy
• Definition : Γ(Q) :
– Vertices are wall components of Q, namely V (Γ(Q)) = U .
– U, U ′ ∈ U are joined if dQ(U,U ′) ≤ L.
– J = G/Kerφ acts on Γ(Q).
• Remark : k = max degree of Γ(Q) is finite.
Proof : Because each wall is compact.
• Corollary : Q admits a hierarchy of length k + 1.
Proof :
– Γ(Q) can be colored by k + 1 colores.
– Cut Q along the walls with the same color in order, and
we obtain a hierarchy
Q ⊃ Q1 ⊃ Q2 ⊃ · · · ⊃ Qk+1.
– The complement of ∪U∈UU , that is Qk+1, consists of con-
tractible pieces which will be called cubical polyhedra.
– A cubical polyhedra inherits a NPC cube complex struc-
ture with a distinguished vertex by cubical barycentric sub-
division of Q.
• We want of find a finite cover of P with a hierarchy modeled
on the hierarchy of Q.
10.4 Invariant Coloring Measure
• n-coloring : The map c : V (Γ) → 1, . . . , n such that if
u, v ∈ E(Γ), then c(u) 6= c(v).
Cn(Γ) ⊂ 1, . . . , nV (Γ) will be the set of n colorings on Γ.
• 1, . . . , nV (Γ) with product topology becomes a Cantor set.
• M(Ω) ; a space of probability measures on Ω,
Since J acts on 1, · · · , nV (Γ), we have
MJ(Cn(Γ)) ⊂M(Cn(Γ)) ⊂M(1, . . . , nV (Γ)).
• Theorem 5.2 in [2] :
If max degree of Γ is k, then MJ(Ck+1(Γ)) 6= ∅.In particular, J -invariant measure on the space of k+1-colorings
exists.
Proof : Refer to [2].
10.5 Gluing equation
• The last stage Qk+1 of the hierarchy consists of cubical poly-
hedra and each facet is included in some hyperplane.
• Equivalence relation : on V × Ck+1(Γ).
1. (v, c) ∼ (w, d) only if c(v) = d(v) and c(v) = d(w).
2. (v, c) ∼ (v, d) if c(v) = 1 = d(v).
3. (v, c) ∼ (v, d) if c(v) = j = d(v), 2 ≤ j ≤ k + 1 and
moreover for each neighbor vertex w ∈ V , (w, c) ∼ (w, d)
if c(w) < j or d(w) < j.
Remark : If c(v) = j, then the equivalence class containing
(v, c) depends only on the values of c on the ball of radius j−1
about v.
Thus an equivalence class in v×Ck+1(Γ) becomes a clopen
set.
• Extension of ∼ to : F × Ck+1(Γ) and C × Ck+1(Γ) in
obvious way, where
C ; the set of cubical polyhedra in Q and
F ; the set of their facets in Q, namely each facet faces exactly
two cubical polyhedra.
• Action of J : For g ∈ J ,
g(v, c) = (g · v, c g−1)
on V × Ck+1(Γ)
and similarly on F × Ck+1(Γ) and P × Ck+1(Γ).
Remark : There are only finitely many J -orbits of equiva-
lence classes.
• Suppose we have a finite cover of P = X/G which is a quotient
of Q,
then it will consists of copies of cubical polyhedra.
• Fix F ∈ F ,
then there are two cubical polyhedra C,C ′ ∈ C which share
F as a facet.
• Fixing a color c ∈ Ck+1(Γ) and
look at the preimage of (F, c) in a finite cover.
Each component share (C, d) and (C ′, d′) for some colors d, d′.
If ω([(C, d)]) denotes the number of copies which belong to
[(C, d)] in the cover, the gluing identity∑[(C,d)]s.t.(F,d)∼(F,c)
ω([(C, d)]) =∑
[(C ′,d)]s.t.(F,d)∼(F,c)
ω([(C ′, d)]),
must be established.
• Gluing eqation : Varying F ∈ F and c ∈ Ck+1(Γ), we
get a system of equations on nonnegative integral valued J -
invariant weight functions
ω : C × Ck+1(Γ)/ ∼ → Z≥0.
• Real valued solution exists : Choose µ ∈MJ(Ck+1(Γ)),
then by additivity,∑[(C,d)]s.t.(F,d)∼(F,c)
µ([(C, d)]) = µ(d|(F, d) ∼ (F, c))
= µ([(F, c)])
=∑
[(C ′,d)]s.t.(F,d)∼(F,c)
µ([(C ′, d)]).
• Integral solution ω0 exists : beause the glueing equation
is a system of integral linear equations (Exercise).
10.6 Virtual gluing
• Hierarchy ; based on the hierarchy of Q, we construct a
hierarchy
Rk+1 ⊂ Rk ⊂ · · · ⊂ R1 ⊂ R0
using an integral solution ω0 as follows :
1. Rk+1 consists of ω0(C, c) copies of [(C, c)] where C runs
over all orbit representatives of C.2. Rk is obtained by pasting members of Rk+1 along faces
colored by k + 1 according to the rule determined by ω0.
3. Do similar steps, however the union of facets may produce
nontrivial topology. Thus to get Rj−1, we might need to
take cover Rj of Rj. This step will be guaranteed by the
next theorem.
Remark : Even, we can reach to a hierarchy by taking
the composition of all covers we needed.
• Theorem 3.1 in Agol [2] : Let R be a compact virtu-
ally special NPC cube complex such that π1(R) is hyperbolic.
Let ∂R ⊂ R be an embedded locally convex acylindrical sub-
complex (which does not mean the boundary) which covers an
NPC cube orbi-complex π : ∂R→ ∂R0.
Then there exists a finite regular cover R → R such that the
preimage of E ⊃ ∂R→ ∂R is a regular orbi-cover ∂R→ ∂R0
(Draw a diagram).
• Proof :
– May assume that R is special already.
– To find a regular cover of ∂R0, glue R and a mapping cone
Cπ of π : ∂R→ ∂R0 along ∂R to get R′ = R ∪∂R Cπ.– Since Cπ is acylindrical, π1(R
′) is hyperbolic by Bestvina-
Feighn and hence π1(R′) ∈ QVH.
– Thus π1(R′) is virtually special by Wise.
– Assume π1(R′) is already special without loss of generality.
– Choose a finite regular cover : Cπ → Cπ.
– We have a local isometry by the composition : Cπ →Cπ → P ′ and hence, there is a finite cover C(Cπ, R
′)→ R′
with an embedding Cπ → C(Cπ, R′) and a contraction
π1(C(Cπ, R′))→ π1(Q).
– Tak a further regular cover of R through R′ will imply the
desired covering.
11 Cubulated hyperbolic groups (Detailed version)
11.1 Main Result
• Theorem (Theorem 1.1 in Agol [2]) : A cocompactly
cubulated hyperbolic group is in QVH. In particular, it it
virtually special.
• Corollary : A cocompactly cubulated hyperbolic group is
either virtually abelian or large. In particular, a fundamental
group of a hyperbolic 3-manifold is large.
• Setting for the proof : A hyperbolic group G is cocom-
pactly cubulated on X .
P = X/G has a wall system by finitely many compact hyper-
planes.
There are finitely many orbits of walls W ⊂ X .
The stabilizer GW of W is quasi isometric to W , and is qua-
siconvex in G since W is convex (totally geodesic in fact).
• Outline of the proof :
1. Choose an infinite regular coverQ of P so that each hyper-
plane in P is covered by compact embedded hyperplanes
in Q.
2. Cutting Q along the hyperplanes provides the base case of
a hierarchy by an infinite number of cubical polyhedra (see
picture !).
3. Find a finite hierarchy of Q by labeling hyperplanes of Q
with finite colors.
4. Construct a finite cover of P so that it has a hierarchy
modeled on the hierarchy of Q.
There will be a technical argument to fit
11.2 Quotient Complex of Compact Walls
• Theorem (Theorem A.1 in Agol-Glove-Manning [2])
:
G ; a hyperbolic group,
H < G ; a quasiconvex virtually special subgroup.
For any g ∈ G−H ,
there is a homomorphism φ : G→ J onto a hyperbolic group
J such that
1. φ(g) 6∈ φ(H),
2. φ(H) is finite.
Proof : Very long and difficult !
• Setting
– A hyperbolic group G is cocompactly cubulated on X .
– P = X/G has a wall system by hyperplanes.
– It lifts to a wall system W in X .
– There are finitely many orbits of walls in X since P is
compact.
– The stabilizer GW of W ∈ W is quasi isometric to W .
– It is quasiconvex in G since W is convex (totally geodesic
in fact).
• Aim 1 : We want to construct a infinite cover Q→ P using
Theorem above such that each wall is covered by an embedded
compact wall in Q.
• Lemma : There exists R > 0 such that
if d(W,W ′) > R, then |GW ∩GW ′| <∞.
Proof : If |GW ∩ GW ′| = ∞, then the orbits on W and
W ′ are parallel. This could happen only within a bounded
distance because G is hyperbolic.
• Let W1, . . . ,Wm be the orbit representatives of the walls of
X under the action of G.
Induction on the maximal dimension of a cube :
GWiis virtually special for 1 ≤ i ≤ m,
• By the above lemma,
Ai = GWigGWi
| d(g(Wi),Wi) ≤ R − GWi
is a finite set for all i.
• Lemma 4.1 in [1] :
∃ a surjective homo φ : G→ J such that
for all 1 ≤ i ≤ m and for all GWigGWi
∈ Ai,φ(g) 6∈ φ(GWi
) and φ(GWj) is finite for all j.
Moreover,
the action of GWi∩ Kerφ preserves the co-orientation,
Kerφ is torsion fee
and X(1)/Kerφ contains no closed loops.
• Proof :
– Fix an elemant g such that GWigGWi
∈ Ai.– Choose elements g1, g2, . . . , gm such that gi = 1 and
H = 〈Gg1W1, . . . , Ggm
Wm〉 ' Gg1
W1∗ · · · ∗GWi
∗ · · · ∗GgmWm.
and g 6∈ H and H quasiconvex.
This can be done by ping-pong argument.
– H is virtually special since it is a free product of virtually
spacial groups.
– Apply Theorem to conclude that there is a surjective ho-
momorphism φg : G→ Jg such that φg(g) 6∈ H and φ(H)
finite.
Two remarks :
1. φg(GWj) is finite for all j.
2. May assume that Kerφg ∩GWiis contained in the sub-
group preserving the co-orientation.
– A : the finitely many double coset representatives for ∪iAiappeared in this construction.
– T ⊂ G : a finite set of representatives for each conjugacy
class of torsion elements in G such that T ∩ GWj= ∅
for all j, and for each conjugacy class of group elements
identifying endpoints of edges of X(1).
– Apply the same technique to obtain ψg : G → J ′g such
that ψg(g) 6= 1 and ψg(GWj) is finite for all j.
– Define φ to be a quotient map to
J = G/ ∩g∈A Kerφg ∩g∈T Kerψg
• Notation and Properties: Q = X/Kerφ.
1. Q is a NPC cube complex.
2. NR(Wj)/(GWj∩ Kerφ) embeds in Q under the natural
covering map.
3. Q admits a wall system U so that each wall is compact,
embedded, co-oriented.
• Definition : Γ(Q) :
Vertices are wall components of Q, namely V (Γ(Q)) = U .
U, U ′ ∈ U are joined if dQ(U,U ′) ≤ R.
J = G/Kerφ acts on Γ(Q).
• Remark : k = max degree of Γ(Q) is finite.
Proof : Because each wall is compact.
• Corollary : Q admits a hierarchy of length k + 1.
Proof :
– Since max degree of Γ(Q) = k, it can be colored by k + 1
colores.
– Cut Q along the walls with the same color in the reverse
order from k + 1.
– The complement of ∪U∈UU consists of contractible pieces.
– This piece inherits a NPC cube complex structure with a
distinguished vertex by cubical barycentric subdivision of
Q, and will be called a cubical polyhedron.
• We want of find a finite cover of P with a hierarchy modeled
on the hierarchy of Q.
11.3 Invariant Coloring Measure
• n-coloring : The map c : V (Γ) → 1, . . . , n such that if
u, v ∈ E(Γ), then c(u) 6= c(v).
Cn(Γ) ⊂ 1, . . . , nV (Γ) will be the set of n colorings on Γ.
• 1, . . . , nV (Γ) with product topology becomes a Cantor set.
• M(Ω) ; a space of probability measures on Ω,
If F acts on Ω,
then we have
MJ(Cn(Γ)) ⊂M(Cn(Γ)) ⊂M(1, . . . , nV (Γ)).
• Theorem 5.2 in [2] :
If max degree of Γ is k, then MJ(Ck+1(Γ)) 6= ∅.In particular, J -invariant measure on the space of k+1-colorings
exists.
Proof : Refer to [2].
11.4 Gluing equation
• Q \ ∪U∈U U consists of cubical polyhedra and each facet is
included in some wall.
• Equivalence relation : on V × Ck+1(Γ).
1. (v, c) ∼ (v, d) if c(v) = 1 = d(v).
2. (v, c) ∼ (v, d) if c(v) = j = d(v), 2 ≤ j ≤ k + 1 and
moreover for each neighbor vertex w ∈ V , c(w) = i < j if
and only if d(w) = i < j.
Remark : If c(v) = j, then the equivalence class containing
(v, c) depends only on the values of c on the ball of radius j−1
about v.
Thus an equivalence class in v×Ck+1(Γ) becomes an clopen
set.
• Extension of ∼ to : F × Ck+1(Γ) and P × Ck+1(Γ) in
obvious way, where
P ; the set of cubical polyhedra in Q and
F ; the set of their facets in Q, namely each facet faces exactly
two cubical polyhedra.
• Action of J : For g ∈ J ,
g(v, c) = (g · v, c g−1)
on V × Ck+1(Γ)
and similarly on F × Ck+1(Γ),
P × Ck+1(Γ).
Remark : There are only finitely many J -orbits of equiva-
lence classes.
• Suppose we have a finite cover of P = X/G which is a quotient
of Q,
then it will consists of copies of cubical polyhedra.
• Fix F ∈ F ,
then there are two cubical polyhedra P, P ′ ∈ P which share
F as a facet.
• Fixing a color c ∈ Ck+1(Γ) and
look at the preimage of (F, c) in a finite cover.
Each component share (P, d) and (P ′, d′) for some colors d, d′.
If ω([(P, d)]) denotes the number of copies of (P, d) in the
cover, the gluing identity∑[(P,d)]s.t.(F,d)∼(F,c)
ω([(P, d)]) =∑
[(P ′,d)]s.t.(F,d)∼(F,c)
ω([(P ′, d)]),
must be established.
• Gluing eqation : Varying F ∈ F and c ∈ Ck+1(Γ), we
get a system of equations on nonnegative integral valued J -
invariant weight functions
ω : P × Ck+1(Γ)/ ∼→ Z≥0.
• Real valued solution exists : Choose µ ∈MJ(Ck+1(Γ)),
then by additivity,∑[(P,d)]s.t.(F,d)∼(F,c)
µ([(P, d)]) = µ(d|(F, d) ∼ (F, c))
= µ([(F, c)])
=∑
[(P ′,d)]s.t.(F,d)∼(F,c)
µ([(P ′, d)]),
11.5 Step 4
11.6 Virtual gluing
• Theorem (Theorem 3.1 in Agol [2]) : Let P be a com-
pact virtually special NPC cube complex such that π1(P ) is
hyperbolic. Let Q ⊂ P be an embedded locally convex acylin-
drical subcomplex which covers an NPC cube orbi-complex
π : Q→ Q0.
Then there exists a finite regular cover P → P such that the
preimage of P ⊃ Q→ Q is a regular orbi-cover Q→ Q0.
• Proof :
– May assume that P is special already.
– To find a regular cover of Q0, glue P and a mapping cone
Cπ of π : Q→ Q0 along Q to get P ′ = P ∪Q Cπ.– Since Cπ is acylindrical, π1(P
′) is hyperbolic by Bestvina-
Feighn and hence π1(P′) ∈ QVH.
– Thus π1(P′) is virtually special by Wise.
– Assume π1(P′) is already special without loss of generality.
– Choose a finite regular cover : Cπ → Cπ.
– We have a local isometry by the composition : Cπ →Cπ → P ′ and hence, there is a finite cover C(Cπ, P
′)→ P ′
with an embedding Cπ → C(Cπ, P′) and a contraction
π1(C(Cπ, P′))→ π1(Q).
– Tak a further regular cover of P through P ′ will imply the
desired covering.
12 Many Surfaces Exist
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