Discrete, Algebraic and Geometric Structures II …Discrete, Algebraic and Geometric Structures II...

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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