IV Topics in Geometric Group Theory

2Van Kampen diagrams

2 Van Kampen diagrams

Recall we previously tried to understand the Dehn function of

in terms of

filling in squares on a grid. In this chapter, we pursue this idea further, and

come up with an actual theorem.

To make this more precise, recall that in the case of

, we could embed a

null-homotopic word into the plane of the form

We saw, at least heuristically, the algebraic area of

is just the geometric area

of this figure. Of course, sometimes it is slightly more complicated. For example,

if we have the word w = [a, b]

, then we have to use the cell [a, b] twice.

Definition

(Singular disc diagram)

A (singular) disc diagram is a compact,

contractible subcomplex of R

, e.g.

We usually endow the disc diagram D with a base point p, and define

Area

(D) = number of 2-cells of D

and

Diam

(D) = length of the longest embedded path in D

(1)

, starting at p.

If we orient

, then

has a well-defined boundary cycle. A disc diagram is

labelled if each (oriented) edge is labelled by an element

s ∈ S

. The set of face

labels is the set of boundary cycles of the 2-cells.

If all the face labels are elements of

, then we say

is a diagram over

hS | Ri.

Definition

(van Kampen diagram)

w ∈ hhRii

, and

is the boundary cycle

of a singular disc diagram

over the presentation

hS | Ri

, we say that

is a

van Kampen diagram for w.

Example. The diagram

is a van Kampen diagram for abababa

−2

−1

Note that in this case, the map

→ X

that represents

factors through

a map D → X

Lemma

(van Kampen’s lemma)

Let

hS | Ri

be a presentation and

w ∈ S

∗

Then the following are equivalent:

(i) w = 1 in Γ presented by P (i.e. w is null-homotopic)

(ii) There is a van Kampen diagram for w given P.

If so, then

Area

(w) = min{Area

(D) : D is a van Kampen diagram for w over P}.

Proof. In one direction, given

w =

i=1

−1

∈ F (S)

such that w = 1 ∈ Γ, we start by writing down a “lollipop diagram”

This defines a diagram for the word

i=1

−1

which is equal in

(

) to

, but is not exactly

. However, note that performing elementary reductions

(or their inverses) correspond to operations on the van Kampen diagram that

does not increase the area. We will be careful and not say that the area doesn’t

change, since we may have to collapse paths that look like

In the other direction, given a diagram

for

, we need to produce an

expression

w =

i=1

−1

such that d ≤ Area(D).

We let

be the first 2-cell that the boundary curve arrives at, reading

anticlockwise around

∂D

. Let

be the path from

, and let

be the relator

read anti-clockwise around e. Let

= D − e,

and let w

= ∂D

. Note that

w = gr

±1

−1

Since Area(D

) = Area(D) − 1, we see by induction that

w =

i=1

±1

−1

with d = Area(D).

We observe that in our algorithm about, the

’s produced have length

≤ Diam(D). So we see that

Corollary. If w is null-homotopic, then we can write w in the form

w =

i=1

−1

where

≤ Diam D

with D a minimal van Kampen diagram for w. We can further bound this by

(max |r

) Area(D) + |w|

≤ constant · δ

(|w|

) + |w|

So in particular, if we know

(

|w|

), then we can bound the maximum

length of g

needed.

It is now easy to prove that

Proposition.

The word problem for a presentation

is solvable iff

computable.

Proof.

(⇐)

By the corollary, the maximum length of a conjugator

that we need to

consider is computable. Therefore we know how long the partial algorithm

needs to run for.

(⇒)

To compute

(

), we use the word problem solving ability to find all

null-homotopic words in

(

) of length

≤ n

. Then for each

, go out and

look for expressions

w =

i=1

±1

−1

A naive search would find the smallest area expression, and this gives us

the Dehn function.

It is a hard theorem that

Theorem

(Novikov–Boone theorem)

There exists a finitely-presented group

with an unsolvable word problem.

Corollary. δ

is sometimes non-computable.

Let’s look at some applications to geometry. In geometry, people wanted to

classify manifolds. Classifying (orientable) 2-dimensional manifolds is easy. They

are completely labeled by the genus. This classification can in fact be performed

by a computer. If we manage to triangulate our manifold, then we can feed the

information of the triangulation into a computer, and then the computer can

compute the Euler characteristic.

Is it possible to do this in higher dimensions? It turns out the word problem

gives a severe hindrance to doing so.

Theorem.

Let

n ≥

4 and Γ =

hS | Ri

be a finitely-presented group. Then we

can construct a closed, smooth, orientable manifold M

such that π

∼

Γ.

This is a nice, little surgery argument.

Proof. Let S = {a

, . . . , a

} and R = {r

, . . . , r

}. We start with

= #

i=0

× S

n−1

Note that when we perform this construction, as n ≥ 3, we have

∼

by Seifert–van Kampen theorem. We now construct M

from M

k−1

such that

∼

, . . . , a

| r

, . . . , r

We realize

as a loop in

k−1

. Because

n ≥

3, we may assume (after a small

homotopy) that this is represented by a smooth embedded map

→ M

k−1

We take

to be a smooth tubular neighbourhood of

. Then

∼

× D

n−1

⊆ M

k−1

. Note that ∂N

∼

× S

n−2

Let

× S

n−2

. Notice that

∂U

∼

∂N

. Since

n ≥

4, we know

simply connected. So we let

k−1

= M

a manifold with boundary

× S

n−2

. Choose an orientation-reversing diffeo-

morphism ϕ

: ∂U

→ ∂M

k−1

. Let

= M

k−1

∪

Then by applying Seifert van Kampen repeatedly, we see that

= π

k−1

/hhr

ii,

as desired.

Thus, if we have an algorithm that can distinguish between the fundamental

groups of manifolds, then that would solve (some variant of) the word problem

for us, which is impossible.

Finally, we provide some (even more) geometric interpretation of the Dehn

function.

Definition

(Filling disc)

Let (

M, g

) be a closed Riemannian manifold. Let

→ M

be a smooth null-homotopic loop. A filling disk for

is a

smooth map f : D

→ M such that the diagram

Since there is a metric

, we can pull it back to a (possibly degenerate)

metric

∗

, and hence measure quantities like the length

(

) of

and

the area Area(f) of D

A classic result is

Theorem

(Douglas, Rad´u, Murray)

is embedded, then there is a least-area

filling disc.

So we can define

Definition (FArea).

FArea(γ) = inf{Area(f ) | f : D

→ M is a filling disc for γ}.

Definition (Isoperimetric function). The isoperimetric function of (M, g) is

F iU

: [0, ∞) → [0, ∞)

` 7→ sup{FArea(γ) : γ : S

→ M is a smooth null-homotopic loop, `(γ) ≤ `}

Theorem

(Filling theorem)

let

be a closed Riemannian manifold. Then

F iU

' δ

By our previous construction, we know this applies to every finitely-presented

group.