2Van Kampen diagrams

IV Topics in Geometric Group Theory

2 Van Kampen diagrams
Recall we previously tried to understand the Dehn function of
Z
2
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
Z
2
, we could embed a
null-homotopic word into the plane of the form
We saw, at least heuristically, the algebraic area of
w
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]
2
, 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
2
, e.g.
We usually endow the disc diagram D with a base point p, and define
Area
g
(D) = number of 2-cells of D
and
Diam
p
(D) = length of the longest embedded path in D
(1)
, starting at p.
If we orient
R
2
, then
D
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
R
±
, then we say
D
is a diagram over
hS | Ri.
Definition
(van Kampen diagram)
.
If
w hhRii
, and
w
is the boundary cycle
of a singular disc diagram
D
over the presentation
hS | Ri
, we say that
D
is a
van Kampen diagram for w.
Example. The diagram
a
b
is a van Kampen diagram for abababa
2
b
1
a
1
b
1
ab
1
a
1
.
Note that in this case, the map
S
1
X
P
that represents
w
factors through
a map D X
P
.
Lemma
(van Kampen’s lemma)
.
Let
P
=
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
a
(w) = min{Area
g
(D) : D is a van Kampen diagram for w over P}.
Proof. In one direction, given
w =
n
Y
i=1
g
i
r
±
i
g
1
i
F (S)
such that w = 1 Γ, we start by writing down a “lollipop diagram”
g
d
r
±
d
g
3
r
±
3
g
2
r
±
2
g
1
r
±
1
.
.
.
This defines a diagram for the word
Q
n
i=1
g
i
r
±
i
g
1
i
which is equal in
F
(
S
) to
w
, but is not exactly
w
. 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
D
for
w
, we need to produce an
expression
w =
d
Y
i=1
g
i
r
±
i
g
1
i
such that d Area(D).
We let
e
be the first 2-cell that the boundary curve arrives at, reading
anticlockwise around
D
. Let
g
be the path from
p
to
e
, and let
r
be the relator
D
0
= D e,
and let w
0
= D
0
. Note that
w = gr
±1
g
1
w
0
.
Since Area(D
0
) = Area(D) 1, we see by induction that
w =
d
Y
i=1
g
i
r
±1
i
g
1
i
with d = Area(D).
We observe that in our algorithm about, the
g
i
’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 =
d
Y
i=1
g
i
r
±
i
g
1
i
,
where
|g
i
|
S
Diam D
with D a minimal van Kampen diagram for w. We can further bound this by
(max |r
i
|
S
) Area(D) + |w|
S
constant · δ
P
(|w|
S
) + |w|
S
.
So in particular, if we know
δ
P
(
|w|
S
), then we can bound the maximum
length of g
i
needed.
It is now easy to prove that
Proposition.
The word problem for a presentation
P
is solvable iff
δ
P
is
computable.
Proof.
()
By the corollary, the maximum length of a conjugator
g
i
that we need to
consider is computable. Therefore we know how long the partial algorithm
needs to run for.
()
To compute
δ
P
(
n
), we use the word problem solving ability to find all
null-homotopic words in
F
(
S
) of length
n
. Then for each
d
, go out and
look for expressions
w =
d
Y
i=1
g
i
r
±1
i
g
1
i
.
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
P
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
n
such that π
1
M
=
Γ.
This is a nice, little surgery argument.
Proof. Let S = {a
1
, . . . , a
m
} and R = {r
1
, . . . , r
n
M
0
= #
m
i=0
(S
1
× S
n1
).
Note that when we perform this construction, as n 3, we have
π
1
M
0
=
F
m
by Seifert–van Kampen theorem. We now construct M
k
from M
k1
such that
π
1
M
k
=
ha
1
, . . . , a
m
| r
1
, . . . , r
k
i.
We realize
r
k
as a loop in
M
k1
. Because
n
3, we may assume (after a small
homotopy) that this is represented by a smooth embedded map
r
k
:
S
1
M
k1
.
We take
N
k
to be a smooth tubular neighbourhood of
r
k
. Then
N
k
=
S
1
× D
n1
M
k1
. Note that N
k
=
S
1
× S
n2
.
Let
U
k
=
D
2
× S
n2
. Notice that
U
k
=
N
k
. Since
n
4, we know
U
k
is
simply connected. So we let
M
0
k1
= M
k
\
˚
N
k
,
a manifold with boundary
S
1
× S
n2
. Choose an orientation-reversing diffeo-
morphism ϕ
k
: U
k
M
0
k1
. Let
M
k
= M
0
k1
ϕ
k
U
k
.
Then by applying Seifert van Kampen repeatedly, we see that
π
1
M
k
= π
1
M
k1
/hhr
k
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
γ
:
S
1
M
be a smooth null-homotopic loop. A filling disk for
γ
in
M
is a
smooth map f : D
2
M such that the diagram
S
1
D
2
M
γ
f
Since there is a metric
g
on
M
, we can pull it back to a (possibly degenerate)
metric
f
g
on
D
2
, and hence measure quantities like the length
`
(
γ
) of
γ
and
the area Area(f) of D
2
.
A classic result is
Theorem
.
If
γ
is embedded, then there is a least-area
filling disc.
So we can define
Definition (FArea).
FArea(γ) = inf{Area(f ) | f : D
2
M is a filling disc for γ}.
Definition (Isoperimetric function). The isoperimetric function of (M, g) is
F iU
M
: [0, ) [0, )
` 7→ sup{FArea(γ) : γ : S
1
M is a smooth null-homotopic loop, `(γ) `}
Theorem
(Filling theorem)
.
let
M
be a closed Riemannian manifold. Then
F iU
M
' δ
π
1
M
.
By our previous construction, we know this applies to every finitely-presented
group.