3Bass–Serre theory

IV Topics in Geometric Group Theory

3.2 The Bass–Serre tree
Given a graph of groups
G
, we want to analyze
π
1
G
=
π
1
X
, and we do this by
the natural action of G on
˜
X by deck transformations.
Recall that to understand the free group, we looked at the universal cover of
the rose with r petals. Since the rose was a graph, the universal cover is also a
graph, and because it is simply connected, it must be a tree, and this gives us a
normal form theorem. The key result was that the covering space of a graph is
a graph.
Lemma.
If
X
is a graph of spaces and
ˆ
X X
is a covering map, then
ˆ
X
naturally has the structure of a graph of spaces
ˆ
X
, and
p
respects that structure.
Note that the underlying graph of
ˆ
X
is not necessarily a covering space of
the underlying graph of X .
Proof sketch. Consider
[
vV (Ξ)
X
v
X.
Let
p
1
[
vV (Ξ)
X
V
=
a
ˆvV (
ˆ
Ξ)
ˆ
W
ˆv
.
This defines the vertices of
ˆ
Ξ
, the underlying graph of
ˆ
X
. The path components
ˆ
X
ˆv
are going to be the vertex spaces of
ˆ
X
. Note that for each
ˆv
, there exists a
unique v V (Ξ) such that p :
ˆ
X
ˆv
X
v
is a covering map.
Likewise, the path components of
p
1
[
eE(Ξ)
X
e
× {0}
form the edge spaces
`
eE(Ξ)
ˆ
X
ˆe
of
ˆ
X
, which again are covering spaces of the
edge space of X.
Now let’s define the edge maps
±
ˆe
for each
ˆe E
(
ˆ
Ξ
)
7→ e E
(Ξ). To do so,
we consider the diagram
ˆ
X
ˆe
ˆ
X
ˆe
× [1, 1]
ˆ
X
X
e
X
e
× [1, 1] X
p
By the lifting criterion, for the dashed map to exist, there is a necessary and
sufficient condition on (
ˆ
X
ˆe
×
[
1
,
1]
X
e
×
[
1
,
1]
X
)
. But since this
condition is homotopy invariant, we can check it on the composition (
ˆ
X
ˆe
X
e
X
)
instead, and we know it must be satisfied because a lift exists in this
case.
The attaching maps
±
ˆe
:
ˆ
X
e
ˆ
X
are precisely the restriction to
ˆ
X
e
×{±
1
}
ˆ
X.
Finally, check using covering space theory that the maps
ˆ
X
ˆe
×
[
1
,
1]
ˆ
X
can injective on the interior of the cylinder, and verify that the appropriate maps
are π
1
-injective.
Now let’s apply this to the universal cover
˜
X X
. We see that
˜
X
has a
natural action of G = π
1
X, which preserves the graph of spaces structure.
Note that for any graph of spaces X, there are maps
ι : Ξ X
ρ : X Ξ
such that
ρ ι ' id
Ξ
. In particular, this implies
ρ
is surjective (and of course
ρ
itself is also surjective).
In the case of the universal cover
˜
X
, we see that the underlying graph
˜
Ξ
=
T
is connected and simply connected, because
π
1
˜
Ξ
=
ρ
(
π
1
˜
X
) = 1. So it is a tree!
The action of
G
on
˜
X
descends to an action of
G
on
˜
Ξ
. So whenever we have
a graph of spaces, or a graph of groups
G
, we have an action of the fundamental
group on a tree. This tree is called the Bass–Serre tree of G.
Just like the case of the free group, careful analysis of the Bass–Serre tree
1
(G) to the vertex groups G
v
and edge groups G
e
.
Example. Let
G = F
2
= hai hbi = Z Z.
In this case, we take X to be
X
e
X
u
X
v
Here we view this as a graph where the two vertex spaces are circles, and there
is a single edge connecting them. The covering space
˜
X then looks like
This is not the Bass–Serre tree. The Bass–Serre tree is an infinite-valent bipartite
tree, which looks roughly like
This point of view gives two important results that relate elements of
G
to
the vertex groups G
v
i
and the edge maps
±
e
j
: G
e
j
G
v
±
j
.
Lemma
(Britton’s lemma)
.
For any vertex Ξ, the natural map
G
v
G
is
injective.
Unsurprisingly, this really requires that the edge maps are injective. It is an
exercise to find examples to show that this fails if the boundary maps are not
injective.
Proof sketch.
Observe that the universal cover
˜
X
can be produce by first building
universal covers of the vertex space, which are then glued together in a way that
doesn’t kill the fundamental groups.
More importantly, Bass-Serre trees give us normal form theorems! Pick a
base vertex v V (Ξ). We can then represent elements of G in the form
γ = g
0
e
±1
1
g
1
e
±1
2
· · · e
±1
n
g
n
where each
e
i
is an edge such that
e
±1
1
e
±1
2
· · · e
±1
n
forms a closed loop in the
underlying graph of Ξ, and each
g
i
is an element of the graph at the vertex
connecting e
±1
i
and e
±1
i+1
.
We say a pinch is a sub-word of the form
e
±1
±1
e
(g)e
1
,
which can be replaced by
1
e
(g).
We say a loop is reduced if it contains no pinches.
Theorem
(Normal form theorem)
.
Every element can be represented by a
reduced loop, and the only reduced loop representing the identity is the trivial
loop.
This is good enough, since if we can recognize is something is the identity,
then we see if the product of one with the inverse of the other is the identity.
An exact normal form for all words would be a bit too ambitious.
Proof idea.
It all boils down to the fact that the Bass–Serre tree is a tree.
Connectedness gives the existence, and the simply-connectedness gives the
“uniqueness”.