Last time we proved that if is a graph of groups, then
acts on a tree
, namely the underlying graph of the universal cover of
. Lemmas 16, 17, and 18 describe the vertex and edge sets of
and the action of
. The following is immediate.
Theorem 14 (Serre): Every graph of groups is developable.
Corollary: If is a graph of groups and
, then the map
is injective.
The tree is called the Bass-Serre Tree of
. We will usually equip
with a length metric in which each edge has length 1. The approach we’ve taken is due to Scott-Wall. Here’s a sample application.
Lemma 19: Let . If
is torsion, then
is conjugate into
or
.
Proof. Let be the Bass-Serre Tree. Fix a vertex
amd consider
. This is a finite set.
Exercise 19: There is an that minimizes
(when
is torsion).
Such an is fixed by
. Therefore,
stabilizes a vertex. The stabilizers of the vertices are precisely the conjugates of
and
.
Our next goal is to understand the elements of — which are trivial and which not? We’ll start with a presentation. If
is a tree, then
can be thought of as a sequence of amalgamated free products. If
is a rose, then
can be thought of as a sequence of HNN-extensions. In general, fix a maximal tree
. This determines a way of decomposing
as a sequence of amalgamated products followed by a sequence of HNN-extensions. This process is sometimes called “Excision.” It follows that
has a “presentation” like this:
We can write any in the following form:
where ,
is a loop in
, and
where
is the terminus of
and the origin of
. A priori, we just know that
. Suppose, say, that
doesn’t originate from
. Then insert stable letters corresponding to edges in
that form a path from
to the origin of
. Now repeat.
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