Recall from last time:
Theorem 16: Every is semisimple. If
is loxodromic, then
is isometric to
and
acts on
as translation by
.
In particular, for every ,
is a
-invariant subtree of
.
Exercise 21: If and
, then
is loxodromic,
, and
.
(Hint: It is enough to construct an axis for .)
Lemma 20 (Helly’s Theorem for Trees): If are closed subtrees and
for every
, then
.
Proof. Case : Let
where
. Let
be the center of the triangle with vertices
. Then
.
General case: Let for
. Then
by the
case. Now, by induction,
Corollary: If a finitely generated group acts on a tree
with no global fixed point, then there is a (unique)
-invariant subtree
that is minimal with respect to inclusion, among all
-invariant subtrees. Furthermore,
is countable.
Proof. Let
For any -invariant
, it is clear that
. Therefore
is minimal. Let
be a finite generating set for
. Suppose that every element of
is elliptical, so for all
,
is elliptical. Then for all
,
Therefore by Lemma 20, a contradiction.
Suppose that are loxodromic and
. By Exercise 21,
intersects
and
non-trivially. Thus,
is connected.
It remains to show that is
-invariant. Let
. For any
,
This implies that , and
So we conclude that is
-invariant.
Definition: If is connected, then the graph of groups carried by
is the graph of groups
with underlying graph
such that the vertex
is labelled by
, and the edge
is labelled
(with obvious edge maps). There is a natural map
, where
and
. This map is an injection by the Normal Form Theorem.
Lemma 21: If is countable and
is finitely generated, then there is a finite subgraph
such that
(where
is the graph of groups carried by
).
Proof. Let be an exhaustion of
by finite connected subgraphs. Let
denote the graph of groups carried by
and set
. Since
is finitely generated, there is an
such that
contains each generator of
, and since
, we conclude that
.
Lemma 22: If is countable and
is finitely generated then there is a finite minimal (wrt inclusion) subgraph
that carries
.
Proof. Let be the Bass-Serre tree of
. Let
. It’s an easy exercise to check that this is as required.
We will refer to (and
, the graph of groups carried by
) as the core of
.
Theorem 17: If is finitely generated, then
decomposes as
of a finite graph of groups
. The vertex groups of
are conjugate into the vertex groups of
. The edge groups are likewise.
Proof. Let be the Bass-Serre tree of
, and set
.
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