# More on Free Splittings

December 29, 2021

Let $F$
be a group.
A *free splitting* of $F$
is a simplicial tree $T$
equipped with an $F$
-action
such that edge stabilizers are trivial.
I’ve been thinking a lot about free splittings;
here are some of my musings.
There isn’t really a punch line here,
but since it was clarifying to write this, I may as well post it.

We are interested in free splittings where there are finitely many orbits of edges and some vertex has infinite stabilizer. In this case, Bassâ€“Serre theory shows that $F$ is a free product of the form

$F = A_1*\cdots*A_n*F_k$where the $A_i$ are countable groups (at least one of which is infinite) and $F_k$ is a free group of finite rank $k$ .

## The Gromov boundary of $T$

Fix a basepoint $p \in T$
.
A *geodesic ray* in $T$
based at $p$
is a proper embedding $\xi\colon [0,\infty) \to T$
such that $\xi(0) = p$
.
We are really interested in the image of $\xi$
more than the map.
Similarly we have *geodesics;* embeddings $[0,1] \to T$
.
Every path in $T$
is homotopic rel endpoints to a unique geodesic or to a constant map.
Let $\partial_\infty T$
denote the set of geodesic rays in $T$
based at $p$
.
This is the *Gromov boundary* of $T$
.
We may topologize $T \cup \partial_\infty T$
by giving $T$
the usual topology
and defining basic open neighborhoods of boundary points in the following way.
Given a boundary point $\xi$
, the basic open neighborhood is determined by a finite subpath
$\gamma$
of the ray $\xi$
;
the basic open set $N(\gamma)$
consists of those points $x \in T \cup \partial_\infty T$
such that the geodesic from $p$
to $x$
contains $\gamma$
.
In July, we showed that in our situation,
if all the groups $A_i$
are countable and one is countably infinite,
then the Gromov boundary of $T$
is homeomorphic (in the subspace topology) to the Baire space.

## The observer’s topology

We may give $T \cup \partial_\infty T$
a different topology
that makes it into a compact space:
this is the *observer’s topology,* coined by Coulbois, Hilion and Lustig.
Let $x \in T$
be a point.
A *half-tree* based at $x$
is a component of $T \setminus \{x\}$
.
A subbasis for the observer’s topology on $T \cup \partial_\infty T$
is given by the set of half-trees;
where we say a boundary point $\xi \in \partial_\infty T$
belongs to a half-tree
if the corresponding geodesic ray has infinite intersection with the half-tree.
In our situation, where the groups $A_i$
are countable,
the tree $T$
has countably many edges, so has a countable dense subset.
Since every point supports at most countably many half-trees,
it follows that the observer’s topology on $T \cup \partial_\infty T$
is second countable.
Thus it makes sense to talk about the topology in terms of convergent sequences:
a sequence $\{x_n\} \in T \cup \partial_\infty T$
converges to a point $x$
if for every point $y \ne x$
,
the points $x_n$
belong to the same half-tree at $y$
as $x$
for $n$
sufficiently large.

Coulbois, Hilion and Lustig show that $T \cup \partial_\infty T$ is compact in the observer’s topology. Here is their argument: suppose $\{x_n\}$ is a sequence, and let $\{U_i\}$ be a countable set of half-trees generating the topology on $T \cup \partial_\infty T$ For each $U_i$ , we may, after passing to a subsequence, assume that $x_n$ either belongs to $U_i$ or not for $n$ large. Now note that the intersection of (the images of) two (distinct) geodesics or geodesic rays based at $p$ is either a point or a geodesic. Let $\gamma_n$ denote the geodesic or geodesic ray from $p$ to $x_n$ , and consider the geodesic

$I_k = \bigcap_{m \ge k} \gamma_m.$Observe that $I_k \subset I_{k+1}$ , so

$\overline{\bigcup_k I_k}$is a geodesic or geodesic ray; let its endpoint be $x$ . We will show that our subsequence $\{x_n\}$ converges to $x$ . Suppose for $n \ge N$ we have that $x_n \notin U_i$ . Then for any finite collection of such $x_n$ , the subtree spanned by the collection is entirely outside $U_i$ , since the (closed) complement of a half-tree is convex. The nearest point to $p$ belongs to this spanning subtree, so inductively this shows that the endpoint of $I_k$ is outside $U_i$ for large $k$ . It follows that $x \notin U_i$ . Therefore if $x \in U_i$ , then infinitely many, and thus all but finitely many of the $x_n$ belong to $U_i$ , showing that the sequence $\{x_n\}$ converges to $x$ .

Next we would like to show that the topology induced on $\partial_\infty T$ agrees with the one above. Suppose $\{x_n\}$ converges to $\xi \in \partial_\infty T$ in the observer’s topology, so for any point $y \in T$ , the points $x_n$ belong to the same half-tree at $y$ as $\xi$ for $n$ large. Taking the points $y$ to be on the geodesic ray $\xi$ shows that the geodesic or geodesic ray from $p$ to $x_n$ must contain any finite subpath of $\xi$ for $n$ large. If conversely the geodesic or geodesic ray from $p$ to $x_n$ contains any finite subpath of $\xi$ for $n$ sufficiently large, then if the intersection of the ray $\xi$ with a given half-tree is infinite, we may take a finite subpath $\gamma$ so large that it ends inside the half-tree. If the geodesic or geodesic ray from $p$ to $x_n$ contains $\gamma$ , then since $T$ is a tree, we conclude that $x_n$ belongs to the half-tree.

## The Bowditch Boundary of $T$

It’s not hard to see that if $x \in T$
is a point of finite valence,
(i.e. either a vertex of finite valence or a point in the interior of an edge)
then there is a basic open set containing $x$
that contains only points of finite valence.
If we write $V_\infty(T)$
for the set of vertices of infinite valence,
this argument shows that $\partial T = \partial_\infty T \cup V_\infty(T)$
is closed in the observer’s topology,
hence it is compact.
This is the *Bowditch boundary* of $T$
.
In fact, we showed in July that $\partial T$
is a Cantor set;
one might think of the points $V_\infty(T)$
as being akin to the rational points
of the standard middle-thirds Cantor set.

## Laminations

Consider the space

$\tilde{\mathcal{B}}(T) = ((\partial T \times \partial T)\setminus \Delta )/ \mathbb{Z}/2\mathbb{Z}$where $\Delta$
is the diagonal and $\mathbb{Z}/2\mathbb{Z}$
acts by interchanging the coordinates.
Thus $\tilde{\mathcal{B}}(T)$
is the space of unordered pairs of points
in the Bowditch boundary of $T$
.
The diagonal action of $F$
on $\partial T$
extends to an action on $\tilde{\mathcal{B}}(T)$
.
A *lamination* is an $F$
-invariant closed subset of $\tilde{\mathcal{B}}(T)$
.

Given a point $\xi \in \partial_\infty T$
, there is a lamination $\Lambda(\xi)$
called the *limit set* of $\xi$
defined as follows.
A point $(\alpha,\omega) \in \tilde{\mathcal{B}}(T)$
belongs to $\Lambda(\xi)$
if (possibly after swapping $\alpha$
and $\omega$
),
there is a sequence $\{g_n\}$
of elements of $F$
such that the sequence $\{g_n.p\}$
converges to $\alpha$
and the sequence $\{g_n.\xi\}$
converges to $\omega$
.