# 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 $\omega$ .