Let FF be a group. A free splitting of FF is a simplicial tree TT equipped with an FF -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, Bass–Serre theory shows that FF is a free product of the form

F=A1AnFkF = A_1*\cdots*A_n*F_k

where the AiA_i are countable groups (at least one of which is infinite) and FkF_k is a free group of finite rank kk .

The Gromov boundary of TT

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

The observer’s topology

We may give TTT \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 xTx \in T be a point. A half-tree based at xx is a component of T{x}T \setminus \{x\} . A subbasis for the observer’s topology on TTT \cup \partial_\infty T is given by the set of half-trees; where we say a boundary point ξT\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 AiA_i are countable, the tree TT 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 TTT \cup \partial_\infty T is second countable. Thus it makes sense to talk about the topology in terms of convergent sequences: a sequence {xn}TT\{x_n\} \in T \cup \partial_\infty T converges to a point xx if for every point yxy \ne x , the points xnx_n belong to the same half-tree at yy as xx for nn sufficiently large.

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

Ik=mkγm.I_k = \bigcap_{m \ge k} \gamma_m.

Observe that IkIk+1I_k \subset I_{k+1} , so

kIk\overline{\bigcup_k I_k}

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

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

The Bowditch Boundary of TT

It’s not hard to see that if xTx \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 xx that contains only points of finite valence. If we write V(T)V_\infty(T) for the set of vertices of infinite valence, this argument shows that T=TV(T)\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 TT . In fact, we showed in July that T\partial T is a Cantor set; one might think of the points V(T)V_\infty(T) as being akin to the rational points of the standard middle-thirds Cantor set.

Laminations

Consider the space

B~(T)=((T×T)Δ)/Z/2Z\tilde{\mathcal{B}}(T) = ((\partial T \times \partial T)\setminus \Delta )/ \mathbb{Z}/2\mathbb{Z}

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

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