Let be a group. A free splitting of is a simplicial tree equipped with an -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 is a free product of the form
where the are countable groups (at least one of which is infinite) and is a free group of finite rank .
The Gromov boundary of
Fix a basepoint . A geodesic ray in based at is a proper embedding such that . We are really interested in the image of more than the map. Similarly we have geodesics; embeddings . Every path in is homotopic rel endpoints to a unique geodesic or to a constant map. Let denote the set of geodesic rays in based at . This is the Gromov boundary of . We may topologize by giving the usual topology and defining basic open neighborhoods of boundary points in the following way. Given a boundary point , the basic open neighborhood is determined by a finite subpath of the ray ; the basic open set consists of those points such that the geodesic from to contains . In July, we showed that in our situation, if all the groups are countable and one is countably infinite, then the Gromov boundary of is homeomorphic (in the subspace topology) to the Baire space.
The observer’s topology
We may give a different topology that makes it into a compact space: this is the observer’s topology, coined by Coulbois, Hilion and Lustig. Let be a point. A half-tree based at is a component of . A subbasis for the observer’s topology on is given by the set of half-trees; where we say a boundary point belongs to a half-tree if the corresponding geodesic ray has infinite intersection with the half-tree. In our situation, where the groups are countable, the tree 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 is second countable. Thus it makes sense to talk about the topology in terms of convergent sequences: a sequence converges to a point if for every point , the points belong to the same half-tree at as for sufficiently large.
Coulbois, Hilion and Lustig show that is compact in the observer’s topology. Here is their argument: suppose is a sequence, and let be a countable set of half-trees generating the topology on For each , we may, after passing to a subsequence, assume that either belongs to or not for large. Now note that the intersection of (the images of) two (distinct) geodesics or geodesic rays based at is either a point or a geodesic. Let denote the geodesic or geodesic ray from to , and consider the geodesic
Observe that , so
is a geodesic or geodesic ray; let its endpoint be . We will show that our subsequence converges to . Suppose for we have that . Then for any finite collection of such , the subtree spanned by the collection is entirely outside , since the (closed) complement of a half-tree is convex. The nearest point to belongs to this spanning subtree, so inductively this shows that the endpoint of is outside for large . It follows that . Therefore if , then infinitely many, and thus all but finitely many of the belong to , showing that the sequence converges to .
Next we would like to show that the topology induced on agrees with the one above. Suppose converges to in the observer’s topology, so for any point , the points belong to the same half-tree at as for large. Taking the points to be on the geodesic ray shows that the geodesic or geodesic ray from to must contain any finite subpath of for large. If conversely the geodesic or geodesic ray from to contains any finite subpath of for sufficiently large, then if the intersection of the ray with a given half-tree is infinite, we may take a finite subpath so large that it ends inside the half-tree. If the geodesic or geodesic ray from to contains , then since is a tree, we conclude that belongs to the half-tree.
The Bowditch Boundary of
It’s not hard to see that if 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 that contains only points of finite valence. If we write for the set of vertices of infinite valence, this argument shows that is closed in the observer’s topology, hence it is compact. This is the Bowditch boundary of . In fact, we showed in July that is a Cantor set; one might think of the points as being akin to the rational points of the standard middle-thirds Cantor set.
Laminations
Consider the space
where is the diagonal and acts by interchanging the coordinates. Thus is the space of unordered pairs of points in the Bowditch boundary of . The diagonal action of on extends to an action on . A lamination is an -invariant closed subset of .
Given a point , there is a lamination called the limit set of defined as follows. A point belongs to if (possibly after swapping and ), there is a sequence of elements of such that the sequence converges to and the sequence converges to .