A Stallings’ type theorem
for quasi-transitive graphs
We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings’ splitting theorem for multi-ended finitely generated groups and indeed it implies this theorem. It will also lead to a characterisation of accessible graphs in terms of tree amalgamations. We obtain applications of our results for hyperbolic graphs, planar graphs and graphs without any thick end. The application for planar graphs answers a question of Mohar in the affirmative.
Stallings  proved that finitely generated groups with more than one end are either a free product with amalgamation over a finite subgroup or an HNN-extension over a finite subgroup. The main aim of this paper is to obtain an analogue of Stallings’ theorem for quasi-transitive graphs. The obvious obstacle for this is that free products with amalgamations and HNN-extensions are group theoretical concepts. So in order to obtain a graph-theoretical analogue, we first need to find a graph-theoretical analogue of free products with amalgamations and HNN-extensions. The proposed notation by Mohar  are tree amalgamations and indeed we will prove the following theorem. (We refer to Section 5 for the definition of tree amalgamations.)
Every connected quasi-transitive locally finite graph with more than one end is a non-trivial tree amalgamation of finite adhesion of two connected quasi-transitive locally finite graphs.
On the other side, we can ask if we start with finite or one-ended connected quasi-transitive locally finite graphs and do iterated tree amalgamations of finite adhesion, what class of graphs do we end up with? In the case of finitely generated groups, the answer is the class of accessible groups (by definition). Thomassen and Woess  defined accessibility for graphs: a quasi-transitive locally finite graph is accessible if there is some such that every two ends can be separated by at most edges. They showed in  that a finitely generated group is accessible if and only if each of its locally finite Cayley graphs is accessible. We will show that tree amalgamations and accessibility fit well together in that we prove that the above described class of graphs we obtain is the class of accessible connected quasi-transitive locally finite graphs.
In 1988, Mohar  asked whether tree amalgamations are powerful enough to yield a classification of infinitely-ended transitive planar graphs in terms of finite and one-ended infinite planar transitive graphs. Our theorems enable us to answer his question in the affirmative for quasi-transitive graphs because Dunwoody  proved that they are accessible, see Section 7.3.
Additionally, we obtain as a corollary Stallings’ theorem, see Section 7.1, and a new characterisation of quasi-transitive locally finite graphs that are quasi-isometric to trees, see Section 7.2. In Section 7.4 we apply our theorems to hyperbolic graphs and show that a quasi-transitive locally finite graph is hyperbolic if and only if it is obtained by iterated tree amalgamations starting with finite or one-ended hyperbolic quasi-transitive locally finite graphs.
We follow the general notations of  unless stated otherwise. In the following we will state the most important definitions for convenience.
Let be a graph. A geodesic is a shortest path between two vertices. A ray is a one-way infinite path, the infinite subpaths of a ray are its tails. Two rays are equivalent if there exists no finite vertex set separating them eventually, i. e. two rays are equivalent if they have tails contained in the same component of for every finite set of vertices. The equivalence classes of rays in a graph are its ends. The degree of an end is the maximum number of disjoint rays in that end, if it exists. If that maximum does not exist, we say that this end has infinite degree and call it thick. An end with finite degree is called thin. An end is captured by a set of vertices if every ray of has infinite intersection with and it lives in if every ray of has a tail in .
Let . Let be the graph with vertex set , where is a new vertex, and edges between if and only if and is adjacent to precisely those vertices that have a neighbour in . We call the contraction of in and we say that we contracted . Since edges are just vertex sets of size , the definition carries over to edges.
Let be a group acting on and let . The (setwise) stabilizer of with respect to is the set
An orbit of (or a -orbit) is a set for some . We say acts transitively on if is one -orbit and acts quasi-transitively on if consists of finitely many -orbits.
3 Canonical tree-decompositions
In this section we will look at our main tool for our proofs: tree-decompositions. A tree-decomposition of a graph is a pair where is a tree and is a family of vertex sets of such that the following holds:
For every edge there is a such that contains both vertices that are incident with .
whenever lies on the path in .
The sets are called the parts of and the vertices of the decomposition tree are its nodes. The sets with are the adhesion sets of the tree-decomposition. We say that has finite adhesion if all adhesion sets are finite.
Let be an edge of the decomposition tree of a tree-decomposition . For , let be the component of that contains . It follows from (T3) that separates the vertices in from those in .
We say distinguishes two ends and if there is a finite adhesion set such that one end lives in and the other lives in , where is the maximal subtree of containing . It distinguishes them efficiently if no vertex set in of smaller size than separates them. For , two ends of are -distinguishable if there is a set of vertices of that separates them.
Let be a group acting on . If every maps parts of to parts and thereby induces an automorphism of we say that is -invariant.
The following theorem by Carmesin et al. will be the main result we are building on.
 Let be a locally finite graph, let be a group acting on and let . Then there is a -invariant tree-decomposition of of adhesion at most that efficiently distinguishes all -distinguishable ends.∎
4 Basic tree-decompositions
The aim of this section is first to modify the tree-decomposition of Theorem 3.2 and then to prove some properties of the newly obtained tree-decomposition, in particular, where the tree-decomposition captures the ends of the graph. Our first step in modifying the tree-decomposition of Theorem 3.2 will be to make all adhesion sets connected while keeping the action of on .
Let be a group acting on a locally finite graph and let be a -invariant tree-decomposition of of finite adhesion. Then there is a -invariant tree-decomposition of such that every adhesion set of is finite and connected and such that for every .
Let and be two vertices of an adhesion set of . Assume that is the set of all geodesics between and and assume that is the set of all vertices of that lie on the paths of . For a part let be the union of with all sets where and lie in an adhesion set contained in . Let . We claim that is a tree-decomposition. By construction it has the desired properties, i. e. every adhesion set is connected and and, since is locally finite and since the adhesion sets of are finite, every adhesion set of is finite. Since we made no choices when adding all possible geodesics to the adhesion sets, still acts on .
As every element of is a superset of some element of , we just have to verify (T3) to see that is a tree-decomposition. To see this, let for and let be on the - path in with and . If , then we have as is a tree-decomposition. If , then it lies on a geodesic between two vertices of an adhesion set of in . Since every adhesion set separates from and since , the path must pass through . Thus, either contains two vertices of such that lies on the - subpath of , or lies in . In the first case, we added to the adhesion set because is a geodesic with its end vertices in . Thus, in both cases lies in and thus in . If , let with . By the previous case, lies in for every on the - or - paths in . Since is a tree, these cover the path and hence . This proves that is a tree-decomposition. ∎
We call a tree-decomposition of a graph connected if all parts induce connected subgraphs of .
The step to make the adhesion sets connected is just an intermediate step for us as we aim for connected parts, i. e. we aim for connected tree-decompositions. The next lemma ensures that in connected graphs all parts are connected if all adhesion sets are connected.
If all adhesion sets of a tree-decomposition of a connected graph are connected, then the tree-decomposition is connected.
Let be a graph and let be a tree-decomposition of all of whose adhesions sets are connected. Let and be two vertices of for some . Since is connected, there is a path with and . We choose with as few vertices outside of as possible. Let us suppose that leaves . Let such that and let be the first vertex of after that lies in . As we know that such a vertex always exists. Let such that . Then the adhesion set , where is the neighbour of on the - path in , separates from . Hence, the definition of a tree-decomposition implies that must lie in , too. But then we can replace the subpath of between and by a path in . The resulting walk contains a path between and with fewer vertices outside of than . This contradiction shows that all vertices of lie in and hence is connected. ∎
Most of the time we do not need the full strength of Theorem 3.2 in that it suffices to consider -invariant tree-decompositions with few -orbits that still distinguish some ends.
Let be a group acting on a connected locally finite graph with at least two ends. A -invariant tree-decomposition of is a basic tree-decomposition (with respect to ) if it has the following properties:
distinguishes at least two ends.
Every adhesion set of is finite.
acts on with precisely one orbit on .
If it is clear from the context which group we consider, we just say that is a basic tree-decomposition of . It follows from Theorem 3.2 that basic tree-decompositions always exist:
Let be a group acting on a locally finite graph with at least two ends. Then there is a basic tree-decomposition for .
By Theorem 3.2, we find a -invariant tree-decomposition of bounded adhesion that separates some ends. Let be an edge of such that separates some ends. Let be the orbit of , i. e. the set , and let be obtained from by contracting each component of to a single vertex . We set and set be the set of those sets . It is easy to see that is a basic tree-decomposition with respect to : the only non-trivial requirement is that distinguishes at least two ends. But this follows from the fact that separates two ends. ∎
Let us combine our results on connected and basic tree-decompositions.
Let be a group acting on a connected locally finite graph with at least two ends. Then the following hold.
There is a connected basic tree-decomposition of with respect to .
If is a basic tree-decomposition of with respect to , then there is a connected basic tree-decomposition of with respect to such that for every .
Now we investigate some of the connections between the graphs and the parts of any of the connected basic tree-decompositions. We start by showing that these tree-decompositions behave well with respect to the class of quasi-transitive graphs.
Let be a group acting quasi-transitively on a connected locally finite graph with at least two ends and let be a connected basic tree-decomposition of . Then for each part its stabilizer acts quasi-transitively on .
If does not lie in any adhesion set, then none of its images under elements of lie in an adhesion set. Hence, if maps to , it must fix setwise, as it acts on , so it lies in the stabilizer of . Thus, the intersection of with the -orbit of is the -orbit of .
Now we consider the vertices in an adhesion set . Let be another adhesion set. As is basic, there exists that maps to . If stabilizes , all vertices of lie in -orbits of the vertices of . Let us assume that does not stabilize and let be another adhesion set such that the element that maps to does not stabilize . Then maps to and stabilizes . We conclude that the number of -orbits of vertices in adhesion sets of is at most twice the number of -orbits of vertices in adhesion sets of . ∎
Subtrees of connected basic tree-decompositions that contain a common adhesion set cannot be to large as the following lemma shows.
Let be a group acting quasi-transitively on a connected locally finite graph with at least two ends and let be a connected basic tree-decomposition of with respect to . For an adhesion set let be the maximal subtree of such that for all . Then the diameter of is at most .
Suppose the diameter of is at least . We have for every since is contained in every adhesion set and since they all have the same size. Let be a maximal path in . We shall show that is a double ray.
Let us suppose that is the last vertex on . As is basic, we find such that . Note that fixes setwise. If , then is a neighbour of distinct from that contains , a contradiction to the choice of . If , then fixes the edge but neither of its incident vertices. Let map to . Note that fixes setwise, too. Then either or maps to a neighbour of distinct from . This is again a contradiction, which shows that has no last vertex. Analogously, has no first vertex. So it is a double ray.
Note that the part of some node of contains properly as is finite otherwise. But as acts transitively on , we have at most two -orbits on . Hence infinitely many parts of contain properly. Thus and since each is connected, one vertex of must have infinitely many neighbours. This contradiction to local finiteness shows the assertion. ∎
Our next result is a characterisation of the finite parts of a connected basic tree-decomposition.
Let be a group acting quasi-transitively on a connected locally finite graph with at least two ends and let be a connected basic tree-decomposition of . Then the degree of a node is finite if and only if is finite.
Note that each vertex lies in only finitely many adhesion sets as we only have one orbit of adhesion sets and as is locally finite. So if is finite, then the degree of is finite, too.
Now let us assume that the degree of is finite. Let be a subset of that consists of one vertex from each -orbit that meets . By Proposition 4.5 the set is finite. The vertices in have bounded distance to the union of all adhesion sets in . As they meet all -orbits and fixes setwise, all vertices in have bounded distance to . Note that is finite as has finite degree. Since is locally finite, must be finite. ∎
Let be a tree-decomposition of a graph . We say that an end of captures an end of if for every ray in the union captures and a node of captures if its part does so.
Let us now investigate where the ends of lie in .
Let be a graph and let be a connected tree-decomposition of such that the maximum size of its adhesion sets is at most . Then the following holds.
Each end of is captured either by an end or by a node of .
Every thick end of is captured by a node of .
Every end of captures a unique thin end of , which has degree at most .
Assume that acts quasi-transitively on and that is -invariant with only finitely many -orbits on . Every end of that is captured by a node corresponds to a unique end of .111This shall mean that for every end of that is captured by there is a unique end of with .
Let be an end of and let be two rays in . For an edge let and be the subtrees of with and . If the ray has all but finitely many vertices in and has all but finitely many vertices in or vice versa, then we have a contradiction as and cannot lie in the same end if they have tails that are separated by the finite vertex set . We now orient the edge from to if and lie in eventually and we orient it from to if the rays lie in eventually. Obviously, every node of has at most one outgoing edge. Let be nodes of such that the first vertex of lies in , and the first vertex of lies in , and let and be the maximal (perhaps infinite) directed paths in our orientation of that start at and , respectively. Note that if and meet at a vertex, they continue in the same way. Thus, if they meet, they either end at a common vertex or have a common infinite subpath. We shall show that and meet. Let be the - path in . Then there is a unique sink on it as every node of has at most one outgoing edge. This sink is a common node of and . If and end at a node, this node captures and if they share a common infinite subpath, this is a ray whose end captures . We proved 1.
Now let us assume that has degree at least . Then there are pairwise disjoint rays in . Let be a node and a path of defined for as we defined and for the ray . By an easy induction, we can extend the above argument that and meet to obtain that all have a common node . Let us suppose that is captured by an end of . Let be the node of that is adjacent to and that separates and . Then all rays must contain a vertex of . This is not possible as contains at most vertices and the rays are disjoint. This contradiction shows 2 and the second part of 3.
Let be two rays that lie in ends of that are captured by the same end of . With the notations as above, the intersection is a ray in . As is locally finite and is a connected tree-decomposition, there are infinitely many disjoint paths between and and thus, they are equivalent and lie in the same end of . This proves 3.
To prove 4, let us assume that acts quasi-transitively on and has finitely many orbits on the edges of the decomposition tree . Let be an end of that is captured by a node and let be a ray in that starts at a vertex in . Since captures , there are infinitely many vertices of on . Whenever leaves through an adhesion set, it must reenter it through the same adhesion set by Remark 3.1. We replace every such subpath , where the end vertices of lie in a common adhesion set and the inner vertices of lie outside of , by a geodesic in between the end vertices of . We end up with a walk with the same starting vertex as . We shall see that contains a one-way infinite path. First, we recursively delete closed subwalks of to end up with a path . Since is locally finite and meets infinitely often, contains vertices of that are arbitrarily far away from the starting vertex of . As we only took geodesics to replace the subpaths of that were outside of and as acts on with only finitely many orbits on the edges of , these replacement paths have a bounded length. Hence, eventually leaves every ball of finite diameter around its starting vertex. This implies that is a ray. Obviously, and are equivalent. Thus, contains a ray in . Let be the end of that contains and let be a ray in . Since no finite separator can separate and in , the rays are also equivalent in . Thus, we have shown .
Let be an end in different from , let be a finite subset of that separates from , and let be a path in connecting vertices in different components of . As before, whenever leaves through an adhesion set, it must reenter it through the same adhesion set by Remark 3.1. We again replace every such subpath, where the end vertices lie in a common adhesion set and the inner vertices lie outside of , by a geodesic in to obtain a walk in . Since and have the same endpoints and must meet , we know that either contains a vertex in , or it contains a vertex in an adhesion set which meets . Let be the set containing all vertices of and all vertices contained in adhesion sets that meet . There are only finitely many orbits of vertices in adhesion sets, hence there is an upper bound on the diameter of the adhesion sets. Since is finite and is locally finite, this implies that is finite. By definition, there is no path in connecting vertices in different components of . In particular, separates every ray in from every ray in , and hence 4 holds. ∎
5 Tree amalgamations
In this section, we prove our main result, Theorem 1.1. But before we move on to that proof, we first have to state some definitions, in particular, the main definition: tree amalgamations, a notion introduced by Mohar . After we stated those definitions, we compare tree amalgamations and connected basic tree-decompositions.
For the definition of tree amalgamations, let and be graphs. Let be a family of subsets of . Assume that all sets have the same cardinality and that the index sets and are disjoint. For all and , let be a bijection and let . We call the maps and bonding maps.
Let be a -semiregular tree, that is, a tree in which for the canonical bipartition of the vertices in all have degree . Denote by the set obtained from the edge set of by replacing every edge by two directed edges and . For a directed edge , we denote by the edge with the reversed orientation. Let be a labelling, such that for every , the labels of edges starting at are in bijection to .
For every and for every , take a copy of the graph . Denote by the corresponding copies of in . Let us take the disjoint union of the graphs for all . For every edge with and we identify each vertex in the copy of with the vertex in . Note that this does not depend on the orientation we pick for , since . The resulting graph is called the tree amalgamation of the graphs and over the connecting tree and is denoted by or by if we want to specify the tree.
In the context of tree amalgamations the sets are called the adhesion sets of the tree amalgamation. More specifically, the sets are the adhesion sets of and the sets are the adhesion sets of . If the adhesion sets of a tree amalgamation are finite, then this tree amalgamation has finite adhesion. We call a tree amalgamation trivial if for some the canonical map that maps the vertices to the vertices of that is obtained from by all the identifications is a bijection. Note that if the tree amalgamation has finite adhesion, it is trivial if is the only adhesion set of and for some .
We remark that the map described in the definition of a trivial tree amalgamation does not induce a graph isomorphism : it is a bijection but need not induce a bijection .
The identification length of a vertex is the diameter of the subtree of induced by all nodes for which a vertex of is identified with . The identification length of the tree amalgamation is the supremum of the identification lengths of its vertices. The tree amalgamation has finite identification length if the identification length is finite.
We remark that in Mohar’s definition of a tree amalgamation  the identification length is always at most . But apart from this, our definition is equivalent to his.
It is worth noting that every tree amalgamation gives rise to a tree decomposition in the following sense.
Let be a graph. If is a tree amalgamation of finite adhesion, then there is a naturally defined tree-decomposition of : for let be the set obtained from after all identifications in . Set . Obviously, all vertices of lie in and for each edge there is some containing it. Property (T3) of a tree-decomposition is satisfied as the copies are arranged in a treelike way and as identifications to obtain a vertex take place in subtrees of . So is a tree-decomposition. If has finite adhesion, so does . If the tree amalgamation is non-trivial, then has at least two ends and so does . Also, distinguishes two ends of : those that are captured by ends of .
So far, the tree amalgamations do not interact with any group actions on and . In particular, it is easy to construct a tree amalgamation of two quasi-transitive graphs that is not quasi-transitive: e. g. take as a double ray and as a finite non-trivial graph. Let have precisely two adhesion sets and at least two, all of size . The tree amalgamation is not quasi-transitive.
In the following, we describe some conditions on tree amalgamations which will ensure that tree amalgamations of quasi-transitive graphs are again quasi-transitive, see Lemma 5.3.
Let be a group acting on for , let , let and let . We say that the tree amalgamation respects , if there is a permutation of such that for every there is such that
Note that this in particular implies that . The tree amalgamation respects if it respects every .
Let and let . We call the bonding maps from to and consistent if there is such that
We say that the bonding maps between two sets and are consistent, if they are consistent for any , , and .
We say that the tree amalgamation is of Type 1 respecting the actions of and or is a tree amalgamation of Type 1 for short if the following holds:
The tree amalgamation respects and .
The bonding maps between and are consistent.
We say that the tree amalgamation is of Type 2 respecting the actions of and or is a tree amalgamation of Type 2 for short if the following holds:
, , and ,222Technically this is not allowed, in particular since for the definition of we needed and to be disjoint. These technicalities can be easily dealt with by an appropriate notion of isomorphism the details of which we leave to the reader. and there is such that , if and only if .
The tree amalgamation respects .
The bonding maps between and are consistent.
In this second case also say that is a tree amalgamation of with itself.
We say that is a tree amalgamation respecting the actions of and if it is of either Type 1 or Type 2 respecting the actions and and we speak about the tree amalgamation .
Note that conditions 1 and 2 in both cases do not depend on the specific labelling of the tree. This is no coincidence. In fact we will show that any two legal labellings of give isomorphic tree amalgamations, see Lemma 5.3. Furthermore, any (interpreted as an isomorphism between parts of two such tree amalgamations) can be extended to an isomorphism of the tree amalgamations, which also implies that the tree amalgamations obtained this way are always quasi-transitive.
Before we turn to the proof of these facts, we need some notation. A legally labelled star centred at is a function from to . If the tree amalgamation is of Type 2, we further require that if and only if . Informally, think of this as a star whose labels on directed edges could appear on a subtree of induced by a vertex and its neighbours: for with label , the value tells us the label of .
An isomorphism of two legally labelled stars is a triple consisting of some , a permutation of , and a family of elements of such that for every
In our interpretation of legally labelled stars as subtrees of , this corresponds to an isomorphism of the corresponding subgraphs of the tree amalgamation.
Let be two legally labelled stars with respect to a tree amalgamation centred at and let . Then extends to an isomorphism of and . Furthermore, if we are given and such that
then we can choose and .
Since the tree amalgamation respects , there are and such that
Let be such that , and let be such that . These exist by 2; for Type 2 recall that by the definition of legally labelled stars if and only if . Now clearly
thus showing that the two stars are isomorphic.
For the second part, let be an isomorphism between and . Let . Define and . Let and let
For the remaining , let and . It is straightforward to check that , , and define an isomorphism between and with the desired properties. ∎
Let and be connected locally finite graphs and let be a group acting quasi-transitively on for . Then the tree amalgamation is quasi-transitive and independent (up to isomorphism) of the particular labelling of .
Let and be two labelled trees giving rise to tree amalgamations and , respectively, such that the adhesion sets as well as the bonding maps for both tree amalgamations are the same. Let and let such that and are both isomorphic to . Let . We claim that there is an isomorphism such that
where and denote the canonical isomorphisms from to and respectively. Clearly, the lemma follows from this claim.
For the proof of the claim define the star around by the map mapping to the label of , where is the unique edge with label starting at . By Proposition 5.2, there are a bijection and a family which extend to an isomorphism of the stars around and . Iteratively apply Proposition 5.2 to vertices at distance from . We obtain an isomorphism and maps for each such that the restriction of to and its neighbours and the corresponding maps form an isomorphism between the stars at and .
For , define . Note that for any edge the two concurring definitions given for vertices of and that get identified for the tree amalgamation coincide. Hence is well defined, and since it obviously maps edges to edges and non-edges to non-edges, it is the desired isomorphism. ∎
A closer inspection of the proof of Lemma 5.3 together with Remark 5.1 shows that tree amalgamations respecting the actions of quasi-transitive groups give rise to basic tree-decompositions of . The following lemma shows that the converse also holds, that is, basic tree-decompositions of quasi-transitive graphs give rise to tree amalgamations respecting the actions of some quasi-transitive group on the parts.
Let be a group acting quasi-transitively on a connected locally finite graph and let be a connected basic tree-decomposition of with respect to . Then one of the following holds.
There are such that is a non-trivial tree amalgamation
of Type 1 respecting the actions of the stabilisers of and in .
There is such that is a non-trivial tree amalgamation
of Type 2 respecting the actions of the stabiliser of in .
Choose an oriented edge . We say that