Infinite primitive and distance transitive directed graphs of finite out-valency

We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.


Background and main results
We begin with an overview of the paper. Most of the terminology is standard and definitions can be found in the next subsection.
We are interested in the construction and classification of infinite, vertex transitive directed graphs of finite out-valency whose automorphism groups have additional transitivity properties, such as primitivity, distance transitivity or high arc transitivity. In contrast to the finite case where powerful tools from finite group theory are available, there is no possibility of a complete description of such digraphs. Instead, our results will focus on the structure of the descendant set of a vertex in such a digraph: this is the induced subdigraph on the set of vertices reachable from the given vertex by an outward-directed path. Much of the motivation for the work comes from questions of Peter M. Neumann on infinite permutation groups, and work on highly arc transitive digraphs originating in [6].
In [11], Neumann asked whether there exists a primitive permutation group having an infinite suborbit which is paired with a finite suborbit. This amounts to asking whether there is a digraph with infinite in-valency and finite out-valency whose automorphism group is transitive on edges and primitive on vertices. Countable digraphs of this sort were constructed in [8] using amalgamation methods developed in model theory (cf. [5] for background on such methods). In these examples, the descendant sets are directed trees, and the resulting examples are also highly arc transitive. Similar methods were used in [7] to construct continuum-many non-isomorphic countable, primitive, highly arc transitive digraphs all with isomorphic descendant sets. So this suggests that a classification of such digraphs is out of the question, even under the very strong assumption of high arc transitivity. Nevertheless, Neumann (private communication) suggested that a classification of the descendant sets in these digraphs might be possible, at least under stronger hypotheses on the automorphism group of the digraph.
Descendant sets in highly arc transitive digraphs of finite out-valency were studied by the first Author in [1,2], following on from results obtained by Möller for locally finite, highly arc transitive digraphs in [10]. This work isolates a small number of quite simple properties (essentially P0, P1, P3 of Section 3 here) satisfied by such descendant sets and shows that these properties have rather strong structural consequences. In particular, the descendant set admits a non-trivial, finite-to-one homomorphism onto a tree. Digraphs having the given properties, but which are not trees are constructed in [2,10]. Moreover (imprimitive) highly arc transitive digraphs having these as descendant sets are constructed in [2][3][4].
It was conjectured in [2] that there are only countably many directed graphs with the properties for a descendant set isolated in [2]. In Section 2 we reprove some of the results of [2] in a slightly wider context and prove the conjecture. In particular, we have the following (note that a highly arc transitive digraph is distance transitive). The proof of this is given at the end of Section 2. Together with Corollary 4.4 of [4], it gives a reasonable picture of the descendant sets in distance transitive digraphs of finite out-valency and infinite in-valency: conditions P0, P1, P3 are necessary and sufficient conditions for a digraph to be a descendant set in such a digraph, and there are only countably many digraphs satisfying these conditions.
In Section 3 we are interested in descendant sets under the additional assumption of primitivity. Then main result is: The construction of D Γ is as in the paper [8], where the descendant set Γ is a tree. However, the proof of primitivity in the general case is much harder than in [8], and this is where the novelty lies in the above result. We do not know whether the condition P2 on Γ is a necessary condition here, but it is satisfied by all of the examples constructed in Section 5 of [2]. So this gives new examples of descendant sets in primitive (and even highly arc transitive) digraphs of finite out-valency and infinite in-valency. It would of course be interesting to find necessary and sufficient conditions on the descendant set in a primitive distance transitive digraph of finite out-valency and infinite in-valency. It would be even more interesting to know whether anything can be said without the assumption of distance transitivity.

Notation and terminology
A digraph (D; E(D)) consists of a set D of vertices, and a set E(D) ⊆ D×D of ordered pairs of vertices, the (directed) edges. Our digraphs will have no loops and no multiple edges. We will think of a subset X of the set D of vertices as a digraph in its own right by considering the full induced subdigraph on X (so E(X) = E(D) ∩ X 2 ). Throughout this paper, 'subdigraph' will mean 'full induced subdigraph'. Thus henceforth, we will not usually distinguish notationally between a digraph and its vertex set. In particular, we will usually refer to the digraph (D; E(D)) simply as 'the digraph D'. Note that this is a different convention from the usual notation D = (V D; ED). Furthermore, we will use notation such as 'α ∈ D' to indicate that α is a vertex of the digraph D.
We denote the automorphism group of the digraph D by Aut(D). We say that D is transitive (respectively, edge transitive) if this is transitive on D (respectively, E(D)). We say that D is primitive if Aut(D) is primitive on D, that is, there are no non-trivial Aut(D)-invariant equivalence relations on D.
The out-valency of a vertex α ∈ D is the size of the set {u ∈ D : (α, u) ∈ E(D)} of out-vertices of α; similarly, the in-valency of α is the size of the set {u ∈ D : (u, α) ∈ E(D)} of in-vertices. Let s ≥ 0 be an integer. An s-arc from u to v in D is a sequence u 0 u 1 . . . u s of s + 1 vertices such that u 0 = u, u s = v and (u i , u i+1 ) ∈ ED for 0 ≤ i < s and u i−1 = u i+1 for 0 < i < s. Usually our digraphs will be asymmetric, in which case this last condition is redundant. We denote by D s (u) the set of vertices of D which are reachable by an s-arc from u. The descendant set D(u) (or desc(u)) of u is s≥0 D s (u). Similarly the set anc(u) of ancestors of u is the set of vertices of which u is a descendant.
In particular, fix α ∈ D, and let Γ = D(α). If Aut(D) is transitive on the set of vertices of D, then D(u) ∼ = Γ for all vertices u, and we shall speak of the digraph Γ as the descendant set of D.
We say that the digraph D is highly arc transitive if for each s ≥ 0, Aut(D) is transitive on the set of s-arcs in D. Following [9], we say that a digraph D is (directed)-distance transitive if for every s ≥ 0, Aut(D) is transitive on pairs (u, v) for which there is an s-arc from u to v, but no t-arc for t < s. Note that this implies vertex and edge transitivity, but is weaker than being highly arc transitive. We generally exclude the case of null digraphs, where there are no edges.
Henceforth, we shall be interested in the structure of a descendant set Γ = Γ(α) of a vertex α in some transitive digraph D with finite out-valency m. We will be considering this as a digraph with its full induced structure from D. We refer to α as the root of Γ and write Γ = Γ(α) to indicate that any vertex of Γ is a descendant of α. Similarly, we write Γ i instead of Γ i (α) for the set of vertices reachable by an i-arc starting at α and if β ∈ Γ(α), then we write Γ(β) = desc(β) ⊆ Γ(α). It is clear that if D is highly arc transitive, then Aut(Γ(α)) is transitive on s-arcs in Γ(α) which start at α. Similarly, if D is distance transitive, then Aut(Γ(α)) is transitive on Γ n (α) for each n ∈ N.
We shall see that conditions G0, G1, G3 hold when Γ is the descendant set in a distance transitive digraph of finite out-valency and infinite in-valency (Corollary 2.5). The minimum possible k in G3 is the parameter k(Γ) which appears in Theorem 1.1. If k(Γ) = 1 then Γ is a directed tree, however Section 5 of [2] constructs digraphs Γ(Σ, t) satisfying G0-G3 with arbitrary value for k(Γ(Σ, t)).
A priori there could be continuum-many isomorphism types of digraphs with these properties. Our main result in this section (Theorem 2.15) is that there are only countably many isomorphism types of digraph Γ which satisfy G0, G1 and G3. To establish this, we show that there is a natural equivalence relation ρ on Γ (refining the 'layering' of Γ given by G0) such that the quotient digraph Γ/ρ is a directed tree. If G2 holds then this is not a directed line and the size of the layers Γ n grows exponentially.

Lemma 2.1. Suppose D is a (non-null) digraph of finite out-valency which has no directed cycles and is distance transitive. Then any descendant set
Proof. This is the same as the proof of Proposition 3.10 in [6], so we omit the details. 2 Lemma 2.2. Suppose D is a digraph of finite out-valency with a directed cycle and whose automorphism group is either primitive on vertices or transitive on edges. Then D has finite in-valency.
But this set is finite, as D has finite out-valency. Now suppose D is vertex-primitive. Consider the relation ∼ on D given by u ∼ v ⇔ u ∈ D(v) and v ∈ D(u). This is an Aut(D)-invariant equivalence relation on D and as D contains a directed cycle, its classes are not singletons. Thus, by primitivity u ∼ v for all u, v ∈ D. In particular, every edge of D is contained in a cycle. We can then argue as in the first case. 2 Lemma 2.3. Suppose Γ satisfies G0, G1 and that for each i ∈ N the automorphism group By the transitivity assumption, this depends only on i. As anc( Choosing k so that t k is as large as possible, the result follows. 2 Remark 2.4. Note that in the above if G2 also holds, then Proof. By Lemma 2.2, if D has infinite in-valency then D has no directed cycles, so by Lemma 2.1, Γ satisfies G0. As D has transitive automorphism group, G1 holds. Distance transitivity implies that Aut(Γ) is transitive on each Γ i , so G3 holds.
Suppose Aut(D) is primitive on vertices of D. If G2 does not hold for some n, then for β, β ∈ Γ 1 (α) we have Γ n (β) = Γ n+1 (α) = Γ n (β ). If m > 1, then this gives a non-trivial equivalence relation on the vertices of D which is preserved by Aut(D), and we have a contradiction to primitivity. So it remains to show that m > 1. But if m = 1, then the underlying (undirected) graph of D has no cycles. This contradicts primitivity of Aut(D), as it implies that being at even distance in the underlying graph is an equivalence relation on the vertices. 2

Structure theory
Throughout this section we assume that Γ satisfies G0, G1, G3. We let k be an integer satisfying the condition in G3. The proofs in this section are all adapted from [2].
Proof. Note that the result holds for = k by G3.
For ≥ k and x ∈ Γ l (α) we write [x] ρ for the ρ-equivalence class containing x.
We use notation such as v, w etc. for such classes and write Γ(u) = x∈u Γ(x) and Γ s (u) = x∈u Γ s (x).
Proof. It suffices to prove this when w ∈ Γ +1 (α). So suppose that (v, w), (v , w ) are directed edges and ρ(w, w ) holds. We need to show that , as ρ(w, w ) holds. It follows (by taking ancestors one level back) that Proof. The statement follows from Lemmas 2.8 and 2.9. 2 Note that for β ∈ Γ(α) we can consider the equivalence relation ρ computed in both Γ(α) and Γ(β), where in the latter we only consider ancestors in Γ(β) when defining ρ: a priori this gives a coarser relation.

Counting isomorphism types
We let T be the class of structures T with the following properties • T is a digraph of finite out-valency and T = T (u) for some finite set u ⊆ T .
• There is an equivalence relation ρ on T such that each ρ-class is contained in a layer T s (u). • The quotient digraph T /ρ is a directed forest. • For every ρ-class w there is a ρ-class v ⊆ u and a ρ-isomorphism between T (v) and T (w).
We show: Theorem 2.14. There are only countably many ρ-isomorphism types of structures in T . Proof. Fix such a Γ. Let T be the disjoint union of digraphs T (v) with the equivalence relation ρ as in the previous section, taking v to be a ρ-class in Γ 2k−1 . So in fact, T = ≥2k−1 Γ . Then T ∈ T , by Corollaries 2.13 and 2.12. Moreover we can recover Γ from T by looking at the descendant set of any vertex in T . Thus there are only countably many possibilities for Γ, by the above Theorem. 2 We now prove Theorem 2.14. Let T = T (u) ∈ T and let v 1 , . . . , v r be the ρ-classes in T 0 = u. We colour a ρ-class v in T with colour C i if i is (as small as possible) such that T (v) is ρ-isomorphic to T (v i ). If d ∈ N, then we denote by B d T the digraph on s≤d T s together with the structure given by the ρ-classes and the colouring on this set. Similarly if v is a ρ-class we denote by B d T (v) the corresponding structure on s≤d T s (v). In the following, by a ρ − C-isomorphism we mean a digraph isomorphism which preserves the relation ρ and the colouring.

Proof. Let
It is easy to see that Ψ is a well-defined bijection between B d+1 T and B d+1 S . As f i , Φ and α i all preserve ρ-classes and the colouring, the same is true of Ψ. So it remains to show that Ψ preserves edges and non-edges.
First we show that if x ∈ B 1 T , then Ψ(x) = Φ(x). If x ∈ T 0 then this is by definition Thus Ψ preserves edges and non-edges in Proof. By assumption and Proposition 2.17, for n ≥ N S the set I n of ρ −C-isomorphisms B n T → B n S is non-empty. Restriction gives a map I n+1 → I n and so, as each I n is finite, König's Lemma implies that there is a ρ − C-isomorphism T → S. 2 Proof of Theorem 2.14. Suppose T ∈ T . As above, consider this with a colouring of the ρ-classes. Let N = N T be as in Lemma 2.16. Then by Corollary 2.18, the (coloured) ball B N +1 T determines T within T up to isomorphism. There are only countably many possibilities for this finite structure, hence the result. 2 We now give the proof of Theorem 1.1.
Proof of Theorem 1.1. Let D and Γ = Γ D be as in the statement of the theorem. First, we define the numbers k(Γ) and M (Γ). By Corollary 2.5, Γ satisfies G0, G1, G3. Let k = k(Γ) be the smallest value of k which satisfies G3 for Γ.
Let w be a ρ-class in Γ 2k−1 . By distance transitivity, Aut(Γ) is transitive on each Γ , so if ≥ 2k − 1 and v is a ρ-class in Γ , then there is a ρ-isomorphism from Γ(w) to Γ(v), by Corollary 2.12. Let T Γ ∈ T consist of Γ(w) together with its ρ-structure. Note that we have only one 'colour' C i used here, so ρ-isomorphisms will be ρ − C-isomorphisms in what follows.
Let M (Γ) = 2k(Γ) +N T Γ , where N T Γ is as in Lemma 2.16 (chosen as small as possible). Thus, from the proof of Lemma 2.16, N T Γ is the smallest value of N such that any permutation of w which extends to a ρ-automorphism of T ≤N Γ extends to a ρ-automorphism of T Γ . Now suppose that D 1 , D 2 are distance transitive digraphs satisfying the hypotheses of the theorem. Let Γ i = Γ D i and suppose that k( , considered also with its ρ-structure. Then N T 1 = N T 2 = N and θ gives a ρ-isomorphism between the balls B N +1 T 2 . By Corollary 2.18 (and the above remark on colours) T 1 and T 2 are isomorphic. It then follows that Γ 1 and Γ 2 are isomorphic, as required. 2

Constructions
In this section, we prove Theorem 1.2. The construction of the digraphs D Γ is as in [8] and we recall briefly some notation and terminology from there.
Suppose D is a digraph and A ⊆ D. We write A ≤ D if for every a ∈ A we have desc(a) ⊆ A. We say that A ≤ D is finitely generated (f.g.) if there is a finite X ⊆ A with A = a∈X desc(a), and say that X is a generating set for A. We It is easy to check (cf. Lemma 2.2 of [8]) that if A ≤ + B ≤ + C then A ≤ + C (and similarly for ≤).
Of course, P0 and P1 are the same as G0, G1 and P3 implies G3 (as in Lemma 2.3). From Section 2.1, if Γ is the descendant set of a vertex in an infinite, distance transitive digraph D of finite out-valency and with no directed cycles then Γ satisfies P0, P1, P3. If D is primitive, then (as noted in [2] under the stronger assumption of high arc transitivity) Γ satisfies the following weaker version of P2: P2 For all a 1 , a 2 ∈ Γ, if Γ(a 1 ) \ Γ(a 2 ) and Γ(a 2 ) \ Γ(a 1 ) are finite, then a 1 = a 2 .
Note in particular that P2 implies that different vertices have different sets of outvertices. Section 5 of [2] gives examples Γ(Σ, k) which satisfy P0, P1, P2 , P3 and it can be checked that these examples also satisfy P2. In this section we prove that if Γ satisfies P0-P3, then there is a primitive digraph D Γ with Γ as its descendant set. If Γ has the property that Aut(Γ) is transitive on n-arcs from γ (as is the case with the Γ(Σ, k) from [2]), then the D Γ which we construct will be highly arc transitive.
The construction of D Γ is essentially the same Fraïssé amalgamation class construction which was used in [8]. We will recall this briefly, making use of results from [4]. Once we have D Γ , the main work of the section will be in proving primitivity of Aut(D Γ ).
So suppose Γ satisfies P0-P3. Let C Γ consist of the digraphs A with the property that for every a ∈ A, desc(a) ≤ + A and desc(a) ∼ = Γ. Let C Γ be the finitely generated elements of C Γ . Note that Γ ∈ C Γ , so in particular, C Γ is non-empty.
If  (Note that C Γ as defined here is a subset of the C Γ defined in Section 4.1 of [4].) The result we want follows: for (1), take A = ∅ and (2) is immediate. 2 It is easy to show that C Γ is closed under free amalgamation over finitely generated ≤ + -subsets (as in Lemma 2.6 of [8]). More formally, if B 1 , B 2 ∈ C Γ and A ≤ + B i is f.g., then the digraph F which has vertices the disjoint union of B 1 and B 2 over A and whose edges are the edges of B 1 and B 2 is also in C Γ . Furthermore, B 1 , B 2 ≤ + F . This gives the following ≤ + -amalgamation property: 2). Then there exist F ∈ C Γ and ≤ + -embeddings g i : B i → F with the property that g 1 (f 1 (a)) = g 2 (f 2 (a)) for all a ∈ A. 2 Note that if f 1 is inclusion, then we can also take g 1 to be inclusion here. Once we have these lemmas, the following existence and uniqueness result is fairly standard and we omit some of the details of the proof. (1) If a ∈ D Γ then desc(a) ≤ + D Γ and desc(a) ∼ = Γ. a for all a ∈ A and g(B) ≤ + D Γ .
Moreover, D Γ is uniquely determined up to isomorphism by these conditions and is ≤ + -homogeneous, meaning that if A 1 , A 2 ≤ + D Γ are f.g. and h : A 1 → A 2 is an isomorphism, then h extends to an automorphism of D Γ .
Proof. Note that (1) here follows from (2). For the existence part, we build a chain of with the property: then there is some j ≥ i and a ≤ + -embedding g : B → D j such that g(f (a)) = a for all a ∈ A.
Once we have this, we let D Γ be the union n∈N D n . Then (2) follows as each D n is in C Γ , and (3) follows from (*). In order to obtain (*) we build the D n inductively. During this process, there will be countably many 'tasks' to be performed: there are countably many choices of f.g. A in each D i and countably many isomorphism types of ≤ + -embeddings f : A → B with B ∈ C Γ (by Lemma 3.1). As we have countably many steps available during the construction, it will suffice to show how to complete one of these tasks: ensuring that they are all completed during some stage of the construction is then just a matter of organisation (see the proof of Theorem 2.8 of [7] for a formal way of doing this).
So suppose D n has been constructed, A ≤ + D n is f.g. and f : A → B is a ≤ + -embedding with B ∈ C Γ . Using amalgamation (Lemma 3.2) we can find D n ≤ + D n+1 and a ≤ + -embedding g : B → D n+1 with g(f (a)) = a for all a ∈ A, as required.
This completes the construction of some countable digraph D Γ with properties (1), (2), (3). For the 'Moreover' part, suppose D Γ is also a countable digraph with properties (1), (2), (3). Suppose A ≤ + D Γ and A ≤ + D Γ are f.g. and h : A → A is an isomorphism. It will suffice to prove that h extends to an isomorphism D Γ → D Γ . As D Γ , D Γ are countable, this follows by a back-and-forth argument (and symmetry) once we show: Claim. If c ∈ D Γ there exist finitely generated B ≤ + D Γ and B ≤ + D Γ with A ≤ + B, A ≤ + B and c ∈ B and an isomorphism g : B → B extending h.
Existence of B here follows from (2) in D Γ . Existence of g and B then follows from (3) It is clear that ≤ + -homogeneity together with property (1) in Theorem 3.3 imply that Aut(D Γ ) is transitive on vertices. Moreover, for every a ∈ D Γ , any automorphism of the descendent set D Γ (a) extends to an automorphism of D Γ (necessarily fixing a) and so by P3, Aut(D Γ /a) (the stabiliser of a) is transitive on D n Γ (a) (vertices reachable by an n-arc from a). Thus D Γ is distance transitive.
The remainder of this section is devoted to showing: Theorem 3.4. With the above notation, Aut(D Γ ) is primitive on the vertices of D Γ .
By the above remarks, Theorem 1.2 then follows.
The following lemma is a simple application of free amalgamation and the extension property (3) in Theorem 3.3, but we shall give the details. If a, b ∈ D Γ and a = b, let Δ(a, b) be the orbital digraph which has (a, b) as an edge. So this is the digraph with vertex set D Γ and edge set the Aut(D Γ )-orbit which contains (a, b). By D.G. Higman's criterion, to prove the primitivity, it will be enough to show that each such Δ(a, b) is connected (meaning that its underlying undirected graph is connected).  If (a, b) and (a, b ) are in the same Aut(D Γ )-orbit, and Δ(b, b ) is connected, then Δ(a, b) is connected (of diameter at most twice that of Δ(b, b )).
Proof. There is an (undirected) Δ(a, b)-path bab from b to b . Thus if (b 1 , b 2 ) is an edge in Δ(b, b ) there is a Δ(a, b)-path of length 2 from b 1 to b 2 . As Δ(b, b ) is connected, if x, y ∈ D Γ there is a Δ(b, b )-path from x to y, and therefore there is a Δ(a, b)-path from x to y with at most twice as many edges. 2 Proof. First suppose that desc(a) ∪ desc(b) ≤ + D Γ . Given x 1 , x 2 ∈ D Γ , by Theorem 3.3(2) there is a finitely generated Z ≤ + D Γ with x 1 , x 2 ∈ Z. Let X be the disjoint union of Z and a copy C of Γ. So X is the free amalgam of Z and C over the empty set and X ∈ C Γ . By Theorem 3.3(3) we may assume that X ≤ + D Γ . Let c ∈ D Γ be the generator of C. Then desc(c) ∩ desc(x i ) = ∅ and desc(x i ) ∪ desc(c) ≤ + D Γ (for i = 1, 2). It follows (using the 'Moreover' part of Theorem 3.3) that (c, x 1 ) and (c, x 2 ) are edges in Δ(a, b). So Δ(a, b) has diameter 2.
In general, there is b ∈ D Γ such that (a, b ) is an edge in Δ(a, b) and f.g. B, B ≤ + D Γ with a, b ∈ B and a, b ∈ B satisfying B ∪ B ≤ + D Γ and B ∩ B = desc(a) (again, this uses free amalgamation and property (3) of D Γ ; alternatively we can apply Lemma 3.5). Then b, b are as in the first part of the proof and so Δ(b, b ) has diameter 2. Therefore by Lemma 3.6, Δ(a, b) has diameter at most 4. 2 The main work is in proving: Proposition 3.8. Suppose a, b ∈ D Γ are distinct vertices and a / ∈ desc(b) and b / ∈ desc(a). Then there exist r ∈ N and g 1 , . . . , g r ∈ Aut(D Γ ) such that, setting b 0 = a, b 1 We postpone the proof for now, and carry on with the proof of Theorem 3.4. Proof. Let r, g i and b i be as in Proposition 3.8. By Lemma 3.7, Δ(b r , b r+1 ) is connected. So by Lemma 3.6, Δ(b r−1 , b r ) is connected. Proceeding in this way, we obtain that Δ(b 0 , b 1 ) is connected. 2 The remaining case to consider is: ∈ desc(b) (and a = b). Then Δ(a, b) is connected.
We tend to write Γ(a) for desc(a) in order to emphasise the isomorphism with Γ. We write Γ n (a) for the descendants reachable by an n-arc, and say that b ∈ Γ n (a) is at level n with respect to a. Variations such as Γ ≥n (a) (for m≥n Γ m (a)) will also be used. that σ b i−1 (x, x ) holds. But x ∈ Z i−1 and Z i−1 is a union of σ b i−1 -classes, so x ∈ Z i−1 and therefore x ∈ Y i . So y has an ancestor in Z i , as required. 2 Suppose now that we can choose the g i so that if Z i = ∅ then g i Z i = Z i , and therefore Z i+1 is a proper subset of Z i . As these sets are finite, for some r we must have Z r = ∅. It then follows from Claim 4 that Y r = ∅, as required by the proposition.
It remains to explain how to construct the g i , for i ≥ 2. We do this inductively, so suppose we have constructed up to g i−1 and have b 1 , . . . , b i with the required properties. Suppose Z i−1 = ∅. We need to find g i ∈ Aut(D Γ /b i−1 ) such that (with the above notation) Y i ⊆ Γ(b i−1 ) and g i Z i−1 = Z i−1 . First, note that Z i−1 is a proper subset of Γ n (b i−1 ). If not, then Γ n (b i−1 ) ⊆ Γ n (b i ) and as these are finite sets of the same size, we obtain that Γ n (b i−1 ) = Γ n (b i ). So b i−1 and b i have the same descendants in level n and as b i−1 = b i , this contradicts property (1) in the definition of D Γ in Theorem 3.3. It then follows by property P3 of Γ that there is an automorphism h of Γ(b i−1 ) with hZ i−1 = Z i−1 . We claim that h extends to an automorphism g i of D Γ with the property that Γ(g i b i ) ∩ Γ(b i ) ⊆ Γ(b i−1 ), and this will suffice.
To do this, let B ≤ + D Γ be finitely generated and contain b i−1 , b i (by property (2) of D Γ in Theorem 3.3). Applying Lemma 3.5 (with the given h and A = Γ(b i−1 )) gives the required g i . This completes the proof of Proposition 3.8. 2