Homogeneous Actions on the Random Graph

We show that any free product of two countable groups, one of them being infinite, admits a faithful and homogeneous action on the Random Graph. We also show that a large class of HNN extensions or free products, amalgamated over a finite group, admit such an action and we extend our results to groups acting on trees. Finally, we show the ubiquity of finitely generated free dense subgroups of the automorphism group of the Random Graph whose action on it have all orbits infinite.


Introduction
The Random Graph (or Rado graph or Erdős-Rényi graph) is the unique, up to isomorphism, countable infinite graph R having the following property : for any pair of disjoint finite subsets (U, V ) of the set of vertices there exists a vertex adjacent to any vertex in U and not adjacent to any vertex in V .Using this property and a model theoretic device called "back-andforth" one can show that R is homogeneous: any graph isomorphism between finite induced subgraphs can be extended to a graph automorphism of R. Hence, the Random Graph plays the same role in graph theory as the Uryshon's space does in metric spaces.
The Random Graph has been popularized by Erdős and Rényi in a serie of papers between 1959 and 1968.They showed [ER63] that if a countable graph is chosen at random, by selecting edges independently with probability 1 2 from the set of 2-elements subsets of the vertex set, then almost surely the resulting graph is isomorphic to R. Erdős and Rényi conclude that this Theorem demolishes the theory of infinite random graphs (however, the world of finite random graphs is much less predictable).
Since almost all countable graphs are isomorphic to R, Erdős and Renyi did not give an explicit construction of the Random Graph.However, by using the uniqueness property of R, it is clear that one may give many different explicit constructions.Such an explicit description was proposed by Rado [Ra64].The uniqueness property of R may also be used to show many stability properties (if small changes are made on R then the resulting graph is still isomorphic to R) and to construct many automorphisms of R as well as group actions on R.
The homogeneity of R means that its automorphism group Aut(R) is large: it acts transitively on vertices, edges and more generally on finite configurations of any given isomorphism type.We will view it as a closed subset of the Polish group (for the topology of pointwise convergence) of the bijections of the vertices.Hence, it has a natural Polish group topology.
The goal of this paper is to understand the countable dense subgroups of Aut(R).The first construction of such subgroups was given in [Mac86], where Macpherson showed that Aut(R) contains a dense free subgroup on 2 generators.More generally, he showed that if M is a ℵ 0categorical structure, then Aut(M ) has a dense free subgroup of rank ℵ 0 .Melles and Shelah [MS94] proved that, if M is a saturated model of a complete theory T with |M | = λ > |T |, then Aut(M ) has a dense free subgroup of cardinality 2 λ .By using the extension property for graphs, Bhattacharjee and Macpherson showed in [BM05] that Aut(R) has a dense locally finite subgroup.
We call a group action Γ R homogeneous if, for any graph isomorphism ϕ : U → V between finite induced subgraphs U, V of R, there exists g ∈ Γ such that g(u) = ϕ(u) for all u ∈ U .The homogeneity of R means exactly that Aut(R) R is homogeneous.Moreover, it is easy to check that a subgroup G < Aut(R) is dense if and only if the action G R is homogeneous.
Hence, to understand the countable dense subgroups of Aut(R) one has to identify the class H R of all countable groups that admit a faithful and homogeneous action on R.Besides free groups ( [Mac86], [MS94], [GK03] and [GS15]) and a locally finite subgroup ( [BM05]), little is known on groups in H R .There are some obvious obstructions to be in the class H R : it is easy to deduce from the simplicity of Aut(R), proved in [Tr85], that any Γ ∈ H R must be icc and not solvable (see Corollary 2.14).Our first positive result is the following.We shall use the notion of highly core-free subgroups, which is a strengthening of core-freeness, introduced in [FMS15] and recalled in Section 1.2.As an example, let us note that a finite subgroup in an icc group is highly core-free.
Theorem A. If Γ 1 , Γ 2 are non-trivial countable groups and Γ 1 is infinite then Γ 1 * Γ 2 ∈ H R .If Σ < Γ 1 , Γ 2 is a common finite subgroup such that Σ is highly core-free in Γ 1 and, either Γ 2 is infinite and Σ is highly core-free in Γ 2 , or Γ 2 is finite and To prove Theorem A, we first show that any infinite countable group Γ admits a "nice" action on R (Corollary 2.8).To produce this explicit action on R we use an inductive limit process.Then starting from an action of Γ := Γ 1 * Γ 2 R and an automorphism α ∈ Aut(R), we construct a natural action π α : Γ R and we show that the set {α ∈ Aut(R) : π α is faithful and homogeneous} is a dense G δ in Aut(R) whenever the initial action is "nice enough" (Theorem 3.1).We follow the same strategy for amalgamated free products but we need to be more careful since we also have to realize the amalgamated free product relations.
Using the same strategy, we prove an analogous result for HNN-extensions.
Theorem B. Let H be an infinite countable group, Σ < H a finite subgroup and θ : Σ → H an injective group homomorphism.If both Σ and θ(Σ) are highly core-free in H then HNN(H, Σ, θ) ∈ H R .
By Bass-Serre theory we obtain the following result.
Corollary C. Let Γ be a countable group acting, without inversion, on a non-trivial tree T in the sense of [Se77].If every vertex stabilizer of T is infinite and, for every edge e of T the stabilizer of e is finite and is a highly core-free subgroup of both the stabilizer of the source of e and the stabilizer of the range of e, then Γ ∈ H R .
Finally, we study the ubiquity of dense free subgroups of Aut(R).Gartside and Knight [GK03] gave necessary and sufficient conditions for a Polish topological group to be "almost free1 ", and gave applications to permutation groups, profinite groups, Lie groups and unitary groups.In particular, they showed that if M is ℵ 0 -categorical then Aut(M ) is almost free.There are abundant results on the ubiquity of free subgroups in various classes of groups.In particular, almost freeness of various oligomorphic2 groups has been shown in [Dr85], [Ka92], [GMR93], [Ca96] and [GK03].We prove the following result.For k ≥ 2 and ᾱ = (α 1 , . . ., α k ) ∈ Aut(R) k , we denote by ᾱ the subgroup of Aut(R) generated by α 1 , . . ., α k , and we set Theorem D. For all k ≥ 2, the set of ᾱ = (α 1 , . . ., α k ) ∈ Aut(R) k such that ᾱ is a free group with basis ᾱ and ᾱ R is homogeneous with all orbits infinite is a dense G δ in A k .
To prove Theorem D we use the "back-and-forth" device (Theorem 6.6).
The paper is organized as follows.In Section 1 we introduce the notations used in the paper about graphs, random extensions and inductive limits.In Section 2 we introduce the Random Graph and we show how to extend any group action on a finite or countable graph to a group action on the Random Graph.We study the basic properties of the extension and the properties of groups acting homogeneously of R. We prove Theorem A in Section 3, Theorem B in Section 4, Corollary C in Section 5 and Theorem D in Section 6.
1. preliminaries Any subset U ⊂ V (G) has a natural structure of a graph, called the induced graph structure on U and denoted by G U , defined by Note that the inclusion ι of a subset U ⊂ V (G) gives an open and injective graph homomorphism ι : G U → G.Moreover, an injective graph homomorphism π : G 1 → G 2 induces an isomorphism between G 1 and the induced graph on π(V (G 1 )) if and only if π is open.
A partial isomorphism of a graph G is an isomorphism between induced subgraphs of G.For a partial isomorphism ϕ, we write d(ϕ) its domain and r(ϕ) its range.We denote by P (G) the set of finite partial isomorphisms i.e. those partial isomorphism for which d(ϕ) is finite (hence r(ϕ) is also finite).
Given a countable graph G, we write Aut(G) the group of isomorphisms from G to G.An action of a group Γ on the graph G is a group homomorphism α : Γ → Aut(G).We write Γ G for an action of Γ on G. Let S(X) be the group of bijections of a countable set X.It is a Polish group under the topology of pointwise convergence.Note that Aut(G) ⊂ S(V (G)) is a closed subgroup hence, Aut(G) is a Polish group.1.2.Group actions.The main purpose of this section is to introduce the Random Extension of a group action on a graph.This notion will be crucial to produce explicit actions on the Random Graph (Section 2.2).Many properties of an action are preserved by its Random Extension (Proposition 1.1).However, the freeness property is not preserved; except for actions of torsion-free groups.Nevertheless, a weaker -but still useful-property, that we call strong faithfulness, is preserved.

An action Γ
X of a group Γ on a set X is called strongly faithful if, for all finite subsets When X is finite, an action on X is strongly faithful if and only if it is free.When X is infinite, an action Γ X is strongly faithful if and only if for all g ∈ Γ \ {1}, the set {x ∈ X : gx = x} is infinite.Note also that, for an action Γ X with X infinite, one has: almost free (i.e.every non-trivial group element has finitely many fixed points) =⇒ strongly faithful =⇒ faithful.

An action Γ
G of a group Γ on a graph G is called homogeneous if for all ϕ ∈ P (G) (recall that d(ϕ) and r(ϕ) are supposed to be finite), there exists g ∈ Γ such that gu = ϕ(u) for all u ∈ d(ϕ).It is easy to see that Γ G is homogeneous if and only if the image of Γ in Aut(G) is dense.
We say that Γ G has infinite orbits (resp. is free, resp. is strongly faithful) if the action Γ V (G) on the set V (G) has infinite orbits i.e. all orbits are infinite (resp. is free, resp. is strongly faithful).

We say that Γ
G disconnects the finite sets if for all finite subsets F ⊂ V (G) there exists g ∈ Γ such that gF ∩ F = ∅ and, for all u, v ∈ F , gu ≁ v.Note that if Γ G disconnects the finite sets then it has infinite orbits and the converse holds when E(G) = ∅.

We say that Γ
G is non-singular if for every u ∈ V (G) and g ∈ Γ we have gu ≁ u.
The notion of a highly core-free subgroup Σ < Γ has been introduced in [FMS15]: it is a strengthening of the notion of core-free subgroup.Recall that, given a nonempty subset S ⊂ Γ, the normal core of a subgroup Σ < Γ relative to S is defined by Core S (Σ) = ∩ h∈S h −1 Σh.Then, Σ is called core-free if Core Γ (Σ) = {1}.Now, Σ is called highly core-free if, for every finite covering of Γ with non-empty sets, up to finitely many Σ-classes, there exists at least one set in the covering for which the associated normal core is trivial.More precisely, it means that, for every finite subset F ⊂ Γ, for any n ≥ 1, for any non-empty subsets S 1 , . . ., Many examples of highly core-free subgroups are given in [FMS15].Let us mention some of them bellow.
• For any countable field K, any d ≥ 2, the stabilizer in PSL d (K) of any point in P 1 (K d ) for the natural action PSL d (K) P 1 (K d ) is a highly core-free subgroup of PSL d (K).• Any finite subgroup of an icc group is highly core-free (a group is called icc if the conjugacy class of every non-trivial group element is infinite).• A finite malnormal subgroup of an infinite group is highly core-free (Σ < Γ is called malnormal if ∀g ∈ Γ − Σ one has Σ ∩ gΣg −1 = {1}).
• If Λ and Σ are non trivial groups and Σ is abelian then Σ is highly core-free in Γ = Λ * Σ.More generally, an abelian malnormal subgroup of infinite index is highly core-free.• For any infinite (commutative) field, K * < K * ⋉ K is malnormal, in particular highly core-free by the previous example.
Let us finally note that being highly core-free is a strictly stronger property than being corefree.Indeed, let us denote by S ∞ the group of finitely supported bijections of N.Then, the stabilizer of any n ∈ N in S ∞ is a core-free but not highly core-free subgroup of S ∞ .
The notion of a highly core-free subgroup Σ < Γ with respect to an action Γ X on the set X has also been introduced in [FMS15].It is defined in such a way that Σ < Γ is highly core-free if and only if it is highly core-free with respect to the left translation action Γ Γ.We will use here a similar notion for an action Γ G on a graph G. Let Σ < Γ be a subgroup and Γ G be an action on the graph G.We say that Σ is highly core-free with respect to Γ G if for every finite subset F ⊂ V (G) there exists g ∈ Γ such that gF ∩ ΣF = ∅, gu ≁ v for all u, v ∈ F , Σgu ∩ Σgv = ∅ for all u, v ∈ F with u = v and σgu ≁ gv for all u, v ∈ F and all σ ∈ Σ \ {1}.In practice, we will use the following equivalent definition (obtained by replacing F by F 1 ∪ F 2 ): for every finite subsets It is clear that if Σ is highly core-free w.r.t.Γ G then the action Γ G disconnects the finite sets.We adopt a terminology slightly different from [FMS15]: if E(G) = ∅ then Σ is highly core-free with respect to the action Γ G on the graph G in the sense explained above if and only if Σ is strongly highly core-free with respect to the action Γ V (G) on the set V (G) in the sense of [FMS15, Definition 1.7].In particular, if E(G) = ∅ and Γ G is free then Σ is highly core-free with respect to Γ G if and only if Σ is highly-core free in Γ (see [FMS15, Definition 1.1 and Lemma 1.6].Note also that the trivial subgroup is highly core-free w.r.t.Γ G if and only if the action Γ G disconnects the finite sets and, if Σ is highly core-free with respect to Γ G then every subgroup of Σ is highly core-free with respect to Γ G.In particular, if there exists a subgroup Σ which is highly core-free with respect to Γ G then the action Γ G disconnects the finite sets. In the sequel we always assume that G is a non-empty graph.The Random Extension of G is the graph G defined by V ( G) = V (G) ⊔ P f (V (G)), where P f (X) denotes the set of non-empty finite subsets of a set X and ⊔ denotes the disjoint union, and: Note that the inclusion ι : V (G) → V ( G) defines an injective and open graph homomorphism ι : G → G.
Given an action Γ G there is a natural action Γ G for which the map ι is Γ-equivariant.Indeed, we take the original action of Γ on V (G) ⊂ V ( G) and, for U ∈ P f (V (G)) ⊂ V ( G) and g ∈ Γ, we define the action of g on U by g • U := gU := {gu : u ∈ U } ∈ P f (V (G)).This defines an action of Γ on G since for all z ∈ V ( G), U ∈ P f (V (G)) and all g ∈ Γ we have It is clear that ι is Γ-equivariant.
(3).Suppose that Γ G has infinite orbits.Since ι is equivariant, it suffices to check that the orbit of any U ∈ P f (V (G)) is infinite.Suppose that there is U ∈ P f (V (G)) with a finite orbit.Then there exists g 1 , . . ., g n ∈ Γ such that for all g ∈ Γ there exists i ∈ {1, . . ., n} such that gU = g i U .Hence ∪ g∈Γ gU is a finite set since ∪ g∈Γ gU ⊂ ∪ n i=1 g i U .However, since U = ∅, there exists x ∈ U and Γx ⊂ ∪ g∈Γ gU is infinite, a contradiction.The converse is obvious.
(4).Suppose that Γ G disconnects the finite sets.Let F ⊂ V ( G) be a finite set and write F = F 1 ⊔ F 2 , where F 1 = F ∩ V (G) and F 2 = F ∩ P f (V (G)).Define the finite subset G disconnects the finite sets there exists g ∈ Γ such that g F ∩ F = ∅ and, for all u, v ∈ F , u ≁ gv.It follows gF 1 ∩ F 1 = ∅ and for all u, v ∈ F 1 , gu ≁ v.Moreover, for all U, V ∈ F 2 , gU ∩ V = ∅ hence gU = V so gF 2 ∩ F 2 = ∅.Obvisouly gF 1 ∩ F 2 = ∅ and, since gF 1 ∩ U = ∅ for all U ∈ F 2 we find that gu ≁ U for all u ∈ F 1 and all U ∈ F 2 .Hence gF ∩ F = ∅ and for all u, v ∈ F , gu ≁ v.This shows that Γ G disconnects the finite sets.The converse is obvious.
(5).Suppose Γ G is free and Γ G is not free.Then, there exists U ∈ P f (V (G)) and g ∈ Γ − {1} such that gU = U .Then g n U = U for all n ≥ 0. Since U = ∅ there exists x ∈ U and the set {g n x : n ≥ 0} is finite since it is a subset of U .Hence, there exists n ≥ 1 such that g n x = x.Since Γ G is free we have g n = 1 and Γ is not torsion free.Suppose now that Γ has torsion.
G is strongly faithful it follows that g = 1.
(7).Suppose that Σ is highly core-free w.r.t.Γ G. Let F ⊂ V ( G) be a finite set and write 1.3.Inductive limits.Let X n be a sequence of countable sets with injective maps ι n : , where [x] denotes the class of the element x ∈ X n ⊂ ⊔ n X n for the equivalence relation described above.Those injections satisfy Given a sequence of actions π n : Γ → S(X n ) of a group Γ on the set X n satisfying π n+1 (g) • ι n = ι n • π n (g) for all n ∈ N and g ∈ Γ, we define the inductive limit action π The next proposition contains some standard observations on inductive limits.Since this results are well known and very easy to check, we omit the proof.
Proposition 1.2.The following holds. (1) Let G n be a sequence of countable graphs with injective graphs homomorphisms We collect elementary observations on the inductive limit graph in the following proposition.
(1) ι ∞,n is a graph homomorphism for all n ∈ N. (2) G n for all n.The converse holds when ι n is open for all n. Proof.
(3).Suppose that Γ G ∞ disconnects the finite sets and let n ∈ N and (5).Suppose that Σ is highly core-free with respect to Γ G ∞ and let n ∈ N and gv) for all u, v ∈ F and all σ ∈ Σ \ {1}.By (1) we have gu ≁ v and σgu ≁ gv for all u, v ∈ F and all σ ∈ Σ \ {1}.Suppose now that ι n is open and Σ is highly core-free w.r.t.Γ G n for all n ∈ N. Let F ⊂ V (G ∞ ) be a finite set and take n ∈ N large enough so that F = ι ∞,n (F ′ ), where 2)) we have gu ≁ v and σgu ≁ gv for all u, v ∈ F and all σ ∈ Σ \ {1}.

The Random Graph
2.1.Definition of the Random Graph.Given a graph G and subsets U, V ⊂ V (G) we define G U,V as the induced subgraph on the subsets of vertices Note that V (G U,V ) may be empty for some subsets U and V .To ease the notations we will denote by the same symbol G U,V the induced graph on V (G U,V ) and the set of vertices V (G U,V ).Definition 2.1.We say that a graph G has property (R) if, for any disjoint finite subsets Note that a graph with property (R) is necessarily infinite.We recall the following well-known result (See e.g.[Ca97] or [Ca99]) that will be generalized later (Proposition 2.5).
Proposition 2.2.Let G 1 , G 2 be two infinite countable graphs with property (R) and A ⊂ V (G 1 ), B ⊂ V (G 2 ) be finite subsets.Any isomorphism between the induced graphs ϕ : There are many ways to construct a countable graph with property (R).Proposition 2.2 shows that a countable graph with property (R) is unique, up to isomorphism.Such a graph is denoted by R and called the Random Graph.Given an infinite countable set V of vertices, Erdős and Rényi proved [ER63] that putting (independently) an edge between any pair {u, v} of vertices with probability 1/2, the resulting graph will have property (R) with probability 1.This result motivates the name Random Graph.
Proposition 2.2 also implies that every ϕ ∈ P (R) admits an extension ϕ ∈ Aut(R) (i.e.ϕ| d(ϕ) = ϕ).Proposition 2.2 is also useful to show stability properties of the graph R as done in the next Proposition.
(1) For every finite subset In particular, every countable graph is isomorphic to an induced subgraph of R.
Proof.(1).Let U, V ⊂ V (R)\A be disjoint finite subsets.Apply property (R) with the disjoint finite subsets such that z ∼ u for all u ∈ U and z ≁ v for all v ∈ V .Hence the countable graph induced on V (R) \ A has property (R).
(2).Since R U,V is at most countable, it suffices to check that R U,V has property (R) and it is left to the reader.
(3).Since G is either finite and non-empty or infinite countable, the inductive limit is infinite countable and it suffices to check that it has property (R).Since ι n is open for all n ∈ N it follows from Proposition 1.3 that we may and will assume that Let U, V be two finite and disjoint subsets of the inductive limit.Let n large enough so that Then, by definition of the Random extension, we have, for all u ∈ U , (u, The last assertion also follows from Proposition 1.3 since the inclusion of G 0 = G in the inductive limit is open. Remark 2.4.The construction of Proposition 2.3, assertions (3), shows the existence of R.This construction may also be performed starting with any countable graph G, even G = ∅, and replacing, in the construction of the Random Extension, P f (X) by all the finite subsets of X (even the empty one).The resulting inductive limit is again isomorphic to R.
We shall need the following generalization of Proposition 2.2.The proof is done by using the "back-and-forth" device.
We define inductively pairwise distinct integers k n , pairwise distinct integers l n , subsets Replacing y l n+1 by an element in π 2 (Σ)y l n+1 we may and will assume that . By freeness, we may define a bijection ϕ n+1 : , for all σ ∈ Σ.By construction, it is an isomorphism between the induced graphs.Indeed if . Since ϕ n is a graph isomorphism and since both actions are non-singular (which means that there is no edges on induced subgraphs of the form π k (Σ)z) it shows that ϕ n+1 is also a graph isomorphism.By construction we also have that ϕ n+1 π 1 (σ) = π 2 (σ)ϕ n+1 for all σ ∈ Σ.
If n is odd define We may and will assume that It suffices to show that {k n : n ≥ 1} = N * .Suppose that there exists s ∈ N, s = k n for all n ≥ 1.Since the elements k n are pairwise distinct, the set {k 2n+1 : n ≥ 0} is not bounded.Hence, there exists n ∈ N such that s < k 2n+1 .By definition, we have k 2n+1 = Min{k ≥ 1 : x k / ∈ A 2n }.However we have x s / ∈ A 2n and s < k 2n+1 , a contradiction.The proof of ∪B n = V (R) is similar.
2.2.Induced action on the Random Graph.Let Γ G be an action of a group Γ on a non-empty countable graph G and consider the sequence of graphs G 0 = G and G n+1 = G n with the associated sequence of actions Γ G n .By Proposition 2.3, assertion (3), the inductive limit action defines an action of Γ on R. We call it the induced action of Γ G on R. We will show in Corollary 2.7 that many properties on the action Γ G are preserved when passing to the induced action Γ R.However, freeness is not preserved and one has to consider a weaker notion that we call property (F ).
Definition 2.6.We say that an action Γ G has property (F ) if for all finite subsets S ∈ Γ \ {1} and F ⊂ V (G), there exists x ∈ V (G) \ F such that x ≁ u for all u ∈ F and gx = x for all g ∈ S.
Note that any action with property (F ) is faithful and any free action on R has property (F ).
Proof.The assertions (1) to (6) follow directly follows from Propositions 1.1, 1.2 and 1.3.Let us prove (7).We recall that and S = {g 1 , . . ., g n } ⊂ Γ \ {1} be finite subsets.Since Γ G is strongly faithful we deduce, by Proposition 1.1 (6) and induction that Γ G N is strongly faithful for all N .Let N ∈ N be large enough so that F ⊂ V (G N ).We can use strong faithfulness (and the fact that G N is infinite) to construct, by induction, pairwise distinct vertices y 1 , . . .y n ∈ V (G N ) \ F such that y i = g j y j for all 1 ≤ i, j ≤ n.Define x := {y 1 , . . .
Corollary 2.8.Every infinite countable group Γ admits an action Γ R that is nonsingular, has property (F ) and disconnects the finite sets.If Γ is torsion-free then the action can be chosen to be moreover free.
Proof.Consider the graph G defined by V (G) = Γ and E(G) = ∅ with the action Γ G given by left multiplication which is free, has infinite orbits and hence disconnects the finite sets and is non-singular since E(G) = ∅.By Corollary 2.7 the induced action Γ R has the required properties.

The Random Extension with paramater.
We now describe a parametrized version of the induced action.As explained before, freeness is not preserved when passing to the action on the Random Extension and also when passing to the induced action.However, it is easy to compute explicitly the fixed points of any group element in the Random Extension.Now, given an action Γ G and some fixed group elements in Γ, one can modify the Random Extension by removing the fixed points of our given group elements to make them act freely on this modified version of the Random Extension.Then, the inductive limit process associated to this modified Random Extension will also produce, in some cases, the Random Graph with an action on it, for which our given group elements act freely.We call this process the parametrized Random Extension.

Let π : Γ
G be an action of the group Γ on the graph G and F ⊂ Γ be a subset.From the action π, we have a canonical action Γ G on the Random Extension.
Remark 2.9.If G is infinite, for any g ∈ Γ, the set of fixed points of g for the action on the Random Extension is either empty or of the form : In particular, when the action π is free one has Fix G (g) = ∅ whenever g has infinite order and, if g has finite order, then Fix G (g) is the set of finite unions of g -orbits.
Consider the induced subgraph on V ( G) \ {Fix G (g) : g ∈ F }. Note that if gF g −1 = F for all g ∈ Γ then, since gFix G (h) = Fix G (ghg −1 ), the induced subgraph on V ( G)\{Fix G (g) : g ∈ F } is globally Γ-invariant and we get an action of Γ on it by restriction for which the elements of F act freely by construction.However, for a general F , we cannot restrict the action and this is why we will remove more sets then the fixed points of elements of F .
Assume from now that G is a graph and l ∈ N * .We define the graph G l , the Random Extension of G with parameter l, as the induced subgraph on (2) π is non-singular if and only if π l is non-singular.
(3) π has infinite orbits if and only if π l has infinite orbits.(4) π disconnects the finite sets if and only if π l disconnects the finite sets.
(5) Let Σ < Γ be a subgroup.Σ is highly core-free w.r.t.π if and only if Σ is highly core-free w.r.t.π l .(6) If π is strongly faithful and G is infinite then π l is strongly faithful.
Proof.Assertions (1) to (6) are obvious.Let us prove (7).Since any non-trivial element of Σ acts freely, it follows from Remark 2.9 that any finite subset U ⊂ V (G) in the set of fixed points of σ ∈ Σ is a finite union of sets of the form σ x, hence its size is a multiple of the order of σ and U / ∈ V ( G |Σ| ).
We can now construct the induced action on R of the action π with paramter l.Define the sequence of graphs G 0 = G with action π 0 = π of Γ on it and, for n ≥ 0, G n+1 = (G n ) l with the action π n+1 = ( π n ) l .Consider the inductive limit G l ∞ with the inductive limit action π l ∞ : Γ G l ∞ on it.We list the properties of π l in the next Proposition.
(1).Since G is infinite it follows that G n is infinite for all n and G ∞ is infinite.Hence, it suffices to check that ∞ ) be finite subsets.Taking a larger S if necessary, we may and will assume that gcd(n, l) = 1.We repeat the proof of Corollary 2.7, assertion (7) and we get pairwise distinct vertices y 1 , . . .
. This concludes the proof.Let Σ < Γ be a finite subgroup of an infinite countable group Γ.By considering the induced action on R with parameter l = |Σ| of the free action by left multiplication Γ Γ and view Γ as a graph with no edges, we obtain the following Corollary.
Corollary 2.12.Let Σ < Γ be a finite subgroup of an infinite countable group Γ.There exists a non-singular action Γ R with property (F ) such that the action Σ R is free, the action H R of any infinite subgroup H < Γ disconnects the finite sets and, for any pair of intermediate subgroups 2.4.Homogeneous actions on the Random Graph.
(2) If N < Γ is normal then either N acts trivially or it acts homogeneously.
Proof.(1).Let F ⊂ V (R) be a finite set and write F = {u 1 , . . ., u n } where the vertices u i are pairwise distinct.We shall define inductively pairwise distinct vertices v 1 , . . .
This concludes the construction of the v i by induction.By the properties of the v i , the map ϕ : is an isomorphism between the induced subgraphs.Since Γ R is homogeneous there exists g ∈ Γ such that gu i = v i for all 1 ≤ i ≤ n.It follows that gF ∩ F = ∅ and gu ≁ u ′ for all u, u ′ ∈ F .
(2).Write N the closure of the image of N inside Aut(R).By [Tr85], the abstract group Aut(R) is simple and, since N is normal in Aut(R) one has either N = {1} or N = Aut(R).
Using the previous Proposition and arguing as in [MS13, Corollary 1.6] we obtain the following Corollary.
Corollary 2.14.If Γ ∈ H R then Γ is icc and not solvable.

Actions of amalgamated free products on the Random Graph
Let Γ 1 , Γ 2 be two countable groups with a common finite subgroup Σ and define Γ = Γ 1 * Σ Γ 2 .
Suppose that we have a faithful action Γ R and view Γ < Aut(R).
R and Σ R is free then the set O = {α ∈ Z : π α is homogeneous and faithful} is a dense G δ in Z.In particular, for every countably infinite groups Γ 1 , Γ 2 we have Γ 1 * Γ 2 ∈ H R and, for any finite highly core-free subgroup Proof.We separate the proof in two lemmas.

Lemma 3.2. If Σ
R is free and non-singular and Σ is highly core-free w.r.t.
End of the proof of the Theorem.The first part of the Theorem follows from Lemmas 3.2 and 3.3 since O = U ∩ V .The last part follows from the first part and Corollary 2.12.
We have a similar result when only one of the factors in the free product is infinite.
In particular, for every countably infinite group Γ 1 , for every finite nontrivial group Γ 2 we have Γ 1 * Γ 2 ∈ H R and, for any common finite subgroup Proof.We first prove the analogue of Lemma 3.2.

Claim. There exists pairwise distinct vertices
Proof of the Claim.We define inductively the vertices z 1 , . . ., Since R U,V is infinite and Γ 2 is finite, we may take Then, for all u ∈ Γ 2 α(F ), u ≁ z l+1 and, for all 1 End of the proof of the Lemma.Write d(ϕ) = {x 1 , . . ., x n } and define, for 1 ≤ k ≤ n, y k = ϕ(x k ).Let z 1 , . . .z n be the elements obtained by the Claim.Take h ∈ Γ 2 \ Σ.Then the sets α(ΣF ), Σz i for 1 ≤ i ≤ n, and Σhz i for 1 ≤ i ≤ n are pairwise disjoint.Moreover, u ≁ σz i , u ≁ σhz i and σ ′ z i ≁ σhz j for all u ∈ α(ΣF ), for all σ, σ ′ ∈ Σ and for all 1 ≤ i, j ≤ n.Define Σhz i ) and consider the induced graph structure on A and B. Note that ΣA = A , ΣB = B and the only vertices in A (resp.B) are the ones with extremities in ΣF (resp.α(ΣF )).Since Σ R is free, we may define a bijection γ 0 : A → B by γ 0 (u) = α(u) for u ∈ ΣF and γ 0 (σg 1 x i ) = σz i , γ 0 (σg −1 2 y i ) = σhz i for all 1 ≤ i ≤ n and for all σ ∈ Σ which is a graph isomorphism satisfying γ 0 σ = σγ 0 for all σ ∈ Σ.By Proposition 2.5, there exists an extension γ ∈ Z of γ 0 .Then γ| F = α| F and with g = g 2 hg 1 ∈ Γ we have, for all 1 End of the proof of the Theorem 3.4.The first assertion of the Theorem follows from Lemmas 3.5 and 3.3.The last part follows from the first part and Corollary 2.12, where the group Σ in Corollary 2.12 is actually our group Γ 2 , the group Σ ′ is our group Σ and the group H ′ is our group Γ 1 .

Actions of HNN extensions on the Random Graph
Let Σ < H be a finite subgroup of a countable group H and θ : Σ → H be an injective group homomorphism.Define Γ = HNN(H, Σ, θ) the HNN-extension and let t ∈ Γ be the "stable letter" i.e.Γ is the universal group generated by Σ and t with the relations tσt −1 = θ(σ) for all σ ∈ Σ.For ǫ ∈ {−1, 1}, we write Suppose that we have a faithful action Γ R and view Γ < Aut(R).Define the closed (hence Polish space) subset Z = {α ∈ Aut(R) : θ(σ) = ασα −1 for all σ ∈ Σ} ⊂ Aut(R) and note that it is non-empty (since t ∈ Z).By the universal property of Γ, for each α ∈ Z there exists a unique group homomorphism π α : Γ → Aut(R) such that In this section we prove the following result.
End of the proof of Theorem 4.1.The first assertion follows directly from Lemmas 4.2 and 4.3 since O = U ∩ V and the last part follows from the first part and Corollary 2.12.

Actions of groups acting on trees on the random graph
Let Γ be a group acting without inversion on a non-trivial tree.By [Se77], the quotient graph G can be equipped with the structure of a graph of groups (G, {Γ p } p∈V(G) , {Σ e } e∈E(G) ) where each Σ e = Σ e is isomorphic to an edge stabilizer and each Γ p is isomorphic to a vertex stabilizer and such that Γ is isomorphic to the fundamental group π 1 (Γ, G) of this graph of groups i.e., given a fixed maximal subtree T ⊂ G, the group Γ is generated by the groups Γ p for p ∈ V(G) and the edges e ∈ E(G) with the relations e = e −1 , s e (x) = er e (x)e −1 , ∀x ∈ Σ e and e = 1 ∀e ∈ E(T ), where s e : Σ e → Γ s(e) and r e = s e : Σ e → Γ r(e) are respectively the source and range group monomomorphisms.
Theorem 5.1.If Γ p is countably infinite, for all p ∈ V(G), Σ e is finite and s e (Σ e ) is highly core-free in Γ s(e) , for all e ∈ E(G), then Γ ∈ H R .
Proof.Let e 0 be one edge of G and G ′ be the graph obtained from G by removing the edges e 0 and e 0 .
Case 2: G ′ is not connected.Let G 1 and G 2 be the two connected components of G ′ such that s(e 0 ) ∈ V(G 1 ) and r(e 0 ) ∈ V(G 2 ).Bass-Serre theory implies that Γ = Γ 1 * Σe 0 Γ 2 , where Γ i is the fundamental group of our graph of groups restricted to G i , i = 1, 2, and Σ e 0 is viewed as a highly core-free subgroup of Γ 1 via the map s e 0 and as a highly core-free subgroup of Γ 2 via the map r e 0 since s e 0 (Σ e 0 ) is highly core-free in Γ s(e 0 ) and r e 0 (Σ e 0 ) is highly core-free in Γ r(e 0 ) by hypothesis.Since Γ 1 and Γ 2 are countably infinite and Σ e 0 is finite, we may apply Theorem 3.1 to conclude that Γ ∈ H R .

Actions of free groups on the random graph
Recall that P (R) denotes the set of isomorphisms of R between finite induced subgraphs d(ϕ), r(ϕ) ⊂ V (R) and note that P (R) has a natural structure of groupoid.
In this Section, we prove Theorem D. The main tool, called an elementary extension, is a refinement of the "back and forth" method used to extend any partial isomorphism ϕ ∈ P (R) to an automorphism of R.
6.1.Elementary extensions.Let Φ ⊂ P (R) and F ⊂ V (R) be finite subsets.Let γ ∈ Φ.We construct a partial isomorphism γ ∈ P (R) which extends γ as follows: first, we set Note that D and R have the same cardinality and write D = {x 1 , . . ., x m } and R = {v 1 , . . ., v m }.We then successively find vertices u 1 , . . ., u m ∈ V (R) \ K such that: • the vertex u j is adjacent to all vertices u ∈ d(γ) such that v j ∼ γ(u) and all vertices u j ′ , with j ′ < j, such that v j ′ ∼ v j ; • it is not adjacent to any other vertices in K.
Proof.(1) It is clear from the construction that K is finite, as it is a finite union of finite sets.Hence, d(γ) = K ⊔ {u 1 , . . ., u m } and r(γ) = {y 1 , . . ., y m } ⊔ K are finite.Let us now check that γ is an isomorphism between finite induced subgraphs.Notice that one has d(γ) = {x 1 , . . ., x m } ⊔ d(γ) ⊔ {u 1 , . . ., u m } and let u, x ∈ d(γ).We are going to check that u ∼ x ⇔ γ(u) ∼ γ(x) by distinguishing cases.If u and x are in the same component of the disjoint union {x 1 , . . ., x m } ⊔ d(γ) ⊔ {u 1 , . . ., u m }, the equivalence follows readily from the construction and the fact that γ again by the choice of the y i 's.Finally, if x = x i and u = u j , then γ(u j ) = v j and γ(x i ) = y i .It follows from the selection of the u j 's and y i 's that u j ∼ x i and v j ∼ y i .Hence the equivalence holds.
the result is trivial.Note that we cannot have x ∈ γ(Ω \ d(γ)) and y ∈ γ −1 (Ω \ r(γ)) because otherwise there exists i and j such that x = y i and y = u j which implies that x ≁ y.Suppose now that x ∈ γ(Ω \ d(γ)) and y ∈ Ω.Then there is some i such that x = y i but then y i ∼ y implies that y ∈ r(γ) and γ −1 (y) ∼ γ −1 (x).Since γ −1 (y) = γ −1 (y) ∈ Ω and γ −1 (x) = γ −1 (y i ) = x i ∈ Ω, we are done.The other cases are proved in the same way.6.2."Treezation" of a free group action.We denote by F k the free group on k generators a 1 , . . ., a k .Given a tuple ᾱ = (α 1 , . . ., α k ) ∈ Aut(R) k , we denote by α : F k → Aut(R) the unique group homomorphism such that α(a j ) = α j for all j.
Given a graph G, for l ≥ 2, a minimal path in G from x 1 ∈ V (G) to x l ∈ V (G) is a finite sequence of pairwise distinct vertices x 1 , . . .x l such that x i ∼ x i+1 for all 1 ≤ i ≤ l − 1.When l ≥ 3 and x 1 ∼ x l , we call it a minimal cycle.
Recall that a 1 , . . .a k are the canonical generators of F k .Proposition 6.5.Any treezation β of ᾱ satisfies the following properties.
(1) For all l ≥ 0 the minimal cycles in G l are all contained in F .
(2) The minimal cycles of G β are all in F .
(3) The minimal paths of G β with extremities in F are contained in F .
(4) If the orbits of α are infinite then the orbits of β are infinite.
Proof.(1) The result is obvious for G 0 since all the edges in G 0 have their source and range in F .Now, observe that the edges of G l are included in the edges of G l+1 with the same label and any edge of G l+1 which is not already an edge on G l has an extremity which is not in any domain or range of any β l,j for 1 ≤ j ≤ k since β l+1,j(l) is an elementary extension of β l,j(l) and for j = j(l) one has β l+1,j = β j,l .Hence, the minimal cycles in G l+1 are contained in G l .This proves the result by induction on l.
(2) It follows from (1) since any minimal cycle in G β is a minimal cycle in G l for some l ≥ 0.
(3) The proof is the same as the one of (1) and (2).
(4) It follows from (2) that the induced graph structure on the complement of F coming from G β is a forest.Hence, every x / ∈ F has an infinite β-orbit.If x ∈ F , we find, by Remark 6.3, g ∈ B 0 and 1 ≤ j ≤ n such that x ∈ d(g) and gx / ∈ d(β 0,j ) ∩ r(β 0,j ).Hence either β k,j (gx) / ∈ F or β −1 k,j (gx) / ∈ F .In both cases, we find an element w ∈ F k such that β(w)(x) / ∈ F .By the first part of the proof, the β-orbit of x is infinite.
(5) Let Ω be the orbit of F under the action β and, for l ≥ 0, Ω l be the orbit of F under the groupoid B l .Since every edge of R which is in Ω is actually in Ω l for some l, it suffices to show that, for all l ≥ 0 and all x, y ∈ Ω l with (x, y) ∈ E(R), there exists g ∈ B l and x 0 , y 0 ∈ F such that gx 0 = x and gy 0 = y.For l = 0 it is trivial since Ω 0 = F and the proof follows by induction by using assertion (4) of Proposition 6.2.
(6) The proof is a direct consequence of the following remark, which is itself a direct consequence of (2) and (3).For all x / ∈ F in the connected component of a point in F there exist a unique path in G β without backtracking x 1 , x 2 , . . ., x l such that Moreover, this path is geodesic.6.3.Proof of Theorem D. Let us fix an integer k ≥ 2. We still denote by F k the free group on k generators a 1 , . . ., a k , and by α the morphism F k → Aut(R) associated to some ᾱ ∈ Aut(R) k .Let us mention that one has ᾱ = {α(w) : w ∈ F k }.The set A = {ᾱ ∈ Aut(R) k : every α -orbit on the vertices is infinite}

Fix
an action Γ G. Since for any g ∈ Γ and any finite subset U ⊂ V (G) one has |gU | = |U |, the subgraph G l is globally Γ-invariant and we get an action π l : Γ G l by restriction.Note that for any u ∈ V (G) one has {u} ∈ V ( G l ).It is clear that the inclusions of G in G l and of G l in G are Γ-equivariant open (and injective) graph homomorphisms.Proposition 2.10.Let π : Γ G be an action and l ∈ N * .(1) π is faithful if and only if π l is faithful.
π is faithful if and only if π l Suppose that Γ and G are infinite.If π is strongly faithful then π l In particular, for any finite subgroup Σ of an infinite countable group H such that Σ ǫ < H is highly core-free for all ǫ ∈ {−1, 1}, we have HNN(H, Σ, θ) ∈ H R .
δ in Z.