Schreier graphs of the Basilica group

With any self-similar action of a finitely generated group $G$ of automorphisms of a regular rooted tree $T$ can be naturally associated an infinite sequence of finite graphs $\{\Gamma_n\}_{n\geq 1}$, where $\Gamma_n$ is the Schreier graph of the action of $G$ on the $n$-th level of $T$. Moreover, the action of $G$ on $\partial T$ gives rise to orbital Schreier graphs $\Gamma_{\xi}$, $\xi\in \partial T$. Denoting by $\xi_n$ the prefix of length $n$ of the infinite ray $\xi$, the rooted graph $(\Gamma_{\xi},\xi)$ is then the limit of the sequence of finite rooted graphs $\{(\Gamma_n,\xi_n)\}_{n\geq 1}$ in the sense of pointed Gromov-Hausdorff convergence. In this paper, we give a complete classification (up to isomorphism) of the limit graphs $(\Gamma_{\xi},\xi)$ associated with the Basilica group acting on the binary tree, in terms of the infinite binary sequence $\xi$.


Introduction
Schreier graphs arise naturally from the action of a group on a set. In this paper, we consider groups acting by automorphisms on rooted trees. Let T be a regular rooted tree and G < Aut(T ) be a finitely generated group of automorphisms of T . By fixing a finite set S of generators of G, we get naturally a sequence {Γ n } n≥1 of finite left Schreier graphs of the action of G on T . The vertex set of Γ n coincides with the set L n of vertices of the n-th level of T , and two vertices v, v ′ ∈ L n are connected by an edge if there exists s ∈ S such that s · v = v ′ . If G is transitive on each level, then the graphs Γ n are connected.
Similarly, the action of G on the boundary ∂T of the tree gives rise to an uncountable family of infinite orbital Schreier graphs {Γ ξ } ξ∈∂T , with V (Γ ξ ) = G · ξ. It turns out that these orbital Schreier graphs, viewed as rooted graphs (Γ ξ , ξ), are exactly the limits in the pointed Gromov-Hausdorff topology of finite Schreier graphs (Γ n , ξ n ) rooted at ξ n , the prefix of length n of ξ (see Subsection 2.2). Moreover, if we choose the root of Γ n uniformly at random for all n ≥ 1, then the uniform measure on the set {(Γ ξ , ξ); ξ ∈ ∂T } is the random weak limit (in the sense of [4]) of this sequence of random rooted graphs.
Particularly interesting examples of such sequences of Schreier graphs come from the class of self-similar, or automata, groups. In this paper, we examine in detail finite and infinite Schreier graphs of the Basilica group B acting on the binary tree in a self-similar fashion. The Basilica group is an example of a group generated by a finite automaton (see Fig. 2). It was introduced by R. Grigorchuk and A.Żuk in [9], where they show that it does not belong to the closure of the set of groups of subexponential growth under the operations of group extension and direct limit. L. Bartholdi and B. Virág further showed it to be amenable, making Basilica the first example of an amenable but not subexponentially amenable group [3]. This group has also been described by V. Nekrashevych as the iterated monodromy group of the complex polynomial z 2 − 1 (see [12]), and there exists therefore a natural way to associate to it a compact limit space homeomorphic to the well-known Basilica fractal (see Fig. 1). Finite Schreier graphs {Γ n } n≥1 form an approximating sequence of this limit space.
The aim of this work is to classify (up to isomorphism of unrooted and unlabeled graphs) all limits {Γ ξ } ξ∈∂T of the sequence {Γ n } n≥1 of finite Schreier graphs of the Basilica group. In other words, the question that we answer here is: given an infinite binary sequence ξ, describe the infinite Schreier graph Γ ξ of the action of B on the orbit of ξ. The limit graph Γ ξ is shown to have one, two or four ends, the case of one end being generic with respect to the uniform measure on ∂T (this latter fact can also be deduced from general arguments, see [6]). The main result of the paper is an explicit classification of infinite Schreier graphs of the Basilica group in terms of the boundary point ξ (Theorems 4.1, 4.6, 4.8, 4.11 and 5.4.) There exist one isomorphism class of 4-ended graphs, all belonging to the same orbit; uncountably many isomorphism classes of 2-ended graphs, each consisting of two orbits; and uncountably many isomorphism classes of 1-ended graphs, almost all of which contain uncountably many orbits. This latter aspect is particularly interesting, as it is not the case in other examples where we were able to perform a similar analysis. For example in the case of the Hanoi towers group H (3) , whose finite Schreier graphs form an approximating sequence for the Sierpiński gasket, there are uncountably many isomorphism classes of graphs with one end, but each of them contains at most 6 different orbits. We intend to pursue this investigation in a future work. It follows (see Section 5.2 below) that in the case of the Basilica group there exist uncountably many isomorphism classes of (unlabeled) limit graphs, each of measure 0. In other words, the random weak limit of the sequence of finite Schreier graphs is a continuous measure with uncountable support. A. Vershik recently raised the question about existence of continuous ergodic probability measures on the lattice L(G) of subgroups of a given group G, invariant under the action of G on L(G) by conjugation [15]. For many self-similar groups including the Basilica (more precisely, for all weakly branched groups), the pull-back of the random weak limit of labeled finite Schreier graphs to L(G) gives such a measure, concentrated on stabilizers of points in the boundary of the tree.
It would be interesting to perform a similar analysis systematically on sequences of finite Schreier graphs associated with self-similar groups, in particular with iterated monodromy groups of quadratic polynomials, whose infinite Schreier graphs are closely related to the Julia set of the polynomial (see the recent work [1] for a detailed description of this relation.) In particular, the inflation recursive process of constructing finite Schreier graphs introduced in [12] (and used in [5] to study growth of infinite Schreier graphs) might also give a possible general approach to classification of infinite Schreier graphs of contracting groups of automorphisms of rooted trees, in the spirit of the present work. For example, though there are examples of groups (the first Grigorchuk group of intermediate growth, the adding machine...) with only one or two isomorphism classes of infinite (unlabeled) Schreier graphs, it seems that those are exceptional cases and that in general the support of the random weak limit of Γ n 's is uncountable. We plan to address this question in a future work. Sequences of Schreier graphs of self-similar groups have been mainly studied in the literature from the viewpoint of spectral computations (see e.g. [2], [8], [13]...) However, in contrast with other well-known examples of self-similar groups, the spectral measure of the Laplace operator on infinite Schreier graphs of the Basilica group is not known. Explicit description of these graphs that we obtain in this paper can be useful in this aspect. Our initial motivation comes from the work [11] that aims at constructing new examples of asymptotic behaviour for the Abelian Sandpile Model (ASM). The infinite Schreier graphs of the Basilica group provide uncountable families of examples, on one hand, of 2-ended graphs non quasi-isometric to Z, on which the ASM is not critical and, on the other hand, of 1-ended graphs of quadratic growth, on which the ASM is critical.

Groups acting on rooted trees
Let T be the regular rooted tree of degree q, i.e., the rooted tree in which each vertex has q children. Given a finite alphabet X = {0, 1, . . . , q − 1} of q elements, let us denote by X n the set of words of length n in the alphabet X and put X * = n≥0 X n , where the set X 0 consists of the empty word. Moreover, we denote by X ω the set of infinite words in X. In this way, each vertex of the n-th level of the tree can be regarded as an element of X n and the set X ω can be identified with the set ∂T of infinite geodesic rays starting at the root of T . The set X ω can be equipped with the direct product topology. The basis of open sets is the collection of all cylindrical sets {wX ω ; w ∈ X * }. The space X ω is totally disconnected and homeomorphic to the Cantor set. The cylindrical sets generate a σ-algebra of Borel subsets of the space X ω . We shall denote by λ the uniform measure on X ω .
We denote by Aut(T ) the group of all automorphisms of T , i.e., the group of all bijections of the set of vertices of T preserving the incidence relation. Clearly, the root and hence the levels of the tree are preserved by any automorphism of T . A group G ≤ Aut(T ) is said to be spherically transitive if it acts transitively on each level of the tree.
The stabilizer of a vertex x ∈ T is the subgroup of G defined as Stab G (x) = {g ∈ G : g(x) = x}; the stabilizer of the n-th level L n of the tree is Stab G (L n ) = x∈Ln Stab G (x); finally, the stabilizer of a boundary point ξ ∈ X ω is Stab G (ξ) = {g ∈ G : g(ξ) = ξ}. The following properties hold.
• For all n ≥ 1, Stab G (L n ) is a normal subgroup of G of finite index.
• The subgroups Stab G (x), for x ∈ L n , are all of index q n . Moreover, if the action of G on T is spherically transitive, they are all conjugate.
• Denote by ξ n the prefix of ξ of length n. Then Stab G (ξ) = n∈N Stab G (ξ n ).
• Stab G (ξ) has infinite index in G and If g ∈ Aut(T ) and v ∈ X * , define g| v ∈ Aut(T ), called the restriction of the action of g to the subtree rooted at v, by g(vw) = g(v)g| v (w) for all v, w ∈ X * . Every subtree of T rooted at a vertex is isomorphic to T . Therefore, every automorphism g ∈ Aut(T ) induces a permutation of the vertices of the first level of the tree and q restrictions, g| 0 , ..., g| q−1 , to the subtrees rooted at the vertices of the first level. It can be written as g = τ g (g| 0 , . . . , g| q−1 ), where τ g ∈ S q describes the action of g on L 1 . In fact, Aut(T ) is isomorphic to the wreath product S q ≀ Aut(T ) where S q denotes the symmetric group on q letters, and thus Aut(T ) ∼ = ≀ ∞ i=1 S q . Definition 2.1. [12] A group G acting by automorphisms on a q-regular rooted tree T is self-similar if g| v ∈ G, ∀v ∈ X * , ∀g ∈ G.
A self-similar group G can be embedded into the wreath product S q ≀G. Consequently, an automorphism g ∈ G can be represented as g = τ g (g| 0 , . . . , g| q−1 ), where τ g ∈ S q describes the action of g on L 1 , and g| i ∈ G is the restriction of the action of g on the subtree T i rooted at the i-th vertex of the first level. So, if x ∈ X and w is a finite word in X, we have g(xw) = τ g (x)g| x (w).

Definition 2.2. [12]
A self-similar group G is self-replicating (or fractal) if it acts transitively on the first level of the tree and, for all x ∈ X, the map g → g| x from Stab G (x) to G is surjective.

Schreier graphs
Consider a finitely generated group G with a set S of generators such that id ∈ S and S = S −1 , and suppose that G acts on a set M . Then, one can consider a graph Γ(G, S, M ) with the set of vertices M , and two vertices m, m ′ joined by an edge (labeled by s) if there exists s ∈ S such that s(m) = m ′ . Clearly, if the action of G on M is transitive, then Γ(G, S, M ) is the Schreier graph Γ(G, S, Stab G (m)) of the group G with respect to the subgroup Stab G (m) for some (any) m ∈ M . If the action of G on M is not transitive, and m ∈ M , then we denote by Γ(G, S, m) the Schreier graph of the action on the G-orbit of m, and we call such a graph an orbital Schreier graph.
Suppose now that G acts spherically transitively on a rooted tree T . Then, the n-th Schreier graph of G is by definition Γ(G, S, X n ) = Γ(G, S, P n ) where P n denotes the subgroup stabilizing some word w ∈ X n . For each n ≥ 1, let π n+1 : Γ(G, S, X n+1 ) −→ Γ(G, S, X n ) be the map defined on the vertex set of Γ(G, S, X n+1 ) by π n+1 (x 1 . . . x n x n+1 ) = x 1 . . . x n . Since P n+1 ≤ P n , π n+1 induces a surjective morphism between Γ(G, S, X n+1 ) and Γ(G, S, X n ). This morphism is a graph covering of degree q.
We also consider the action of G on ∂T ≡ X ω and the orbital Schreier graph Γ(G, S, G · ξ) = Γ(G, S, P ξ ) where P ξ denotes the stabilizer of ξ for the action of G on X ω . Recall that, given a ray ξ, we denote by ξ n the prefix of ξ of length n, and that P ξ = ∩ n P n . It follows that the infinite Schreier graph Γ ξ = Γ(G, S, P ξ ) can be approximated (as a rooted graph) by finite Schreier graphs Γ n = Γ(G, S, P n ), as n → ∞, in the compact space of rooted graphs of uniformly bounded degree endowed with pointed Gromov-Hausdorff convergence ( [10], Chapter 3), provided, for example, by the following metric: given two rooted graphs (Γ 1 , v 1 ) and (Γ 2 , v 2 ), where B Γ (v, r) is the ball of radius r in Γ centered in v.

Self-similar groups and automata
An automaton is a quadruple A = (S, X, µ, ν), where S is the set of states; X is an alphabet; µ : S × X → S is the transition map; and ν : S × X → X is the output map. The automaton A is finite if S is finite and it is invertible if, for all s ∈ S, the transformation ν(s, ·) : X → X is a permutation of X. An automaton A can be represented by its Moore diagram. This is a directed labeled graph whose vertices are identified with the states of A. For every state s ∈ S and every letter x ∈ X, the diagram has an arrow from s to µ(s, x) labeled by x|ν(s, x). A natural action on the words over X is induced, so that the maps µ and ν can be extended to S × X * : where we set µ(s, ∅) = s and ν(s, ∅) = ∅, for all s ∈ S, x ∈ X and w ∈ X * . Moreover, (1) defines uniquely a map ν : S × X ω → X ω . If we fix an initial state s in an automaton A, then the transformation ν(s, ·) on the set X * ∪ X ω is defined by (1); it is denoted by A s . The image of a word x 1 x 2 . . . under A s can be easily found using the Moore diagram (see, for instance, Fig. 2 below). Consider the directed path starting at the state s with consecutive labels x 1 |y 1 , x 2 |y 2 , ...; the image of the word x 1 x 2 . . . under the transformation A s is then y 1 y 2 . . .. More generally, given an invertible automaton A = (S, X, µ, ν), one can consider the group generated by the transformations A s , for s ∈ S; this group is called the automaton group generated by A and is denoted by G(A).
A basic theorem [12] states that the action of a group G on X * ∪ X ω is self-similar if and only if G is generated by an invertible automaton.
Let A be a finite automaton with the set of states S and alphabet X and let us denote α(k, s), for k ∈ N and s ∈ S, the number of words w ∈ X k such that s| w = id. Sidki suggested to call A bounded, if the sequence α(k, s) is bounded as a function of k for each state s ∈ S. He showed in [14] that a finite invertible automaton is bounded if and only if any two non-trivial cycles in the Moore diagram of the automaton are disjoint and not connected by a directed path.
It can be shown moreover, that any group generated by a bounded automaton is contracting [7], which means, for a self-similar group, the existence of a finite set N ⊂ G such that for every g ∈ G there exists k ∈ N such that g| v ∈ N , for all words v of length greater or equal to k.

The Basilica group
The Basilica group B was introduced by R. Grigorchuk and A.Żuk [9] as the group generated by the following three-state automaton. It can be read from the automaton that the Basilica group is an automorphism group of the rooted binary tree generated by two automorphisms a and b of the following self-similar structure: where id denotes the trivial automorphism of the tree, while e and ε are respectively the identity and the nontrivial permutation in S 2 . One can verify directly that the stabilizer of the first level is the subgroup a). This implies in particular, that B is self-replicating. Moreover, since the action of B on the first level of the tree is transitive, its action is also spherically transitive. It follows from the results cited in the previous subsection that the Basilica group is a contracting self-similar group generated by a bounded automaton. We end this section by an observation about the action of the Basilica group on the boundary of the binary tree that we will need in Section 4 below. Definition 2.3. Two right-infinite sequences x 1 x 2 . . ., y 1 y 2 . . . ∈ X ω are called cofinal (denoted x 1 x 2 . . . ∼ y 1 y 2 . . .) if they differ only in finitely many letters. cofinality is an equivalence relation. The respective equivalence classes are called the cofinality classes and they are denoted by Cof (·).

The structure of finite Schreier graphs
For each n ≥ 1, let us denote by Γ n ≡ Γ(B, {a, b}, {0, 1} n ) the n-th Schreier graph of the action of the Basilica group. Recall that the edges of Γ n are labeled by the generators a,b of the group B and that its vertices are encoded by words of length n in the alphabet {0, 1}. We begin this subsection by providing some convenient substitutional rules which allow to construct Γ n 's recursively.
Proposition 3.1. The Schreier graph Γ n+1 is obtained from Γ n by applying to all subgraphs of Γ n given by single edges the following substitutional rules SR: Proof. By definition of a, we can distinguish two types of a-edges: a first type joining vertices 0w and 0b(w), and a second type which are loops at vertices 1w, with w ∈ {0, 1} n . Similarly, we distinguish the following types of b-edges: All b-edges can be partitioned in pairs. Pairs of b-edges of second and fourth type connecting vertices 11w and 01w in Γ n+1 arise from a-loops at the vertex 1w in Γ n ; moreover, there must be an a-loop based at 11w by definition of a, hence SR1.
The b-edges of first and third type are paired in chains of length 2. Such a chain connecting 00w to 10b(w) to 00b(w) in Γ n+1 arises from the a-edge joining vertices 0w and 0b(w); moreover, there must be an a-loop based at 10b(w), hence SR3.
Finally, a-edges between vertices 0u and 0v in Γ n+1 are in bijection with b-edges joining u and v in Γ n , hence SR2.
Below follow the pictures of the Schreier graphs Γ n for some first values of n ≥ 1.     Given k ∈ N, recall that a graph G is k-connected if for every proper subset Y ⊂ V (G) with |Y | < k, G \ Y is connected. A connected graph G is separable if it can be disconnected by removing only one vertex. Such a vertex is called a cut vertex. The biggest 2-connected components of a separable graph are called blocks.
The following is an easy consequence of the substitutional rules (Proposition 3.1): For every n ≥ 1, Γ n is a 4-regular separable graph whose blocks are cycles. Every vertex without a loop in Γ n is a cut vertex. Moreover, removing any cut vertex disconnects Γ n into exactly two components. Finally, the maximal length of a cycle in Γ n is 2 ⌈ n 2 ⌉ .
For the remainder of the paper, it will be convenient to consider Schreier graphs Γ n , n ≥ 1, embedded in the plane in such a way that each cycle is a regular polygon and the graph Γ n has two symmetry axes, a horizontal one and a vertical one. The center of the central cycle of Γ n coincides with the origin of the plane, which is the intersection of the axes of symmetry. By convention, the positive rotation by an angle α around the origin is performed in the counterclockwise direction.
From now on we will forget about the labels on the edges of the Schreier graphs and only consider unlabeled graphs. On one hand, our aim is to classify limits of Schreier graphs up to isomorphism of unlabeled graphs. On the other hand, in the case of the Basilica group the labeling is uniquely determined by the graph, so we are not losing any information by forgetting the labels.
Recall from Subsection 2.2 that the finite Schreier graphs {Γ n } n≥1 form a sequence of graph coverings and that we denote by π n+1 : Γ n+1 → Γ n the covering projection given by In the remaining part of this subsection we shall describe in detail the structure of the graphs Γ n and of the projections π n .
Removing v splits Γ n into two connected components, U 1 and U 2 , one of them (say U 1 ) containing the vertex 0 n . We call the decoration If v has a loop attached to it, then D(v) c is just Γ n with the corresponding loop erased.
The decoration of a given vertex v ∈ V (Γ n ) is called a k-decoration (or a decoration of height k) if it is isomorphic to the decoration of the vertex 0 k in the Schreier graph Γ k for some 1 ≤ k ≤ n. 1. Every decoration in Γ n is a k-decoration for some 1 ≤ k ≤ n.

2.
Let v ∈ V (Γ n )\{0 n }. Then, the decoration D(v) of v is a k-decoration if and only if, while reading v from the left, we first encounter 1 in the k-th position.
Proof. Part 1. Observe that, given v ∈ V (Γ n ) and its decoration D(v) ⊂ Γ n , applying SR to D(v) yields a graph isomorphic to D(0v) ⊂ Γ n+1 . Indeed, if v = 0 n , then the subgraph of Γ n+1 resulting from applying the SR to D(v) does not contain 0 n+1 . Since the vertex 0v is a cut vertex, the statement follows by definition of a decoration. On the other hand, if v = 0 n , we can repeat the previous argument replacing 0 n by 0 n−1 1. We now prove the assertion by induction on n. The decorations of the vertices of Γ 1 are both 1-decorations. Let v ∈ V (Γ n+1 ). If v = 1u for some u of length n, then it follows from the SR that v has a loop, hence the decoration of v is a 1-decoration. If v = 0u, then consider vertex u in Γ n together with its decoration D(u). By induction hypothesis, D(u) is a k-decoration for some 1 ≤ k ≤ n that is, D(u) is isomorphic to D(0 k ). Hence, applying the SR to D(u) yields a subgraph of Γ n+1 isomorphic to the subgraph of Γ k+1 obtained by applying the SR to D(0 k ). On the other hand, applying the SR to Similarly, starting from D(1u) and applying the SR l times yields D(v). Suppose now that D(v) is a k-decoration. This means that D(v) is isomorphic to D(0 k ), thus by the observations we just made, l must be equal to k − 1.
It is convenient to encode the graph Γ n by a diagram, denoted D n , constructed as follows: consider a path containing 2 ⌈ n 2 ⌉ edges. Its vertices are identified with vertices of the central cycle of Γ n so that the left-half of D n encodes the upper-half of the central cycle of Γ n , whereas the right-half of D n encodes the lower-half of the central cycle; and the vertices situated at the extremities of the path both encode the vertex 0 n . Denote by D n the diagram without these two boundary vertices. Label every vertex by the height of the decoration attached to it. Here are some examples: Proposition 3.6. The diagram D n+1 encoding Γ n+1 is obtained from D n−1 by the following recursive rule (denoted by RCR): Proof. We know by Proposition 3.4 that when we project Γ n+1 on Γ n under π n+1 , the decoration of a vertex v ∈ V (Γ n+1 )\{0 n+1 , 0 n 1} is mapped onto a decoration of the same height in Γ n . There are only two (n + 1)-decorations in Γ n+1 , namely those of the vertices 0 n+1 and 0 n 1, and they are both mapped under π n+1 to D(0 n ) c in such a way that D(0 n+1 ) superposes onto D(0 n 1) after a rotation of −180 • around the origin and then a translation (see Fig. 3).
Thus, the cycle in each of these two (n + 1)-decorations which contains respectively 0 n+1 and 0 n 1 is mapped bijectively to the central cycle of Γ n . Moreover, pairs of opposite vertices of the central cycle of Γ n+1 are identified under the projection π n+1 . Hence, the central cycle of Γ n+1 is mapped to the twice shorter cycle in Γ n containing the vertex 0 n but not the vertex 0 n−1 1 in such a way that 0 n+1 is identified with 0 n 1 and the two halves of the central cycle are superposed (see Fig. 4).
Since the decoration D(0 n ) in Γ n is bijectively mapped by π n to D(0 n−1 ) c in Γ n−1 , the statement follows.

Converging sequences of Schreier graphs
Recall that for any n ≥ 1, Γ n is a separable graph whose blocks are cycles.
Definition 3.7. A cycle-path of length k, k ∈ N ∪ {∞}, in a graph Γ is a sequence of k distinct cycles in Γ such that two consecutive cycles intersect at exactly one point.
Given w, w ′ ∈ V (Γ n ), there is a unique cycle-path P 1 , . . . , P r in Γ n of minimal length, such that w ∈ P 1 and w ′ ∈ P r . We will say that P 1 , . . . , P r joins w to w ′ . Similarly, given w ∈ V (Γ n ) and P a cycle of Γ n , there is a unique cycle-path of minimal length P 1 , . . . , P r = P such that w ∈ P 1 . We will say then that P 1 , . . . , P r joins w to P . Notation 3.8. Given an infinite word ξ and its prefix ξ n of length n, write CP ξn = {P n i } t i=1 for the unique cycle-path joining ξ n to the central cycle of Γ n . Then we have t = |CP ξn |, the length of the cycle-path, and we denote by LP ξn = {k n i } t i=1 the sequence of lengths of the cycles in the cycle-path.
In order to identify the limit lim n→∞ (Γ n , ξ n ) in the space of rooted graphs, it is important to keep track of the behaviour of the cycle-path CP ξi as i changes from n to n + 1. As we will see, there are three possibilities for the length of CP ξi : it can either increase/decrease by one, or stay stable. More precisely, we have Proposition 3.9. Let ξ ∈ {0, 1} ω . For any n ≥ 1, consider the cycle-path CP ξn (respectively CP ξn+1 ) in Γ n (respectively in Γ n+1 ). Then, the following situations may occur: 1. "Expansion", that is, |CP ξn+1 | = |CP ξn | + 1; this occurs if and only if ξ n ∈ D(0 n ) c .
3. "Stability", that is, |CP ξn+1 | = |CP ξn |; this occurs if and only if ξ n+1 is either 0 n+1 or 0 n 1. Proposition 3.9 motivates the following definition: Definition 3.10. For every word ξ ∈ {0, 1} ω , we define its index of stability as Moreover, define the SEC-sequence of ξ to be the infinite word in {S, E, C} beginning with S z ξ and such that the (n + 1)-st letter (for n ≥ z ξ ) of this word is either E or C depending on whether in the passage from Γ n to Γ n+1 we observe an expansion or a contraction between the cycle-paths CP ξn and CP ξn+1 .
Remark 3.11. We can consider the SEC-sequence of any finite prefix ξ n of ξ by taking the restriction up to the n-th term of the SEC-sequence of ξ. The occurrence of a block of t expansions beginning at position n in the SEC-sequence of a word ξ encodes the fact that |CP ξn+t−1 |− |CP ξn−1 | = t whereas the occurrence of a block of m/2 expansions-contractions EC beginning at position n means that |CP ξn+m−1 |−|CP ξn−1 | = 0. Moreover, observe (see the proof of Proposition 3.9 below) that the SEC-sequence associated with an infinite word ξ cannot contain two consecutive C's.
We will also need to control the lengths of individual cycles in the cycle-path CP ξi as i changes from n to n + 1. Namely, we have Lemma 3.12. Let ξ ∈ {0, 1} ω . For any n ≥ 1, consider the cycle-path CP ξn (respectively CP ξn+1 ) in Γ n (respectively in Γ n+1 ). Then, the sequence LP ξn+1 is obtained from LP ξn as: if n is even.
Proof of Proposition 3.9 and Lemma 3.12. Let us look at how the graph Γ n+1 is obtained from Γ n under π −1 n+1 . Recall that all Γ n 's are embedded in the plane. Consider the decoration D(0 n ) in Γ n . Take two copies of D(0 n ): the first one is rotated by −90 • around the origin, whereas the second one is rotated by +90 • around the origin. Now split the only vertex of degree 2 (which is 0 n ) into two vertices v 1 , v ′ 1 for the first copy and v 2 , v ′ 2 for the second copy. Then glue v 1 with v 2 into a vertex v and v ′ 1 with v ′ 2 into a vertex v ′ (see Fig. 5).
Finally, after a rotation by +180 • around the origin, attach to v a copy of D(0 n ) c by gluing v with the only vertex of D(0 n ) c of degree 2 and attach to v ′ a copy of D(0 n ) c (see Fig. 6). Accordingly, vertices of Γ n+1 are labeled as follows: • add a 0 at the right of each vertex of the first copy of D(0 n ) except for vertices v 1 and v ′ 1 , • add a 1 at the right of each vertex of the second copy of D(0 n ) except for vertices v 2 and v ′ 2 , • add a 0 at the right of each vertex of the left copy of D(0 n ) c , • add a 1 at the right of each vertex of the right copy of D(0 n ) c .

Infinite Schreier graphs
This section and the next one contain the main results of the paper. Our aim is to give a classification of the infinite Schreier graphs (Γ ξ , ξ), ξ ∈ ∂T , of the Basilica group acting on the boundary of the binary tree.
In this section, we show that a graph Γ ξ has one, two or four ends (Theorem 4.1), and classify the graphs with four and two ends up to isomorphism (Theorems 4. 6, 4.8, 4.11). It happens however that almost every infinite Schreier graph is one-ended (see Subsection 5.2 for the discussion), and the oneended Schreier graphs are classified up to isomorphism in Subsection 5.1 (Theorem 5.4).
Given an infinite graph Γ = (V, E), a ray is an infinite sequence of distinct vertices of Γ such that any two consecutive vertices of this sequence are adjacent in Γ. Consider an equivalence relation on the set of rays in Γ: two rays R and R ′ are equivalent if for any finite set S ⊂ V both R and R ′ have a tail in the same component of Γ\S. If two rays are equivalent, and only then, they can be linked by infinitely many disjoint paths. An end is an equivalence class of rays. Note that every infinite, locally finite graph must have at least one end.
This theorem will be proven in the next subsection. Observe that, by Proposition 2.4, E 4 consists of exactly one orbit, whereas E 1 and E 2 consist of infinitely many orbits. More precisely, our classification results proven below (Theorems 4.1, 4.6, 4.8, 5.4) imply the following: 1. There exists only one class of isomorphism of 4-ended (unrooted) infinite Schreier graphs. It contains a single orbit.
Each of these classes contains exactly two orbits.
3. There exist uncountably many classes of isomorphism of 1-ended (unrooted) infinite Schreier graphs. The isomorphism class of Γ 1 ω is a single orbit, and every other class contains uncountably many orbits.
We end this subsection with an infinite analogue of Propositions 3.2 and 3.4 about the structure of the Basilica Schreier graphs: 1. For any ξ ∈ {0, 1} ω , the infinite orbital Schreier graph Γ ξ is separable and its blocks are either cycles or single edges.
3. The decoration of η is a k-decoration if and only if while reading η from the left, we first encounter 1 in the k-th position.

Limit graph with four ends and Proof of Theorem 4.1
In this subsection, we determine for which infinite words ξ ∈ {0, 1} ω the corresponding orbital infinite Schreier graph Γ ξ has four ends, and we study the shape of these graphs.
The next lemma describes the decorations attached to C(0 n ) and C ′ (0 n ). Lemma 4.5. For n ≥ 2, consider the graph Γ n . Then, for 0 ≤ k ≤ ⌈ n−2 2 ⌉, to every vertex situated on C ′ (0 n ) at distance 2 k + l · 2 k+1 for some l ≥ 0 from 0 n is attached a (2k + 1)-decoration if n is odd and a (2k + 2)-decoration if n is even. On the other hand, for 0 ≤ k ≤ ⌊ n−2 2 ⌋, to every vertex situated on C(0 n ) at distance 2 k + l · 2 k+1 for some l ≥ 0 from 0 n is attached a (2k + 2)-decoration if n is odd and a (2k + 1)-decoration if n is even.
Proof. Observe that the cycle C ′ (0 n ) (which is the central cycle of Γ n ) and the decorations attached to the vertices of C ′ (0 n ) are encoded by the diagram D n (see Subsection 3.1). Similarly, the cycle C(0 n ) and the decorations attached to the vertices of C(0 n ) are encoded by the diagram D n−1 . Induction on n and the RCR yield the statement.
Consider now ξ ∈ V (Γ 0 ω ), ξ = 0 ω . Then ξ = 0 k 1ξ ′ for some k ≥ 0; by Proposition 3.4, a (k + 1)decoration is attached to ξ n for every n ≥ k + 1. Since every such decoration is finite, any ray must have infinitely many vertices in common with one of the four previous rays. We thus conclude that the graph (Γ 0 ω , 0 ω ) has exactly four ends.
Finally, Lemma 4.4 implies that the set of infinite words ξ such that the orbital Schreier graph Γ ξ has four ends is exactly constituted by the orbit of 0 ω . Then the result follows from Proposition 2.4.
We describe now a 4-regular infinite separable graph Γ (4) whose blocks are either cycles or single edges. The number of ends of Γ (4) is 4. It will turn out that Γ (4) is isomorphic (as an unrooted graph) to Γ ξ for each ξ ∈ E 4 . Γ (4) is constructed as follows: consider two copies, R 1 and R 2 , of the double ray whose vertices are naturally identified with the integers. Let these two double rays intersect at vertex 0. For every k ≥ 0, we define the subset of Z A k := n ∈ Z|n ≡ 2 k mod 2 k+1 .
Let CP ξ 2k be the cycle-path joining ξ 2k to the central cycle of Γ 2k and let LP ξ 2k = {k 2k i } t i=1 be as defined in Notation 3.8. By the previous observation, |CP ξ 2k+2r | = |CP ξ 2k | = t for each r ≥ 1. Moreover, it follows from Lemma 3.12, that k 2k+2r i = k 2k i for all 1 ≤ i ≤ t−1 and each r ≥ 1. Thus, we conclude that for N sufficiently large, the distance separating the vertex ξ 2n from the central cycle in Γ 2n is constant for every n > N . On the other hand, the length k 2k+2r t of the central cycle of Γ 2k+2r tends to infinity as r → ∞, so that, in the limit, the central cycle splits into two disjoint rays R and R ′ . Thus, Γ ξ has at least two ends.
By Proposition 4.3, given any η ∈ V (Γ ξ ), the height of the decoration D(η) is finite. Thus, any ray in Γ ξ must be equivalent either to R or to R ′ . Thus, Γ ξ has exactly two ends.
We show now that if ξ ∈ E 1 , then Γ ξ has one end. On one hand, the SEC-sequence associated with ξ cannot contain two consecutive C's (see Remark 3.11). On the other hand, if ξ n belongs to the right part of Γ n and n ′ is the first index greater than n such that ξ n ′ belongs again to the right part of Γ n ′ , then |CP ξ n ′ | = |CP ξn | + 1.
Lemma 4.7. For any ξ ∈ {0, 1} ω , consider the subgraph of (Γ ξ , ξ), (CP ξ , ξ) := lim n→∞ (CP ξn , ξ n ). Then CP ξ is a cycle-path of infinite length, CP ξ = {P i } i≥1 , if and only if ξ ∈ E 1 . In this case, for any n ≥ 1 such that ξ n ends with a 1, the subgraph {P n Proof of the lemma. Observe that, CP ξ is a cycle-path of infinite length if and only if there exist infinitely many indices n such that ξ n is in the right part of Γ n ; otherwise, we would observe from some point in the SEC-sequence associated with ξ an alternating sequence of expansions E and contractions C, so that |CP ξn | would be bounded as n → ∞.
It is easy to check that E 1 coincides with the set of words ξ ∈ {0, 1} ω in which the number of subwords of type 10 k 1, with k ≥ 0 even, is infinite. Hence, if ξ ∈ E 1 , there are infinitely many indices n such that ξ n belongs to the right part of the graph Γ n . Conversely, suppose that ξ / ∈ E 1 , so that the number of subwords in ξ of type 10 k 1, with k ≥ 0 even, must be finite. Thus, the SEC-sequence associated with ξ contains from some point an alternating sequence of expansions E and contractions C. It follows that CP ξ is not a cycle-path of infinite length (it consists of a finite cycle-path joining ξ to a double-ray).
The second part of the lemma follows from the proof of Proposition 3.9.
End of Proof of Theorem 4.1. Let R be a ray contained in CP ξ . Each cycle P i ∈ CP ξ is finite and, by Proposition 4.3, there is a finite decoration attached to each of its vertices. Hence, any other ray R ′ in Γ ξ must have infinitely many vertices in common with R. We thus conclude that Γ ξ has one end.

Limit graphs with two ends
Given ξ ∈ E 2 , let w be the longest prefix of ξ ending by a subword of type 10 k 1, with k even (see Proof of Theorem 4.1, Parts 2 and 3 ). If {α i } ≡ 0, then we set w to be the empty word; if {β i } ≡ 0, then w is the prefix of ξ of length 1. Thus, we can write ξ = w0x 1 0x 2 . . ., x i ∈ {0, 1}. Introduce ξ = w0x 1 0x 2 . . ., where x i = 1 − x i for every i ≥ 1. Note that ξ ∈ E 2 if and only ifξ ∈ E 2 .
Theorem 4.8. Let ξ, η ∈ E 2 . Then Γ ξ and Γ η are isomorphic as unrooted graphs if and only if either ξ ∼ η or ξ ∼ η. In particular, each isomorphism class of 2-ended graphs consists of exactly two orbits.
Proof. ⇐: If ξ ∼ η, then Γ ξ = Γ η by Proposition 2.4. Suppose now that η ∼ ξ. This implies that there exist prefixes v, v, v with |v| = | v| = | v| = n 0 and where v ends by a 1, such that where z i = 1 − z i for each i ≥ 1. It follows from the proof of Proposition 3.9 that ξ n0+2 belongs to D(0 n0+2 ) c ⊂ Γ n0+2 . If we encode Γ n0+2 by the diagram D n0+2 (see Subsection 3.1), let u n0+2 be the vertex of D n0+2 to which is attached the decoration containing ξ n0+2 . Let u ′ n0+2 ∈ D n0+2 be the symmetric vertex with respect to the middle vertex of D n0+2 so that the decorations attached to u n0+2 and u ′ n0+2 have the same height (observe that the vertices of D n are labeled symmetrically with respect to the middle vertex labeled by n.) Finally, let ξ n0+2 ∈ D(u ′ n0+2 ) be symmetric to ξ n0+2 as shown in Fig.  8.
⇒: Let η, ξ ∈ E 2 and suppose that ξ ≁ η and ξ ≁ η. For the sake of contradiction, suppose that there exists an isomorphism ϕ : Γ ξ → Γ η of unrooted graphs. We will show that there exists a radius R such that the balls B Γ ξ (ξ, R) and B Γη (ϕ(ξ), R) are not isomorphic which will yield a contradiction.
By Proposition 2.4, ϕ(ξ) ∼ η. There exist finite words u,û such that and |u| = |û|. If not, then ξ and ϕ(ξ) belong to infinite graphs whose decorations have different parity, which is a contradiction. Indeed, depending on whether it is the sequence {α i } or {β j } that takes only finitely many 1's, the corresponding infinite Schreier graph has decorations of only even or only odd height.
Since η ≁ ξ, η ≁ξ but η ∼ ϕ(ξ), there exist infinitely many indices i such that x i = y i and x i+1 = y i+1 . We can thus suppose without loss of generality that u andû both end with a 1. Moreover, arguments as in Proof of Proposition 3.9 imply that there exists an infinite subsequence {n k } k≥0 ⊂ N such that, for any k ≥ 0, ξ n k and ϕ(ξ) n k belong to decorations attached to vertices in different halves of D n k , while ξ n k +2 and ϕ(ξ) n k +2 belong to decorations attached to distinct vertices v n k +2 and v ′ n k +2 in the same half of D n k +2 . Thus d(v n k +2 , w n k +2 ) = d(v ′ n k +2 , w n k +2 ) where w n k +2 denotes the vertex of D n k +2 encoding 0 n k +1 1 (see Fig. 9).
Suppose without loss of generality, that d Γn k +2 (v n k +2 , w n k +2 ) < d Γn k +2 (v ′ n k +2 , w n k +2 ). By RCR, for each k ≥ 0, the minimal distance between v n k +2 (respectively v ′ n k +2 ) and a vertex in D n k +2 labeled by n 0 + 2 is constant. On the other hand, since ϕ(ξ) ∈ E 2 , the distance between v n k +2 (respectively v ′ n k +2 ) and the boundary vertices of D n k +2 tends to infinity as k → ∞.
Thus, for any sufficiently large k, there exists a radius R(n 0 ) defined as the minimum over all radii r such that the ball B Γn k +2 (ξ n k +2 , r) contains a (n 0 + 2)-decoration attached to a vertex of D n k +2 , and such that both B Γn k +2 (ξ n k +2 , r) and B Γn k +2 (ϕ(ξ) n k +2 , r) do not contain any boundary vertices of D n k +2 .
Hence, there exists N (R(n 0 )) > 1 such that for all n k ≥ N (R(n 0 )), we have For ξ ∈ E 2 , we will now construct explicitly a 4-regular, 2-ended graph Γ(ξ) that we will show to be isomorphic (as unrooted graph) to Γ ξ (see Theorem 4.11 below). Recall from Theorem 4.1 that E 2 = {α 1 β 1 α 2 β 2 ... | α i , β j ∈ {0, 1} such that either {α i } or {β j } has finitely many 1's}. Observe that in the case where the sequence {α i } (respectively {β j }) takes the value 1 a finite number of times, only decorations of even (respectively odd) height will appear in the limit graph. We can thus partition the set E 2 = E 2,even ⊔ E 2,odd . We will only consider the case of ξ ∈ E 2,even , (the other case can be treated similarly.) Suppose therefore that ξ = α 1 β 1 α 2 β 2 α 3 β 3 . . . ∈ E 2,even , so that {α i } i≥1 contains finitely many 1's and define the graph Γ(ξ) associated with ξ. Proposition 4.9. Consider the following subsets of Z: Construct Γ(ξ) as a straight line with integer vertices with, for each k ≥ 0, a (2k + 2)-decoration attached by its unique vertex of degree 2 to every vertex corresponding to an integer in A ′ k . The constructed graph Γ(ξ) is well defined.
Then k≥1 L k is a partition of 2Z + 1.
Proof. We first prove that L k ∩ L h = ∅ if k = h. Suppose that h > k and let s, t ∈ Z be such that We prove now that 2Z + 1 ⊂ k≥1 L k by showing that for any n ∈ Z odd, there exist k ≥ 1 and t ∈ Z such that Let p 1 := min{i ≥ 1 : x i = 1}. If n + 1 = 0, then a solution of (3) is k = p 1 and t = 0. Suppose that n + 1 = 0 and write n + 1 = 2 u1 m 1 , with u 1 ≥ 1 and m 1 odd. If p 1 = u 1 , a solution of (3) is Let p 2 := min{i ≥ 1 : x p1+i = 1}. If m 1 + 1 = 0, then a solution of (4) is k = p 1 + p 2 and t = 0. Suppose that m 1 + 1 = 0 and write m 1 + 1 = 2 u2 m 2 , with u 2 ≥ 1 and m 2 odd. If p 2 = u 2 , then setting k = p 1 + min{p 2 , u 2 }, one can check that (4) has a solution. If p 2 = u 2 , then necessarily k > p 1 + p 2 and (4) becomes We iterate the above argument: at the s + 1 step, let p s := min{i ≥ 1 : x s−1 r=1 pr +i = 1}; if m s + 1 = 0, set k = s r=1 p r and t = 0; if m s + 1 = 0, write m s + 1 = 2 us+1 m s+1 , with u s+1 ≥ 1 and m s+1 odd. If p s = u s , one can check as above that (3) has a solution. Clearly, the sequence {|m s + 1|} s≥1 is strictly decreasing. If n < 0, then there exists s 0 such that m s0 = −1 and (3) has a solution. If n > 0, then there exists s 0 such that m s = 1 for all s ≥ s 0 . In the latter case, u s = 1 for all s > s 0 . On the other hand, there are only finitely many indices s such that p s = u s = 1. Otherwise, there would exist i 0 such that x i = 1 for every i > i 0 which would contradict the hypothesis on the sequence {x i } ≥1 . ξ ξ Fig. 10. A finite part of Γ(ξ).
Proof. We again only consider the case where ξ ∈ E 2,even (the case ξ ∈ E 2,odd can be treated analogously). For any t ≥ 1, consider the diagram D 2t encoding Γ 2t . Consider alsoξ := 010β 2 0β 3 . . . . Note thatξ and ξ are cofinal,ξ ∼ ξ, so that, by Proposition 2.4, Γ ξ = Γξ. Arguments as in Proof of Proposition 3.9 imply thatξ 2t ∈ V (Γ 2t ) is situated on the central cycle of this graph, and therefore can be identified with a vertex of D 2t (also denoted byξ 2t ). Let v 2t be the middle vertex in D 2t ; it is labeled by 2t (see the figures below). It is convenient to embed the diagram D 2t in Z, so that every vertex of D 2t has a coordinate (in addition to its label). Suppose that we embed D 2t in Z so that the coordinate ofξ 2t is 0. Claim: . We prove the claim by induction on t. For t = 2, v 4 has coordinate 1 if β 2 = 0, while it has coordinate −1 if β 2 = 1, and the claim holds.
Suppose that the assertion is true for t − 1 and suppose that β t−1 = 0 (the case β t−1 = 1 is treated similarly). The diagram D 2t−2 iŝ Hence, recalling that the length of D 2t is 2 t , the coordinate of v 2t is On the other hand, if β t = 1, the diagram D 2t is v 2tξ 2t Thus, in this case, the coordinate of v 2t is and the claim is proven.

Classification of infinite Schreier graphs with one end
It follows from Theorem 4.1 that the set of infinite words whose associated infinite orbital Schreier graph has one end is and {β j } j≥1 both contain infinitely many 1's}.
We prove in this subsection Part 3. of Corollary 4.2 by classifying all limit graphs Γ ξ , ξ ∈ E 1 , up to isomorphism (of unrooted graphs). We begin with the following lemma: Proof. Let R 1 ⊂ {P i } i≥1 and R 2 ⊂ {Q j } j≥1 be two rays. Since Γ ξ has one end, R 1 and R 2 are equivalent. Suppose that {P i } i≥1 and {Q j } j≥1 are disjoint. Since Γ ξ is separable with blocks either cycles or single edges, there exists a cut vertex which separates {P i } i≥1 from {Q j } j≥1 ; this contradicts the fact that R 1 and R 2 are equivalent.
If there is only a finite number of vertices belonging both to {P i } i≥1 and {Q j } j≥1 , then there are integers ) is a finite subset of vertices which separates {P i } i≥I from {Q j } j≥J which contradicts the fact that R 1 and R 2 are equivalent.
Suppose that |{P i } i≥1 ∩ {Q j } j≥1 | = ∞ and consider the minimal i 0 , j 0 such that P i0 and Q j0 have a vertex v in common. Let w ∈ P i0+k ∩ Q j0+k ′ for some k, k ′ ≥ 0 be at minimal distance from v. Then, i=i0 and {Q j } j0+k ′ j=j0 must coincide. Indeed, suppose that this is not the case: if k = k ′ = 0, then there are four disjoint paths joining v to w which contradicts Part 1. of Proposition 4.3. If k = 0 or k ′ = 0, then there are three disjoint paths joining v to w which is again a contradiction. Finally, if k, k ′ = 0, then there are two disjoint paths joining v to w which again contradicts Proposition 4.3.
Proposition 5.2. An element ξ ∈ {0, 1} ω , ξ not cofinal to 1 ω , belongs to E 1 if and only if there exists a unique triple (k, {m l } l≥0 , {t l } l≥0 ) where k ≥ 1 and m 0 ≥ 0 are integers (and m 0 is even); t 0 = 0; and {m l } l≥1 , {t l } l≥1 are sequences of strictly positive integers (and the m l 's are even), such that ξ can be written as with x j i ∈ {0, 1} for all i, j. If ξ = w1 ω for some w ∈ X * , then there exists a unique triple (k, {m l } l0 l=0 , {t l } l0 l=0 ) where k ≥ 1 and m 0 ≥ 0 are integers (and m 0 is even); t 0 = 0; and {m l } l0 l=1 , {t l } l0 l=1 are finite sequences of strictly positive integers (and the m l 's are even), such that ξ can be written as Corollary 5.3. Observe that the index of stability of ξ is equal to k + m 0 in notation of (6). Hence, the SEC-sequence corresponding to ξ ∈ E 1 , ξ = w1 ω , is while the SEC-sequence corresponding to ξ = w1 ω is In order to prove Theorem 5.4, we need to study in more detail the geometry of the limit graphs Γ ξ . It turns out that it is determined by the infinite cycle-path CP ξ in Γ ξ . In Propositions 5.6 and 5.8 below we describe these infinite cycle-paths completely by listing the lengths of the cycles and finding their relative position, in terms of the infinite ray ξ ∈ E 1 represented in the canonical form (6).
Definition 5.5. Given ξ ∈ E 1 , we associate with it the sequence of integers a i = a ξ i , i ≥ 1, defined as follows: • if ξ = 1 ω , then a i := i for all i ≥ 1; • if ξ = 1 ω , then Proposition 5.2 provides a triple (k, {m l }, {t l }) associated with ξ. Define for all Proposition 5.6. Given ξ ∈ E 1 , let CP ξ = {P i } i≥1 be the infinite cycle-path associated with ξ. Then, Proof. By induction on i: if i = 1, then by definition, a 1 = k + m 0 , that is the index of stability of ξ (see (2) and Corollary 5.3). In other words, the prefix ξ k+m0 of ξ is situated on the central cycle of Γ k+m0 . Thus, CP ξ k+m 0 is constituted of a single cycle of length 2 ⌈ k+m 0 2 ⌉ = 2 ⌈ a 1 2 ⌉ . Moreover, since ξ a1+1 = ξ a1 1, then this cycle is the first one of the cycle-path CP ξ (see Remark 3.11).
Suppose that the statement is true for i; the cycle P i of CP ξ is then isomorphic to the central cycle of Γ ai . Consider the prefix ξ ai of ξ as well as the cycle-path CP ξa i in Γ ai . Let j ≥ 1, 0 ≤ s < t j be such that T j−1 + s + 1 = i so that a i = a Tj−1+s+1 . We distinguish two cases: • if s ≤ t j − 2, then a i+1 = (a Tj−1+s+2 =) k + M j−1 + T j−1 + s + 1 = a i + 1. On one hand, ξ ai+1 = ξ ai 1, so that the central cycle of Γ ai is isomorphic to the penultimate cycle of the cyclepath CP ξa i +1 in Γ ai+1 . On the other hand, ξ ai+2 = ξ ai+1 1 which implies that the central cycle of Γ ai+1 is isomorphic to the penultimate cycle of the cycle-path CP ξa i +2 (see Remark 3.11). Hence, by induction hypothesis, the penultimate cycle of CP ξa i +2 is the (i + 1)-st cycle of CP ξa i +2 which is isomorphic (by Lemma 4.7) to the (i + 1)-st cycle of CP ξ . Its length is • if s = t j −1, then a i+1 = (a Tj−1+s+2 =) k+M j−1 +T j +m j = a i +1+m j . Moreover, ξ ai+1+1 = ξ ai+1 1. Thus, as in the previous case, the last cycle of CP ξa i+1 is isomorphic to the (i + 1)-st cycle of CP ξ . By induction hypothesis, its length is if a i is odd.
It follows that α i = π 2 = π 2 1 + 1 h=1 π 2 m d −h · 0. Suppose now that the assertion is true for m d − 2; consider the graph Γ ai−2 and For CP ξa i −2 ⊂ Γ ai−2 , define the angle α ′ as shown on Fig. 17. Then α ′ =ᾱ i whereᾱ i is the ith angle in the infinite cycle-path (as in Definition 5.7) for anyξ ∈ E 1 withξ ai−1 = ξ ai−2 1 (i.e., a i −ā i−1 = (m d − 2) + 1 = m d − 1.) Therefore we can apply the induction hypothesis to α ′ , and we have The following picture explains how α i is computed from α ′ . The two cases correspond to x d m d /2 being 0 or 1.
Hence, there are no indices i, j such that a ξ i+n = a η j+n for all n ≥ 0. Proposition 5.6 together with Lemma 5.1 imply that the graphs Γ ξ and Γ η are not isomorphic.
Part 2. Let ξ = v1 ω and η = w1 ω and suppose that there exist r, s such that (a), (b), and (c) hold. It follows from Definition 5.5, that there exist i, j such that a ξ i+n = a η j+n , for all n ≥ 0. Writing Γ ξ ⊃ CP ξ = {P h } h≥1 and Γ η ⊃ CP η = {P ′ h } h≥1 , let {v i } = P i ∩ P i+1 and similarly {v ′ j } = P ′ j ∩ P ′ j+1 . Consider the decoration X i (respectively X ′ j ) attached to v i (respectively v ′ j ). Since a ξ i = a η j , these decorations are isomorphic via an isomorphism φ 0 : X i −→ X ′ j . We can extend the isomorphism φ 0 : . Indeed, by Proposition 5.8 and Lemma 5.9, . Note that decorations of v i+1 and v ′ j+1 are both (a ξ i+1 + 1)decorations which means that they are both isomorphic to the decoration D(0 a ξ i+1 +1 ) of 0 a ξ i+1 +1 in Γ a ξ i+1 +1 . This decoration is encoded by the diagramD a ξ i+1 where boundary vertices correspond to v i+1 and v ′ j+1 respectively. By Proposition 3.6, any two vertices situated at the same distance from the boundary vertices of the diagram have isomorphic neighbourhoods.
Repeating the above argument, one can construct a sequence of isomorphisms {φ n } n≥0 , φ n : X i+n −→ X ′ j+n , such that for every n ′ > n, the restriction of φ n ′ to X i+n is equal to φ n . Thus, there is a welldefined isomorphism φ : Γ ξ −→ Γ η given by φ := lim n→∞ φ n .
Conversely, if we suppose that the conditions (a), (b), and (c) are not all satisfied, then three possibilities may occur.
• There are no indices i, j such that a ξ i+n = a η j+n for all n ≥ 0. By Proposition 5.6 and Lemma 5.1, the graphs Γ ξ and Γ η are not isomorphic. ) determines how P i−1 is attached to P i . In particular, the choices x d h and 1 − x d h for each h = 1, . . . , m d 2 correspond to vertices {v i−1 } = P i−1 ∩ P i and {v ′ i−1 } = P i−1 ∩ P i situated at the same distance from v i . On the other hand, an isomorphism φ between Γ ξ and Γ η must map v i+n onto v ′ j+n for each n ≥ 0. This is a contradiction.

Random weak limit of {Γ n } n≥1
Given a sequence of finite connected graphs {Γ n } n≥1 , one can consider it as a sequence of random rooted graphs {(Γ n , v n )} n≥1 by choosing for each n ≥ 1 a root v n ∈ V (Γ n ) uniformly at random. Then, one says that (Γ, v) is the random weak limit (or distributional limit) of the sequence {(Γ n , v n )} n≥1 as n → ∞ if, for every r > 0 and for every finite rooted graph (H, o), the probability that (H, o) is isomorphic to the ball in Γ n centered in ξ n and of radius r converges to the probability that (H, o) is isomorphic to the ball in Γ centered in ξ and of the same radius. This is equivalent to say that the law of (Γ n , v n ) weakly converges to the law of (Γ, v) as probability measures on the space X of connected rooted graphs (see Subsection 2.2).
If {(Γ n , ξ n )} n≥1 are the Schreier graphs of a spherically transitive action of a finitely generated group of automorphisms of a regular rooted tree T ; and if the root ξ n is chosen uniformly at random in each Γ n , then the random weak limit of this sequence is the uniform measure on the set {(Γ ξ , ξ), ξ ∈ ∂T } of rooted orbital Schreier graphs of the group action on ∂T .
Observe that, since the action is spherically transitive, it is ergodic on ∂T . This implies that almost all infinite Schreier graphs of a given group have the same number of ends. We discuss this typical value of the number of ends in more detail in the forthcoming paper [6]. Here, we show that in the case of the Basilica group, the set {(Γ ξ , ξ), ξ ∈ E 1 } is of full measure. Moreover, we also show that if we partition {(Γ ξ , ξ), ξ ∈ E 1 } into classes of isomorphisms of unrooted graphs, then each isomorphism class is of measure 0.
So, we consider again the Basilica group B acting by automorphisms on the binary tree, and we denote by Γ ξ , ξ ∈ {0, 1} ω , infinite Schreier graphs of the induced action of B on the boundary of the binary tree identified with the set of infinite binary words and equipped with the uniform measure λ. Recall that E 4 is the set of infinite words defining Schreier graphs with four ends, E 2,even and E 2,odd are the sets defining Schreier graphs with two ends and, respectively, with odd or even decorations. Finally, E 1 denotes the set of infinite words defining Schreier graphs with one end.