Letd≥2,m≥1andω ∈Ω, and letGωbe the spinal group defined by these parameters.
Let alsoSbe the spinal generating set forGω. We now want to describe the Schreier graphs associated with the action ofGωon the boundary of the treeXN. More precisely, for every ξ ∈XN, we are interested in the graphΓξ= Sch(Gω,StabGω(ξ), S).
Notice thatXNis endowed with the shift operatorσ :XN→XN, which removes the first letter of a sequence. We say that two pointsξ, η ∈ XN arecofinal if they differ at most in finitely many letters, that is, if there exists somer ≥0such thatσr(ξ) = σr(η).
Cofinality is an equivalence relation, and we call the equivalence class ofξ itscofinality class, denotedCof(ξ).
Proposition 3.2.1. For everyξ ∈XN,Gωξ= Cof(ξ).
Proof. Becauseaonly changes the first digit of the sequence and anyb∈Beither fixes the sequence or changes the first digit after a prefix(d−1)n0, any generator changes at most one digit of the sequence. Letη ∈Gωξ, so there existsg ∈Gsuch thatη =gξ. Then,g changes at most|g|Sdigits inξ, which implies thatηandξare cofinal.
Conversely, we can check that starting fromξ and performing transformations corre-sponding to the generators (changing the first letter and changing the first letter after a specific pattern), we can obtain any pointηcofinal withξ.
Proposition 3.2.2. Letξ ∈XN. The Schreier graphΓξ, associated with the action ofGω
on the orbit ofξ, has vertex setCof(ξ)the following edges:
a
Figure 3.4: Finite Schreier graphs for the Fabrykowski-Gupta group. For clarity, only generatorsaandbare drawn. Edges labeled by their inverses are the same but reversed.
• ∀η=η0η1· · · ∈Cof(ξ),∀k= 1, . . . , d−1, there is an edge fromηto(η0+k)η1η2. . . labeled byak.
• ∀η=η0η1· · · ∈Cof(ξ),∀b∈B\ {1},
– If then-prefix ofηis(d−1)n−10for somen≥1, then there is ab-edge fromη to(d−1)n−10(ηn+k)σn+1η, wherekis such thatωn−1(b) =ak.
– Otherwise there is a loop atηlabeled byb.
Γ1
Γ2
Figure 3.5: Finite Schreier graphs for the spinal group withd= 5,m= 1andω =πN. For clarity, vertex and edge labelings have been omitted. Black edges correspond to powers ofa, colored edges correspond to powers ofb.
Proof. Proposition 3.2.1 shows that the orbit ofξis its cofinality class, and the edges follow from the definition of the action ofAandBonXN.
Definition 3.2.3. Letξ ∈ XNandη ∈Cof(ξ). We define the following subgraphs ofΓξ, for everyn≥0:
Γnη =Xnσnη={vσnη|v∈Xn},
Λnη = (d−1)n0Xσn+2η={(d−1)n0iσn+2η|i∈X}.
Notice thatΓnη hasdnvertices and containsη, for everyn≥0, Moreover, the subgraph Γnη does not depend onη0. . . ηn−1, the prefix ofηof lengthn. The subgraphs{Γnη}η∈Cof(ξ) are either disjoint or coincident, and every vertex ofΓξ belongs to exactly one of them.
Furthermore, no vertex inΓnη has outgoing edges toΓξ\Γnη except for(d−1)n−10σnηand possibly(d−1)nσnη, depending on whether it is fixed byBor not.
The subgraphΛnη has exactlydvertices, and it does not depend onη0. . . ηn+1, the prefix ofη of length n+ 2. If η is fixed by B, then it does not belong to Λnη for anyn ≥ 0.
Otherwise,ηbelongs to exactly oneΛnη, for somen≥0. Observe thatΛnη contains only B-edges. In fact,Λnη is isomorphic toΛωnfrom Proposition 3.1.3 and Figure 3.1.
Proposition 3.2.4. Letξ ∈XN,η∈Cof(ξ)andn≥1. LetΓ¯nη be the graph obtained from Γnη after removing all loops on the vertices(d−1)n−10σnη and(d−1)nσnη. Similarly, letΓ¯nbe the graph obtained fromΓnafter removing all loops on the vertices(d−1)n−10 and(d−1)n. ThenΓ¯nη andΓ¯nare isomorphic. The position ofηinΓnη depends only on its n-prefixη0. . . ηn−1.
Proof. Consider the bijection between the vertex sets given byϕ:vσnη7→v. Letv∈Xn ands∈S, and let us prove thatvσnηhas an outgoings-edge tos(v)σnηinΓnη iffvhas an outgoings-edge tos(v)inΓ¯n.
Suppose first thatvis different from(d−1)n−10and(d−1)n, so no loops are removed onv nor vσnη, and therefore v has an outgoing s-edge to s(v). For such v, we have s(vσnη) =s(v)σnη, and soϕ(s(vσnη)) =ϕ(s(v)σnη) =s(v).
Suppose now thatvis(d−1)n−10or(d−1)n. Ifs∈A, agains(vσnη) = s(v)σnη, and soϕ(s(vσnη)) =ϕ(s(v)σnη) = s(v). Becauses(v) 6=v, the edge is not a loop and hence is not removed in neither of the graphs.
Assumes∈B. Asv= (d−1)n−10orv = (d−1)n,s(v) =vand sovhas as-loop inΓn, which is removed inΓ¯n, sovhas no outgoings-edge inΓ¯n.
InΓξ, vσnηhas an outgoings-edge tos(v)s|v(σnη) = vs|v(σnη). Two things may happen. Ifs|v(σnη) =σnη, thenvσnηhas ans-loop inΓnη, which is later removed inΓ¯nη. If s|v(σnη)6=σnη, then thes-edge goes fromvσnηto a vertex inΓξ\Γnη, hence the vertex vσnη has no outgoing s-edge in the subgraphΓnη. In any of the two cases,vσnη has no outgoings-edge inΓ¯nη.
Proposition 3.2.4 is a powerful tool in order to describe the graphΓξ. The subgraphsΓnη are isomorphic toΓnup to the loops at two vertices. We will often abuse notation and say that these subgraphsΓnη are copies ofΓninΓξ.
For everyn≥ 1, we can then regardΓξas the disjoint union of copies ofΓnη, joined together through the subgraphsΛrη,r≥n−1.
Remark 3.2.5. Letξ∈XN,η∈Cof(ξ)andn≥0. We denote byBv(r)the ball centered at a vertexvof radiusr ≥0. Combining Propositions 3.2.4 and 3.1.3, we can check the following properties.
[
v∈Λnη
Bv(2n+1−1) = Γn+2η . (3.1)
[
v∈Λnη
Bv(2k−1) =Xk(d−1)n−k0Xσn+2η, 0≤k≤n. (3.2) Remark 3.2.6. After we fixd≥ 2andm ≥1, the Schreier graphsΓnfor spinal groups Gω ∈ Sd,m only differ in the labeling of the B-edges, for every n ≥ 0. Indeed, by Proposition 3.1.3 the construction depends onωonly in the graphsΛπ, forπ ∈Epi(B, A), which implies that these graphs are all isomorphic as unlabeled graphs.
As we will see later in Section 3.5, for everyd≥2andm≥1, onceξ ∈XNis fixed all graphsΓξare isomorphic as unlabeled graphs for every spinal groupGω ∈ Sd,m. This allows to extend the results about the Schreier graphs of one particular exampleGω ∈ Sd,m to all groups inSd,m, as long as these results do not depend on the edge labeling.
Similarly, if we only fixd≥2, the unlabeled Schreier graphs of spinal groups inSd,m are obtained from those for spinal groups inSd,1, but with additionalB-loops and multiple B-edges. Again this is true because the same holds for the graphsΛπ, forπ∈Epi(B, A), which are the building blocks of the Schreier graphs.
Example 3.2.7. There are two infinite Schreier graphs up to isomorphism for the infinite dihedral group (d= 2,m= 1), shown in Figure 3.6. One corresponds to the orbit of the point1N, while the other is the Schreier graph of any other pointξinXN.
a a a a
b
b b b b
1 0 00 10 100 000 010 110 1100
Γ1N
a a a
b b b b
Γξ
Figure 3.6: Infinite Schreier graphs for the infinite dihedral group. Vertex labels inΓ1N are to be concatenated with1N.
Example 3.2.8. We exhibit two examples of infinite Schreier graphs for Grigorchuk’s group (d = 2, m = 2). The graphs Γ1N and Γ0N are a one-ended line and a two-ended line, respectively. They are depicted in Figure 3.7.
a a a a
Figure 3.7: Infinite Schreier graphs for Grigorchuk’s group. Vertex labels inΓ1N orΓ0N are to be concatenated with1Nor0N, respectively.
Example 3.2.9. Figure 3.8 illustrates two infinite Schreier graphs for the Fabrykowski-Gupta group (d= 3,m= 1). As we will see later in this chapter (Theorem 3.3.1), the graphΓ2N is one-ended andΓ0N is two-ended.