2.4 Spectral properties
2.4.3 Spectral measures
Spectral measures constitute another important object of study of this thesis. LetHbe a Hilbert space andT :H → Hbe a bounded, linear self-adjoint operator. For everyv, w∈ H, there is a unique measureµv,w onsp(T)satisfying, for everyn≥0,
Z
sp(T)
xndµv,w(x) =hTnv, wi.
We shall call this measure thespectral measure ofT associated withvandw. Notice that µv,w(sp(T)) =hv, wi, soµv,w is a finite measure, but could in general take negative values.
Forw=v, the measureµv,vis positive. In that case, we call it thespectral measure ofT associated withv, and denote it simplyµv =µv,v.
IfH=`2(Γ)for some graphΓ, we may consider the Dirac delta functionδp, forp∈Γ, which takes value1onpand vanishes everywhere else. The spectral measure associated with such a function, denotedµp =µδp, is sometimes called theKesten spectral measure of p.
Spectral measures are interesting as they provide more detailed information than just the spectrum of an operator as a set. The spectral measures are supported on the spectrum.
According to a refinement of Lebesgue’s decomposition Theorem, for any v ∈ H, the spectral measureµ= µv can be decomposed into three mutually singular measuresµ = µpp+µac+µsc, whereµppis a pure point or discrete measure, consisting only of atoms with positive measure,µacis absolutely continuous with respect to the Lebesgue measure, andµscis called the singular continuous part. This allows us to define three subspacesHpp, HacandHscofH, containing vectors whose spectral measures are absolutely continuous with each ofµpp,µacandµsc, respectively. These subspaces are invariant underT, and in virtue of the spectral theorem, we haveH=Hpp⊕ Hac⊕ Hsc.
In this thesis we study how different Schreier graphs of spinal groups provide different outcomes in this decomposition for the adjacency operator. We exhibit for instance Schreier graphs where the spectral measures are all absolutely continuous with respect to the Lebesgue measure. We also show Schreier graphs with pure point spectrum, or, equivalently, for which there exists a basis of eigenfunctions. Furthermore, we present examples of Schreier graphs with nontrivial singular continuous spectrum.
Construction of the Schreier graphs
In this chapter we present the construction of Schreier graphs of spinal groups. We shall start by describing the Schreier graphs associated with the action of a spinal group on the n-th level of the treeXn, and then we will provide some tools to study the action onXN, the boundary of the tree.
We discuss the number of ends of any given Schreier graph, and we characterize when two Schreier graphs are isomorphic as marked graphs, both as edge-labeled and unlabeled graphs. We also compute the size of the isomorphism classes in terms of the uniform Bernoulli measureµonXN. Moreover, we provide a way of encoding any Schreier graph Γξas a tree with two equivalent definitions, one purely combinatorial and the other purely geometrical.
Finally, one can transfer an action of a finitely generating group on a topological space to an action on the so-called space of marked Schreier graphs. We do so for the action of any spinal group onXN, and we study the new action with respect to the original one.
3.1 Schreier graphs on X
nLetd≥2,m≥1andω ∈Ω, and letGωbe the spinal group defined by these parameters.
Let alsoSbe the spinal generating set forGω. Our goal is to describe the graphsΓnfor everyn≥1. We recall thatΓn= Sch(Gω,StabGω(v), S)for everyv∈Xn, as the action ofGωonXnis transitive. Let us start by describing the Schreier graph on the first level.
Lemma 3.1.1. The Schreier graphΓ1, associated with the action ofGω onXand spinal generating setS, is the graph with vertex setXand edges:
• ∀i∈X,∀k= 1, . . . , d−1, anak-edge fromitoi+k.
• ∀i∈X,∀b∈B, ab-loop oni.
Proof. For everyi∈X,iis mapped toi+ 1bya, so it is mapped toi+kbyak. Moreover, everyb∈B stabilizes all vertices in the first level, sob(i) =ifor everyi.
We describe the rest of graphsΓnrecursively. In order to do so, let us define the following operation on graphs.
Definition 3.1.2. LetΓbe a graph andv ∈Γbe a vertex. We setΓ˜to be a copy ofΓbut with all loops onvremoved. LetΛbe a finite graph onkvertices, withk≥2, which we call p0, . . . , pk−1.
LetΓ˜0, . . . ,Γ˜k−1bekdisjoint copies ofΓ, with˜ v0, . . . , vk−1be the vertices correspond-ing tovinΓ, respectively. We define the graph˜ Star(Λ,Γ, v)as the disjoint union ofΛand allΓ˜i, identifyingpiwithvifor everyi= 0, . . . , k−1. Namely,
Star(Λ,Γ, v) = Λt
k−1
G
i=0
Γ˜i
!
{pi=vi|i= 0, . . . , k−1}.
Observe that no edges are identified in the process. We will slightly abuse notation by callingΓithei-th copy ofΓinStar(Λ,Γ, v). If the graphsΓandΛare edge-labeled, so is Star(Λ,Γ, v). The vertex set ofStar(Λ,Γ, v)is in bijection withΓ× {0, . . . , k−1}.
TheStargraph operation is similar to the inflation of graphs used in [9] and [12], but taking loops into account. We may now use it in order to recursively describe the graphsΓn. Proposition 3.1.3. The Schreier graphΓn, associated with the action ofGω onXn and spinal generating setS, is defined by
Γ1 = Star(Θ,Ξ,∅), Γn+1= Star(Λωn−1,Γn,(d−1)n−10) ∀n≥1, whereΞ,ΘandΛπ are the following finite graphs:
• Ξis the graph with vertex set{∅}which hasdm−1loops, each labeled with a different elementb∈B\ {1}.
• Θis the graph with vertex setX, which has an edge labeled byakfromitoi+k, for everyi∈Xand for everyk= 1, . . . , d−1.
• Ifπ ∈ Epi(B, A),Λπ is the graph with vertex setX with the following edges: for everyb∈B\ {1}, letkbe such thatπ(b) = ak. Then, for everyi∈X, there is an edge fromitoi+k. Notice that this implies adding loops for allb∈Ker(π).
Some examples of these graphs can be found in Figure 3.1.
d= 2
Figure 3.1: Blocks composing the Schreier graphs of spinal groups for some values ofd≥2 andm≥1.
Proof. The claim about Γ1 is the object of Lemma 3.1.1. Now assumen ≥ 1, and let Γ0n+1 = Star(Λωn−1,Γn,(d−1)n−10). By construction, their vertex sets are bothXn+1, so let us prove that their edges are the same. Letv=v0. . . vn∈Xn+1, we will prove that its set of outgoing edges is the same in bothΓn+1andΓ0n+1. Let us callw=v0. . . vn−1 ∈Xn, so thatv=wvn.
InΓn+1, there is exactly one outgoing edge for every generators∈S fromvtos(v).
So, for everys∈S, we must prove that inΓ0n+1there is exactly one outgoings-edge fromv tos(v). Notice thats(v) =s(wvn) =s(w)vnunlessw= (d−1)n−10ands∈B.
Suppose first that w 6= (d−1)n−10. In that case, inΓn, whas an outgoing s-edge towards s(w). Moreover, asw 6= (d−1)n−10, the outgoing s-edge from v cannot go outside its copy ofΓn. Therefore,vmust have an outgoings-edge tos(w)vn, and in fact s(w)vn=s(v), again becausew6= (d−1)n−10.
Assume now thatw= (d−1)n−10. Ifs∈A,s(w)6=w, so the outgoings-edge is not a loop, and so it is not removed in the construction ofΓ0n+1, hence there is an outgoings-edge
fromvtos(w)vninΓ0n+1. Becauses∈Achanges only the first digit of any vertex, we have s(v) =s(w)vn.
Finally, supposew= (d−1)n−10ands∈B. In this cases(w) =w, so the edge inΓn is a loop and is indeed removed in the construction ofΓ0n+1. But the vertexvis identified with the vertex labeled byvninΛωn−1, which has an outgoings-edge towards the vertex labeled byvn+kinΛωn−1, whereωn−1(s) =ak. Therefore, inΓ0n+1, the vertexvhas an outgoings-edge towards the vertexw(vn+k). As it turns out,s(v) =s((d−1)n−10vn) = (d−1)n−10ωn−1(s)(vn) =w(vn+k), which completes the proof.
Remark 3.1.4. With this characterization, notice thatdiam(Γn) = 2n−1.
Let us use Proposition 3.1.3 to obtain the Schreier graphs of some notable examples of spinal groups.
Example 3.1.5. The finite Schreier graphs of the infinite dihedral group (d= 2,m = 1), depicted in Figure 3.2 are segments with labels alternating betweenaandb.
a
b b
1 0
Γ1
a a
b
b
b
11 01 00 10
Γ2
a a a a
b
b b b
b
111 011 001 101 100 000 010 110
Γ3
Figure 3.2: Finite Schreier graphs for the infinite dihedral group.
Example 3.1.6. The finite Schreier graphs for Grigorchuk’s group were already described in [4], and more generally for any spinal group withd = 2 andm = 2in [55]. Some examples of such graphs can be found in Figure 3.3.
Example 3.1.7. The simplest example withd= 3andm = 1is the Fabrykowski-Gupta group, some finite Schreier graphs of which, described in [4], are displayed in Figure 3.4.
Example 3.1.8. In order to give an example with higherd, Figure 3.5 shows some finite Schreier graphs for the spinal group withd= 5,m= 1and constant sequenceω=πN, with π(b) =a.
c a
111 011 001 101 100 000 010 110
Γ3
Figure 3.3: Finite Schreier graphs for Grigorchuk’s group.