Dependence on the generating set

All the results discussed so far in this chapter concerned spinal groups with the spinal generating set S = (A∪B)\ {1}. One might also wonder what are the spectra like if we consider different generating sets, for instance minimal ones. Some progress has been achieved, for example in [38] the authors characterize whether the spectra of the weighted Markov operator on the Schreier graphs of Grigorchuk’s group are either a union of intervals or a Cantor set, depending on the weights on its usual generators.

For spinal groups acting on the binary tree (d= 2), the infinite Schreier graphsΓ_ξhave linear shape. The Schreier graphs of a minimal generating set can then be obtained by erasing double edges in the Schreier graphΓ_ξcorresponding to the spinal generatorsS. This can

be translated into considering a Markov operator onΓ_ξ with non-uniform distribution of probabilities onS. The spectra of such anisotropic Markov operators were studied in [39].

Their analysis implies the next result. As in Section 4.3,M_ξdenotes the Markov operator on the Schreier graphΓ_ξwith respect to the spinal generating setS, forξ ∈X^N, andM_Gω denotes the Markov operator on the Cayley graph also with respect to the spinal generating setS. For other generating setsT, we will explicitly writeΓ^T_ξ,M_ξ^T andM_G^T

ω.

Proposition 4.4.1. LetGωbe a spinal group withd= 2,m≥2andω∈Ωd,m, and letM_ξ^T be the Markov operator on the graphΓ^T_ξ, with generating setT ⊂S. Forπ ∈Epi(B, A), we defineqπ =|T ∩B\Ker(π)|. Two cases may occur:

• If the numbersq_π are all equal overπ ∈Epi(B, A)appearing infinitely often inω, thensp(M_ξ^T)is a union of intervals.

• Otherwise,sp(M_ξ^T)is a Cantor set of zero Lebesgue measure.

Proof. First notice thata∈T, or elseT would generate a finite group. We observe that we can relabel the vertices inΓ^T_ξ byZin such a way that the number of edges between them is the following:

• There is onea-edge between any vertexv∈2Zandv+ 1.

• There are|T∩B\Ker(ω0)|=qω0 edges, between any vertexv∈4Z+ 1andv+ 1, and|T∩Ker(ω₀)|loops on each ofv,v+ 1.

• There are|T∩B\Ker(ω₁)|=q_ω1 edges between any vertexv∈8Z+ 3andv+ 1, and|T∩Ker(ω1)|loops on each ofv,v+ 1.

...

• In general, for everyi≥ 0, there are|T ∩B\Ker(ωi)|edges between any vertex v ∈2ⁱ⁺²Z+ 2ⁱ⁺¹−1andv+ 1, and|T∩Ker(ωi)|loops on each ofv,v+ 1.

Hence, the simple random walk onΓ^T_ξ is given by a weighted random walk onZ, defined by the following probabilities:

• Probability of _|T¹_| of transitioning between any vertexv∈2Zandv+ 1.

• For everyi≥0, probability of^q_|T^ωi_| of transitioning between any vertexv∈2ⁱ⁺²Z+ 2ⁱ⁺¹−1andv+ 1.

• For everyi≥0, probability of1−^q_|T^ωi_|−_|T¹_|of staying at any vertexv∈2ⁱ⁺²Z+2ⁱ⁺¹−1 orv+ 1.

These probabilities follow a periodic pattern if and only if the numbersq_π are all equal for everyπ ∈Epi(B, A)occurring infinitely often inω. If this is not the case, we may use Corollary 7.2 in [39] to obtain thatsp(M_ξ^T)is a Cantor set of Lebesgue measure zero.

Suppose now thatqπ are all equal for everyπ ∈Epi(B, A)occurring infinitely often in ω, so the probabilities are periodic onZ, letlbe that period. We may computesp(M_ξ^T)using Floquet-Bloch theory. To do so, we first compute the spectrum of a fundamental domain, parametrized byk∈[−π, π], with some boundary conditions. This gives a set of eigenvalues {x₁(k), . . . , x_l(k)}, which are the roots of a polynomial of degreel. These polynomials only depend oncos(k). Nowsp(M_ξ^T)is just the union of these sets of eigenvalues for all k∈[−π, π]. Since the roots will vary continuously ascos(k)∈[−1,1], this union will be a union of at least one and at mostlintervals.

This result provides, for any generating set T ⊂ S, a characterization of the type of spectrum on the Schreier graphs in terms of the numbersqπ. We already know from Theorem 4.1.10 that the second option in Proposition 4.4.1 is realized whenT =S, the spinal generating set. The following result states that, except one degenerate example corresponding toGω =D∞(d= 2,m = 1), every spinal group on the binary tree has a generating set which gives a Cantor spectrum.

Corollary 4.4.2. For every spinal groupGωwithd= 2,m≥2andω∈Ωd,mthere exists a minimal generating setT ⊂Sfor whichsp(M_ξ^T)is a Cantor set of Lebesgue measure zero.

Proof. Letπ, π⁰ ∈Epi(B, A)be two different epimorphisms occurring infinitely often in ω. Recall thatBis a vector space overZ/2Z, and letK = Ker(π)andK⁰ = Ker(π⁰). We know that[B :K] = [B :K⁰] = 2, and[K :K∩K⁰] = 2becauseπ⁰ surjectsKontoA with kernelK∩K⁰, sinceK 6=K⁰. Hence, we have[B :K∩K⁰] = 4. In particular, we can choosem−2elementsx1, . . . , xm−2 ∈ K∩K⁰ which generateK∩K⁰. Moreover, we can choose elementsy∈K\K⁰andy⁰ ∈K⁰\Kto complete the generating set to one ofKorK⁰, respectively, and such that{x₁, . . . , xm−2, y, y⁰}generateB.

If we now defineT ={a, x₁, x2, . . . , xm−2, y, yy⁰} ⊂S, it is clear that it is a minimal generating set forG_ω, since|T|=m+1. Moreover, we haveq_π =|T∩B\K|=|{yy⁰}|= 1 andq_π⁰ =|T∩B\K⁰|=|{y, yy⁰}|= 2. By Proposition 4.4.1,sp(M_ξ^T)is a Cantor set of Lebesgue measure zero.

We can also find a condition on the generating setT ⊂Sunder which the spectrum on the Schreier graphs is one interval, for certain spinal groupsGω.

Proposition 4.4.3. Let G_ω be a spinal group withd = 2, m ≥ 2 and ω ∈ Ω_2,m, with generating setT ⊂S. Ifqωi = 1for everyi≥0, thensp(M_ξ^T)is the interval[1−_|T⁴_|,1].

Proof. In the proof of Proposition 4.4.1, we established that the simple random walk onΓ^T_ξ is given by a weighted random walk onZ. Sinceqωi = 1for everyi≥0, the probabilities are reduced to:

• Probability _|T¹_|of transitioning between any vertexv∈Zandv+ 1.

• Probability1−_|T²_|of staying at any vertexv∈Z.

The fact that the probabilities are periodic allows us to use Floquet-Bloch theory in order to findsp(M_ξ^T), and since the period is1, the computation is rather simple. The only eigenvalue of a fundamental domain, parametrized byk∈[−π, π], is the solution of the equation 0 = 1− 2

|T|+ 1

|T|e^ik+ 1

|T|e^−ik−x= 1− 2

|T|+ 2

|T|cos(k)−x=⇒x= 1− 2

|T|+ 2

|T|cos(k).

Finally,

sp(M_ξ^T) = [

k∈[−π,π]

sp(M_ξ^T(k)) = [

k∈[−π,π]

1− 2

|T|+ 2

|T|cos(k)

=

1− 4

|T|,1

.

Recall that self-similar groups within the family of spinal groups were studied by Šuni´c [65]. For everydandm, there are finitely many of them, and they can be specified in terms of an epimorphismα∈Epi(B, A)and an automorphismρ∈Aut(B). The groups in Šuni´c’s family are then the spinal groups defined by the periodic sequenceω= (ωn)ngiven byωn=α◦ρⁿ. Moreover, it was shown that any of these groups admits a natural minimal Šuni´c generating setT ={a, b₁, . . . , b_m}, contained in the spinal generating setS, such that

a= (1,1)σ b1= (1, b2) b2 = (1, b3) . . . bm−1 = (1, bm) bm= (a, b⁰), for someb⁰ ∈ B. Notice that, fori = 1, . . . , m−1, α(b_i) = 1andρ(b_i) = b_i+1, while α(bm) = a and ρ(bm) = b⁰. The choice of this b⁰ ∈ B in such a way that ρ is an automorphism determines the group. It was also shown in [65] that a Šuni´c group is infinite torsion if and only if allρ-orbits intersectKer(α).

Example 4.4.4. Grigorchuk’s group is the groupGin Šuni´c’s family withd= 2,m = 2, A = {1, a}, B = {1, b₁, b₂, b₁b₂} andρ(b₂) = b₁b₂. With the standard notation b, c, d for the generators, we haveb1 = d, b2 = band b1b2 = c. The only nontrivialρ-orbit is b1 7→ b2 7→ b1b2 7→ b1, which intersects Ker(α) at b1, hence the group is infinite torsion. The minimal Šuni´c generating set isT ={a, b₁, b₂}and the spinal generating set is S={a, b₁, b2, b1b2}, with

a=τ(1,1) b₁= (1, b₂) b₂ = (a, b₁b₂) b₁b₂= (a, b₁),

whereτ is the nontrivial element ofSym(X). By Theorem 4.3.2, we have these minimal generating sets, what is the spectrum on the Cayley graph. So far, we only know it must contain this Cantor set.

Example 4.4.5. One natural choice in the construction of Šuni´c groups above is to take d= 2andρsuch thatb⁰ =b₁. This gives one group for eachm≥2, which we callG_m. We have

a=τ(1,1) b1 = (1, b2) b2 = (1, b3) . . . bm−1= (1, bm) bm = (a, b1), again forτ being the nontrivial element ofSym(X). The elementab₁. . . b_mis of infinite order. We consider two generating sets: the spinal generating setS= (A∪B)\ {1}, of size 2^m, and the Šuni´c minimal generating setT ={a, b₁, . . . , bm}, of sizem+ 1. On the one

Indeed, for any two-endedΓ^T_ξ, the simple random walk translates into the weighted random walk onZwith probability _m+1¹ of moving to a neighbor and probability ^m−1_m+1 of staying on any vertex. By taking a one-vertex fundamental domain parametrized byk∈[−π, π]and using Floquet-Bloch theory, we have: generating setT. Then the spectrum on the Schreier graph with respect toT is

hm−3 m+1,1

i if G=G_mor a Cantor set of zero Lebesgue measure otherwise.

Proof. We found above the spectrum on the Schreier graphs for the groupsG_m. Suppose now thatsp(M_ξ^T)is a union of intervals. By Proposition 4.4.1, we have that the numbersqπ

are all equal overπ∈Epi(B, A)occurring infinitely often inω. By definition of the minimal Šuni´c generating setT, we know thatq_ω0 =m−1, so, asωis periodic,q_ωn =m−1for everyn≥0.

Now, for anyk= 0, . . . , m−1, we know thatω_k(bm−k) =ω₀ρ^k(bm−k) =ω₀(b_m) =a, sobm−k 6∈Ker(ω_k). Asq_ωk =m−1, the only possibility is that, for everyj = 0, . . . , m−1, bj ∈Ker(ωk)if and only ifj6=k. In particular, this implies thatbm = (a, b1), so that we are in fact in the case of the groupG_m.

We also have

Lemma 4.4.7. The Cayley graph ofGmwith generating setT is bipartite, for anym≥2.

Proof. We only have to show that all relations in the groupGmhave even length. Letwbe a freely reduced word onT, and let|w|represent its length and|w|_tthe number of times the generatort∈T occurs inw.

Suppose thatwrepresents the identity element ofGm. In that case,|w|_amust be even, or otherwise its action on the first level would be nontrivial. This allows us to write the word was a product ofbiandb^a_i. Letw0andw1be the two projections of the wordwinto the first level, before reduction. Let us look at the decomposition of a generatort∈T. Ift=a, then it decomposes as1on both subtrees. Ift=b_i, then it decomposes asb_i+1on the right and as 1on the left, or asaifi=m. Notice that the decomposition ofb^a_i is that ofbi exchanging the two projections.

It is clear that bothw₀andw₁represent the identity, too. Hence,|w₀|_aand|w₁|_amust both be even as well. But|w₀|_a+|w₁|_a=|w|_bm, so|w|_bmmust also be even.

By iterating this argument we can conclude that|w|must be even. In general, letw_ube the projection ofwonto the vertexuinX^k, thek-th level of the tree, for1≥k≥m. For anyu∈X^k,wumust represent the identity, and hence|w_u|_amust be even. But tracing back thea’s occurring inw_uwe obtain

X

u∈L_k

|w_u|_a= X

u∈Lk−1

|w_u|_bm =· · ·= X

u∈L1

|w_u|_bm−k+2=|w|_bm−k+1.

This shows that|w|_tis even, for everyt ∈ T, which implies that|w|is indeed even and hence that the Cayley graph with the generating setT is bipartite.

The spectrum of a bipartite graph is symmetric about0. At the same time, for amenable groups, the spectrum on any Schreier graph is contained in the spectrum on the Cayley graph.

Hence we have,

sp(M_G^T_m)⊃ −sp(M_ξ^T)∪sp(M_ξ^T) =

−1,3−m m+ 1

∪

m−3 m+ 1,1

.

Two cases are of special interest: m= 2andm= 3. In these cases, the union of the two intervals above is the whole interval[−1,1]and we can thus conclude that the spectrum of the Cayley graphs ofG₂ andG₃with respect to the minimal Šuni´c generating set is the whole interval[−1,1]. Form≥4, the union of intervals is actually disjoint, so we can only conclude that the spectrum of the Cayley graph contains two intervals[−1,−β]and[β,1], withβ >0.

Note that the group G₂ was studied in [27] and is therefore sometimes called the Grigorchuk-Erschler group. It is the only self-similar group in the Grigorchuk family (spinal groups withd = 2andm = 2) besides Grigorchuk’s group. The groupG₃ is known as Grigorchuk’s overgroup [4] because it contains Grigorchuk’s group as a subgroup. Indeed, the automorphismsb2b3,b1b3,b1b2are the generatorsb,c,dof Grigorchuk’s group.

Corollary 4.4.8. For the Grigorchuk-Erschler groupG2and Grigorchuk’s overgroupG3

the spectrum of the Cayley graph is a union of two disjoint intervals with respect to the spinal generating set and the interval[−1,1]with respect to the minimal Šuni´c generating set.

The groups G_m with respect to the minimal Šuni´c generating set T satisfy the first condition in Proposition 4.4.1. In particular, for everyn≥0, the numbersq_ωnare all equal tom−1, hence the spectrum on the Schreier graphs is a union of intervals, as we computed above. Nevertheless, they are the only Šuni´c groups to satisfy this condition forT, as shown in Proposition 4.4.6.

Example 4.4.9. Another non-torsion example in Šuni´c’s family of self-similar groups is the groupGgiven byd= 2,m= 3andρsuch thatρ(b3) =b1b2b3. We have

a= (1,1)σ b1 = (1, b2) b2= (1, b3) b3 = (a, b1b2b3).

Indeed, the elementb₁b₃ decomposes as(a, b₁b₃), so it constitutes aρ-orbit which does not intersectKer(α). Hence, the elementab1b3 is of infinite order. This group contains the subgroupha, b₂b₃, b₁b₂, b₁b₃i, which is isomorphic to the Grigorchuk-Erschler group. For the spinal generating setS, we have

sp(M_ξ) = sp(M_G) =

−1 4,0

∪ 3

4,1

.

Its minimal Šuni´c generating set is{a, b₁, b₂, b₃}. AsGis notG_m, by Proposition 4.4.6 we can conclude thatsp(M_ξ^T)is a Cantor set of zero Lebesgue measure. More precisely, the sequenceωis4-periodic and we haveqω0 =qω3 = 2whileqω1 =qω2 = 1.

Spectral measures

This chapter extends the study of the spectral properties of adjacency operators on Schreier graphs of spinal groups by computing their spectral measures. The spectrum of the adjacency operator∆ξ on the Schreier graphΓξ associated with a point ξ ∈ X^N is a subset of R.

Spectral measures are measures with support in the spectrum, and provide more information than just the spectrum as a set. For instance, a function in`<sup>2</sup><sub>ξ</sub>=`²(Γ_ξ)projects trivially onto a given eigenspace of eigenvaluexif and only if its associated spectral measure vanishes for the set{x}.

The chapter is divided in four sections. We first compute the so-called empirical spectral measure, a measure which only depends on the finite Schreier graphsΓnand represents an averaging of all spectral measures. Then we exhibit three different types of spectral measures that may occur for infinite Schreier graphs of spinal groups. For spinal groups withd= 2, we show that all spectral measures on the graphsΓξare absolutely continuous with respect to the Lebesgue measure and give their density explicitly. For spinal groups withd ≥3, we construct a complete basis of eigenfunctions for`<sup>2</sup><sub>ξ</sub>=`²(Γ_ξ)for allξin a subset ofX^N of uniform Bernoulli measure one, which implies that all spectral measures on the graphs Γ_ξ of these groups are discrete. Finally, we consider the Schreier graphsΓ_π that occur as accumulation points of graphs{(Γ_ξ, ξ)}_ξ in the space of Schreier graphsG_Gω,XN (see Theorem 3.7.3). We decompose the space`<sup>2</sup><sub>π</sub> =`²(Γπ)as the direct sum of two nontrivial subspaces, whose functions have associated spectral measures which are singular continuous and discrete, respectively, thus providing new examples of Schreier graphs with nontrivial singular continuous component in the spectral measures.

5.1 Empirical spectral measure

In the proof of Theorem 4.1.7 we actually found the explicit multiplicities of the eigenvalues of∆_n. These multiplicities allow us to compute the so-called empirical spectral measure (or

density of states) of the graphs{Γ_n}_n.

Definition 5.1.1. Let{Y_n}_nbe a sequence of finite graphs, and, for everyn≥0let∆nbe the adjacency operator onYnandνnbe the counting measure onsp(∆n). Namely,

ν_n= 1

|Y_n| X

x∈sp(∆_n)

δ_x,

whereδ_xtakes value1atxand vanishes anywhere else, and the eigenvalues are counted with multiplicity. Following [46], if the measuresνnweakly converge to a measureν, we callνtheempirical spectral measureordensity of statesof{Y_n}_n.

Theorem 5.1.2. LetG_ωbe a spinal group withd≥2,m≥1andω ∈Ω_d,m. Letν be the empirical spectral measure of{Γ_n}_n. Ifd= 2,νis absolutely continuous with respect to the Lebesgue measure. Its density is given by the function

g(x) = |x−2^m−1+ 1|

πp

x(x+ 2)(2^m−x)(x−2^m+ 2). (5.1) Ifd≥3, thenνis discrete. More precisely,

ν = d−2

From the multiplicities computed in the proof of Theorem 4.1.7, we have thatν0 =δ_|S|, ν2= ¹_d(δ|S|+ (d−1)δ|S|−d)and, forn≥2,

Ford= 2, all the multiplicities of the eigenvalues in the finite graphs are1, or equiva-lently, every eigenvalue of∆_nhas the same mass, _d¹n. When taking the limit, the measure of each atom tends to zero and the set of eigenvalues becomes dense in either one (m= 1) or two (m≥2) intervals. Therefore, any set of positive empirical spectral measure is the union

−4 −2 0 2 4

−4 −2 0 2 4

Figure 5.1: Density functions of the empirical spectral measureνfor Grigorchuk’s group (d= 2, m= 2, above) and the Fabrykowski-Gupta group (d= 3, m= 1, below).

of cones of the set of preimages ofF(x) = x² −d(d−1)plus their closure, which has positive Lebesgue measure. Hence,νis absolutely continuous with respect to the Lebesgue measure. We can find its precise density if we notice the following, ford= 2andn≥1:

sp(∆n) =

2^m−2 2 +p

4^m−1+ 1 + 2^mcosθ

∈ {±1}, θ∈ 2πZ 2ⁿ

\ {0,−2}.

Indeed, from the proof of Theorem 4.1.7 we recover the two branches of the inverse of ψ:

ψ⁻¹(x) = 2^m−2 2 +p

4^m−1+ 1 + 2^m−1x.

Any x ∈ F^−k(0)can be written asx = ±√

2 +y, withy ∈ F^−(k−1)(0). We can hence complete the proof of the equality above by induction, using the trigonometric identity 2 cos ^θ₂

=±p

2 + 2 cos(θ). This allows us to find an injective, measure-preserving map χ: [0, π]× {0,1} →R, defined by

χ(θ, ) = 2^m−2 2 +p

4^m−1+ 1 + 2^mcosθ,

with the spectrum uniformly distributed with respect to the Lebesgue measureλon[0, π]× {0,1}. The empirical spectral measure of any subsetE ⊂ R is then given byν(E) =

λ(χ⁻¹(E)). The densityg(x)ofνis thus given by g(x) = 1

2π d

dxχ⁻¹(x), which coincides with the expression in the statement.

Dans le document Structural and Spectral Properties of Schreier Graphs of Spinal Groups (Page 94-105)