Spectra of Infinite Graphs Via Freeness with Amalgamation

Spectra of Inﬁnite Graphs via Freeness with Amalgamation

Jorge Garza-Vargas Archit Kulkarni [email protected] [email protected] UC Berkeley UC Berkeley

January 15, 2021

Abstract We use tools from free probability to study the spectra of Hermitian operators on inﬁnite graphs. Special attention is devoted to universal covering trees of ﬁnite graphs. For operators on these graphs we derive a new variational formula for the spectral radius and provide new proofs of results due to Sunada and Aomoto using free probability. With the goal of extending the applicability of free probability techniques beyond universal covering trees, we introduce a new combinatorial product operation on graphs and show that, in the non-commutative probability context, it corresponds to the notion of freeness with amalgamation. We show that Cayley graphs of amalgamated free products of groups, as well as universal covering trees, can be constructed using our graph product. arXiv:1912.10137v4 [math.CO] 14 Jan 2021

1 Contents

1 Introduction 3 1.1 Results ...... 5 1.1.1 Universal Covers ...... 5 1.1.2 Amalgamated Graph Products ...... 8 1.2 Structure of the Paper ...... 9 1.3 Deﬁnitions and Conventions ...... 10

2 Motivation and related work 10 2.1 Density of States of Operators on Universal Covers ...... 10 2.2 Random Lifts ...... 12 2.3 Spectral Radii of Operators on Universal Covers ...... 13 2.4 Graph Products and Noncommutative Probability ...... 14 2.5 Random Walks on Groups and Free Probability ...... 15

3 Preliminaries on Free Probability 15 3.1 Free Probability ...... 16 3.2 Operator-valued Probability Spaces ...... 16 3.3 Freeness with Amalgamation ...... 17 3.4 The Amalgamated Free Product of Hilbert -modules ...... 18 B 3.5 The Operator-Valued R-Transform ...... 19

4 Band Structure of Universal Covers 20 4.1 Sunada’s Theorem via Asymptotic Freeness ...... 20 4.2 Spectral Splitting ...... 22

5 The Amalgamated Free Product for Graphs 25 5.1 Combinatorial Description ...... 25 5.2 Proof of Theorem 5.8...... 29 5.3 Amalgamated Free Product of Groups as Amalgamated Free Product of Graphs . . . 31 5.4 Implications for A ( ) ...... 33 T G 6 Universal Covers: Algebraic Description of the Spectrum 35 6.1 Aomoto’s Equations via the R-transform ...... 35 6.2 Behavior of the Density of States at the Right Edge of the Spectrum ...... 36 6.3 Formula for the Spectral Radius ...... 38

7 Future Research 40

2 1 Introduction

In the present work, by a graph we mean a locally finite undirected graph allowing loops and multi-edges, unless otherwise specified. Given a graph , we denote its vertex set by V( ) and its G G edge multiset by E( ). V( ), E( ) G G G The adjacency matrix, the graph Laplacian matrix, and transition matrices for symmetric random walks all fall under the umbrella of so-called “Jacobi matrices on graphs” [ABS20]. These are Hermitian operators A acting on = `2(V( )) defined by G H G X X (A f )(u) = bu f (u) + ae f (v) + 2 al f (u) (1) G e E( ) l E( ) e= u∈,v G,v,u l=∈ u,Gu { } { } 1 for real coefficients bu u V( ), and nonzero real coefficients ae e E( ) . (The factor 2 appears for loops { } ∈ G { } ∈ G because it will be convenient in our application to universal covers, but see Section 2.1 for further discussion involving half-loops.) Jacobi matrices on graphs will be the main operators considered in this work. We now recall the notion of spectral measure. Given a vertex u V( ) one can define the state ∈ G τu : B( ) C as τu H → def τu(X) = δu, Xδu , X B( ), (2) h i ∀ ∈ H 2 where δu ` (V( )) denotes the indicator function of the singleton u . Since A is self-adjoint, its δu ∈ G { } G spectrum, which we denote by σ(A ), is contained in R. Moreover, by the functional calculus, τu σ( ) G · induces a probability measure µA ,u supported on σ(A ) with the property that µA ,u G G G Z k k x dµA ,u(x) = τu A R G G for all k Z 0. Throughout this work we refer to µA ,u as the spectral measure of A with respect to the ∈ ≥ G G root u V( ). In terms of functional analysis, µA ,u is the composition of the usual projection-valued ∈ G G spectral measure of A with τu. The spectra and spectralG measures of operators on infinite graphs have been extensively studied in the last several decades in different contexts. But despite significant progress in the area, current mathematical tools are still unable to answer simple questions. Since our goal is to provide a new perspective on problems of current interest, the content of this paper is a combination of new results, new proofs of existing results, and new tools for the analysis of infinite graphs, all through the lens of free probability. Our discussion includes the following families of graphs.

Cayley Graphs. Let G be a finitely generated group and S G be a finite symmetric generating ⊂ set; i.e. we assume that if s S then s 1 S. We denote by Γ = Γ(G, S) the Cayley graph of Γ(G, S) ∈ − ∈ G with respect to S. Since S is symmetric, Γ is undirected. By definition of Γ, any weighting a : S R+ induces a weighting on the edges of Γ in the obvious way. The canonical scalar → measure associated to AΓ is the spectral measure of Γ with respect to the root e G and the given ∈ 1 Note that we are deviating from the standard definition of Jacobi operator which requires the ae to be strictly positive. When the graph is a tree, gauge invariance allows to “turn” negative weights into positive without affecting the spectrum. However, in this work, allowing negative coefficients sometimes simplifies our exposition.

3 weighting. Both σ(AΓ) and µΓ,e have been studied thoroughly in the context of random walks on groups [Kes59, Woe86, Woe87, McL88, CS86]. However, several basic spectral questions about some natural Cayley graphs remain open [KFSH19]. The amalgamated free product of groups inspired the notions of free independence and freeness with amalgamation [Voi85], and since Voiculescu’s seminal work it became apparent that tools from free probability can provide important insights into the spectral theory of certain Cayley graphs. In this work we focus on a new connection, namely the role of freeness with amalgamation in the study of universal covering graphs.

Universal Covering Graphs. Let be a connected graph with n vertices. One can form its universal G covering graph2 (also called universal covering tree or universal cover), which we denote by ( ). T G The universal covering graph comes with a covering map φ : ( ) . For simplicity, when is T G → G G clear from context, we will write instead of ( ). Recall that is an infinite tree if has at least T T G T G one cycle or loop and is itself when is a tree. G G Any Jacobi matrix A on can be lifted via φ to a Jacobi matrix A on , called a “periodic G G T T Jacobi matrix with period n”[ABS20] or a “pulled-back local operator” [AFH15]. We can then associate a canonical measure to A as follows. For any u V( ) we fix a representative u˜ V( ) 1 T ∈ G ∈ T in the fiber φ− (u), and then consider the respective state τu˜ as in (2). Now let τ T def 1 X τ = τu˜, T n u V( ) ∈ G and denote by µA the induced scalar measure. We refer to µA as the density of states of A in µA accordance with theT physics and spectral theory literature. T T T It is a well known fact that is the limit, in the Benjamini-Schramm sense, of random lifts of T G (see Section 2.2). Hence, A can be regarded as a limit of random matrices. A bit of thought from the free probability perspectiveT shows that in fact A can be viewed as an operator-valued matrix with free entries (see Section4). This is the starting pointT of the present work. Previous results give an explicit description of σ(A ) and µ when has a particular structure T T G [Kes59, McK81, GM88, FTS94]. Others have made some progress in the case when is an arbitrary G graph [A+88, Aom91, SS92, Sun92, ABS20]. However, as we discuss in the last section, many fundamental questions remain open (also see[ABS20]).

Amalgamated Free Product for Graphs. As mentioned above, since generic universal covering graphs are not Cayley graphs, the emergence of freeness with amalgmation in this context might seem somewhat fortuitous. Here, we clarify this connection by introducing a graph product that corresponds to the notion of freeness with amalgamation. Here is some context. Inspired by the combinatorial description of Cayley graphs of free products of groups, Quenell [Que94] introduced the free product of graphs. Only later it was understood by Accardi, Lenczewski, and Sałapata [ALS07] that this graph product is equivalent Voiculescu’s free product of Hilbert spaces [VDN92] and that free probability can be used to compute the spectra of graphs arising from Quenell’s graph product. We refer the reader to Section 2.4 for a detailed and more precise discussion. In the spirit of [Que94] and [ALS07], in this paper we deﬁne a combinatorial graph product and show that the machinery of freeness with amalgmation can be used to compute the spectra

2See Section 2.1 for a precise deﬁnition and a subtlety involving covers of loops.

4 of graphs arising from this product. In particular, universal covering trees and Cayley graphs of amalgamated products of groups can be constructed using our product.

Bibliographic Note After posting the first version of this article to the arXiv, we became aware of the work of Avni, Breuer and Simon [ABS20] posted six weeks prior. Upon reading their work, we have revised our article in a few ways. First, in place of our previous notion of “adjacency operators on weighted graphs,” we have adopted their terminology of Jacobi matrices on graphs, both to provide consistency in the literature and because it provides a useful distinction between diagonal elements bv and loops (which behave differently upon taking covers). Second, our theorem on the number of bands in the spectrum was demonstrated in [ABS20] to be implicit in work of Sunada [Sun92], which we had not realized. Both our independent proof and the proof given in [ABS20] argue that the Jacobi operator of a universal cover can be viewed as a specific element of a matrix C∗-algebra, and then use a standard K-theory argument which relies on a theorem of Pimsner and Voiculescu. However, these two proofs differ in the way in which the connection with the C∗-algebra is made. The proof we present here uses random lifts and the fact that independent random permutation matrices are asymptotically free (an idea that has previously been exploited in [BC19] for different purposes), while the proof in [ABS20] deals directly with the Jacobi operator on a concrete Hilbert space. Given the relevance of this result, we have decided to keep our alternative argument in this work, but we no longer state the result as ours. Finally, as we are able to answer some questions left open in [ABS20], we include these answers in in Section 4.2 of this revision. To the best of our knowledge, other than what is mentioned in the above paragraph, there is no further overlap between our work and [ABS20].

1.1 Results 1.1.1 Universal Covers In this section will denote a ﬁnite graph with n vertices. Let denote the universal cover of , G T G and φ the covering map φ : . T → G Roughly speaking, the proofs of our results regarding universal covers use free probability in two diﬀerent ways.

(1) Free probability techniques are used to argue that Jacobi operators on can be represented as T n n matrices with entries on a certain C -algebra. For different applications it is convenient to × ∗ use different C∗-algebras. (2) Once a given Jacobi operator of is viewed as an element of a matrix C -algebra, we argue that T ∗ such element can be decomposed as a sum of simpler operators that are free with amalgamation over the algebra Mn(C) (see Section3 for precise definitions). This decomposition allows the use of tools from free probability to understand the spectrum of the original Jacobi operator.

Regarding step (1), it is worth noting that once one has represented the Jacobi operator as an speciﬁc element of a matrix C∗-algebra, one might be able to ﬁnd elementary proofs that show that such a representation is correct. However, one should appreciate that it is not clear a priori that this connection with C∗-algebras exists, and neither is it easy to ”guess” what the correct representation

5 of the Jacobi operator on a given C∗-algebra is. It is for this reason that we decided to include the free probability arguments for step (1), since we believe they provide a conceptual way to arrive at the C∗-algebra representations.

Spectral Radius. Let m be the number of edges in where loops are counted twice and let Γm be G the discrete group obtained by taking the free product of m copies of Z2. Using our combinatorial graph product, which is discussed below in Section 1.1.2, we show that Jacobi operators on can T be represented as elements of the tensor product of Mn(C) with the reduced C∗-algebra of Γm (see Section3 for deﬁnitions). We then use a theorem of Lehner [ Leh99] to prove the following min-max formula for the spectral radius (actually, both spectral edges) of Jacobi operators.

Theorem 1.1 (Formula for the spectral radius). Let A be a Jacobi matrix on a graph with vertex set G G [n] and parameters bi , ae as in (1), and let A be its pullback to the universal cover ( ). Let ρr(A ) { } { } T T G T denote the right edge (i.e. maximum element) of the spectrum of A . Then ρr T     1  X q    2  ρr(A ) = inf max bi + 2 deg(i) + 1 + 4a i,j yi yj . (3) T y1,...,yn>0 i [n]  2yi  −  ∈ j: i,j E( ) { } { }∈ G where deg(i) is the degree of the vertex i in , and where loops each contribute 2 to the degree, and each loop G is understood to appear twice in the summation. Moreover, the infimum can be restricted to those n-tuples (y1,..., yn) for which the n expressions inside the max symbol are equal to each other. Remark 1.2. Using the fact that A is also a Jacobi matrix on , one may obtain a similar expression for − G G the left edge of the spectrum ρ . The spectral radius is equal to max ρr, ρ . l { − l} Remark 1.3. In the case where bv = 0 for all v V( ), the spectrum of A is symmetric about zero, because ∈ G T is bipartite. T One can use Lagrange multipliers on the above variational problem to find an explicit algebraic description of ρ( ) from equation (3); see Corollary 6.7. T Aomoto’s Equations. For u V( ), we can take a representative u˜ V( ) in the fiber φ 1(u), and ∈ G ∈ T − consider the spectral measure with respect to the vertex u˜, namely µA ,u˜. We may then form the T Cauchy transform of this measure: wu(z) Z def 1 1 wu(z) = τu˜[(z A )− ] = dµA ,u˜(t). (4) − T z t T R − The Cauchy transform is also known as the Stieltjes transform, and is closely related to the Green 1 1 function or resolvent (z A )− , as well as to the walk generating function Qu˜(z) = wu˜(1/z) which − T z counts closed walks on based at u˜. It is a standard fact in analysis that the spectral measure µA ,u˜ T T can be fully recovered from wu(z) via the Stieltjes inversion formula. Using the operator-valued version of Voiculescu’s R-transform, we recover the following system of functional equations for the wu(z) presented below in Theorem 1.4.

+ Theorem 1.4 (Aomoto [A 88]). Let A be a Jacobi matrix on a ﬁnite graph with parameters ae , bv as G G { } { } in (1) and let wu(z) be the Cauchy transforms for A as in (4). Then, the following system of equations holds T

6 for all z C in a neighborhood of inﬁnity and for all real z outside the convex hull of the spectrum σ(A ): ∈ T        1  X q X q    2 2 2 wu(z) = 2 deg(u) + 1 + 4ae wu(z)wv(z) + 2 1 + 4ae wu(z)  u V( ).  2(z bu)  −  ∀ ∈ G  −  e E( ) e= u,u E( )   e= u∈,v G, v,u { }∈ G { } (5) where each loop at u contributes 2 to the degree deg(u).

Remark 1.5. Since in Theorem 1.4 we are restricting z to be in a neighborhood of infinity and Cauchy transforms vanish at infinity, the expressions inside the radicals of (5) are always on the right side of the complex plane, and hence the square roots are globally defined. Moreover, by analyzing the behaviour of the wu(z) at infinity, one sees that one should take the positive branch for each square root for the system of equations to hold.

The above system of equations was first discovered by Aomoto [A+88] using techniques from the literature of random walks on groups. See [KLW13, §5 and §6], [KLW15, §4] and [ABS20, §6] for a survey of similar results related to algebraicity of Cauchy transforms. In particular see [ABS20, §6] for a discussion of the role of algebraicity in showing that A has no singular continuous spectrum. In [Aom91] Aomoto used (5) to obtain necessary conditionsT on for the existence of point G spectrum in σ(A ). Here, in Section 6.2, we take a brief detour to show that (5) can be used to establish a connectionT between the behavior of the density of states at the right edge of σ(A ) and the growth rate of . We prove the following. T T Theorem 1.6 (Vanishing density at the right edge). Let A be a Jacobi matrix on a finite graph with at G G least two cycles, and let A be its pullback to the universal cover ( ). Furthermore assume that ae > 0 T T G for all e E( ). Let µ and ρ be the density of states and the maximum element of the spectrum of A , ∈ G dµ T respectively. Then µ is absolutely continuous in a neighborhood of ρ and limx ρ (x) = 0. → dx dµ The behavior of dx at the edge is tied to the type of singularity of the Cauchy transforms wu(z) at z = ρ. On the other hand, the latter have been extensively studied for many classes of infinite graphs (e.g. [NW02, Lal01, Woe00, GL13]), as an intermediate step to understand the asymptotic behavior of transition probabilities and escape rates of random walks. In particular, Theorem 1.6 can be deduced from the results in [NW02, §2], where the wu(z) are related to an auxiliary family of transforms, and sophisticated methods (that seek to prove stronger results) from the theory of random walks on infinite graphs are used. The argument in [NW02, §2] analyzes, as z varies, the evolution of the Perron eigenvalue of the non-backtracking matrix of 3 with entries G weighted as a function of the auxiliary transforms. In contrast, our proof of Theorem 1.6 is short and self-contained, and uncovers a nice relation between Aomoto’s equations and the Perron eigenvalue of the adjacency matrix of . G Sunada’s Theorem on the Band Structure. Let m be the number of edges in where each loop 4 G is counted once and let Fm denote the free group on m generators. In [BC19], large random lifts

3The non-backtracking matrix of G is a non Hermitian matrix whose rows and columns are indexed by the directed edges of G, see [ST96]. 4Note that, unlike in the discussion of the spectral radius where two generators were used for every loop, here, since we are using a diﬀerent C∗-algebra, only one generator per loop is needed.

7 of graphs were studied in the context of free probability. In Section4 we recall from [ BC19] how random lifts can be used to obtain a representation of A as elements of the tensor product of Mn(C) T with the reduced C∗-algebra of Fm. This representation can be combined with a theorem of Pimsner and Voiculescu [PV82] to conclude the following result:

Theorem 1.7 (Sunada). Let A be a Jacobi matrix on a graph with n vertices, and let A be its lift to G G T the universal cover of . Then the density of states of A assigns a positive integer multiple of 1/n to any G T connected component of the spectrum of A . Consequently, the spectrum of A contains at most n connected components. T T

Remark 1.8 (Tightness). As mentioned above, if is a tree then , and hence, if has distinct G G T G eigenvalues (e.g. is a path and A is its adjacency matrix), Sunada’s bound on the number of components G G is tight. A more interesting example of tightness is any finite graph with A having distinct bv and P G e E( ) ae < minu,v bu bv ; see [ABS20, §10.2]. ∈ G | − | The trick needed to reduce the proof of the above theorem to the theorem in [PV82] is standard in K-theory and was first used in the context of graph theory by Aomoto and Kato in [AK88]. The upper bound on the number of bands of the spectrum of A is implicit in the work of Sunada [Sun92]. It was then highlighted and proved explicitly in [ABS20T ]. This technique was also used in [KFSH19] to upper bound the number of bands for certain infinite lattices. In relation to questions regarding the number of bands in the spectrum of A , in Section 4.2 we discuss the phenomenon of spectral splitting and provide answers to several questionsT in [ABS20] regarding possible extensions of theorems of Borg and Borg–Hochstadt.

1.1.2 Amalgamated Graph Products Inspired by a series of results of different authors [Oba04, ABGO04, ALS07] in which it is shown that each notion of non-commutative stochastic independence corresponds to a combinatorial graph product, we investigate the possibility of associating a graph product to the notion of freeness with amalgamation [VDN92, 3.8]. Recall that freeness with amalgamation is not an independence in the sense of Muraki or Speicher [Mur02, Spe97], but it shares many desired properties with the five independences. With this purpose in mind we consider the following setting. Let ,..., n be finite rooted G1 G graphs, and let 1,..., n be disjoint sets of colors. Assume each i is equipped with an edge C C Sn G coloring ci : E( i) i. Let = i and let be a finite rooted graph with an edge coloring G → C C i=1 C G c : E( ) . G → C In this work we define a graph called the free product of ,..., n with amalgamation over . G1 G G This product can be viewed as a procedure to construct an infinite graph by iteratively copying the local structure of the i and where the way in which these neighborhoods are combined is dictated G by the structure of the graph and by how the colorings ci relate to c. G The upshot here is that if is a graph constructed through this procedure, the Jacobi operator K A can be written as an amalgamated free convolution of much simpler non-commutative random variables.K Hence, in this situation much of the machinery developed around freeness with amalgamation (e.g. [Leh99, Spe98, Voi95, BBL19]) can be used to understand the spectral measure of A . ItK turns out that the amalgamated free product of graphs is general enough to be able to construct both

8 (i) any Cayley graph of an amalgamated free product of groups, and

(ii) any universal cover of a graph.

1.2 Structure of the Paper In Section2 we discuss related work and motivate the study of the spectral properties of A . The latter has already been done impeccably by Avni, Breuer and Simon in [ABS20]. So, to minimizeT redundancy, we very briefly survey some of the important results mentioned in [ABS20], and limit our detailed discussion to a topics that were not touched upon in [ABS20]: the relevance of σ(A ) in the theory of relative expanders, with particular emphasis on the right and left edges of σ(A )T and the role of random lifts in this context. We also discuss the connections between noncommutativeT probability and spectral graph theory and spectral results about Cayley graphs in the context of free probability. Section3 contains all the machinery from the theory of free probability that will be needed in the present work, as it is our hope that this paper be of interest to multiple audiences. For example, the reader who is well-acquainted with free probability may skip Sections3, but might find Section 2 informative, while a reader coming from a background of spectral graph theory may want to skip Sections 2.1-2.3, review parts of Section3 and proceed to the sections on the spectra of universal covers (Section4 and Section6). The subsequent sections draw from di fferent parts of Section3 and are developed in a more or less parallel fashion. For this reason, the reader may jump straight to any of the sections that are of their interest and go back as needed. In Section4 we prove Theorem 1.7 using asymptotic freeness of random permutation matrices and an argument from operator K-theory. The knowledge from free probability required for this section is contained in 3.1. In Section5 we develop the theory behind our graph product, which we call the amalgamated free graph product. This discussion is based on the construction of Hilbert bimodules given in Section 3.4 and a good understanding of the content of Sections 3.1-3.3 is recommended. In Section6 we adopt an algebraic approach to the description of the spectral measure of A . The analysis of this section is based on interpreting A as an operator-valued matrix with freeT entries. This interpretation coincides with the frameworkT used by Lehner in [Leh99] and his results are used in the proofs of Theorem 1.4 and Theorem 1.1. In Section7 future directions are discussed.

Note for the Free Probability Expert. Our contribution to the theory of free probability is the deﬁnition of the amalgamated free product of graphs presented in Section5. The generality of this graph product has the potential to allow a free probability approach in contexts that go beyond what is discussed here. In some sense, our construction provides a combinatorial interpretation of Voiculescu’s amalgamated free product for Hilbert bimodules when the underlying algebra is Mn(C). Moreover, our discussion from Section 5.3 has the non-trivial implication that for groups G1, G2 with a common ﬁnite subgroup H, operators in C (G H G2) that are free with amalgamation over C (H), have red∗ 1 ∗ red∗ explicit copies in distribution in an algebra of the form Mn(C) B( ) where n is the order of H. ⊗ H This could facilitate numerical computations of amalgamated free convolutions. Sections4 and6 on the other hand contain applications of well understood tools in free probability and their value resides in providing a new perspective to the independently interesting

9 problem of understanding the spectral properties of A . T

1.3 Deﬁnitions and Conventions Let us be precise about our notion of universal cover for graphs, which is not identical to the standard topological notion of universal covering space. For this, it is useful to work with the directed edge set Edir( ) which consists of ordered pairs of vertices, and which is equipped with G an edge-reversal involution ı : Edir(G) Edir(G). An edge u, v in E( ) with u , v contributes → { } G two directed edges (u, v) and (v, u) to Edir(G), and ı((u, v)) = (v, u). We treat a loop as two elements e1 = (v, v) and e2 = (v, v) with ı(e1) = e2, where ı is the involution which for non-loop edges simply reverses the edge. This is the useful notion of whole-loop, introduced by Friedman [Fri93] alongside the contrasting notion of half-loop, for which one places a single element e = (v, v) in Edir( ) with G ı(e) = e. Whole-loop corresponds to the standard notion of loop used in graph theory. We will occasionally allow half-loops, for example in the deﬁnition of universal cover below, but unless explicitly stated otherwise, in this work graphs will not permitted to have half-loops, and “loops” will refer exclusively to whole-loops.

Definition 1.9 (Universal cover of a graph). Let be a finite undirected graph, possibly with half-loops, G and fix a root v0 V( ). The universal cover of is the tree ( ) constructed as follows: ∈ G G T G (1) We place one vertex in ( ) for every non-backtracking walk in starting at v0, including the empty T G G walk, where non-backtracking means that no directed edge e is immediately followed by ı(e).

(2) We connect two vertices in ( ) by an edge if the walk corresponding to one of them can be obtained by T G appending an edge to the walk corresponding to the other.

We will say that is a base graph of ( ). G T G Example 1.10 (Bouquets). Let k 0. The universal cover of a single vertex with k half-loops is the k-regular ≥ tree, which is a single vertex for k = 0, a single edge for k = 1, and inﬁnite otherwise. The universal cover of a single vertex with k whole-loops is the 2k-regular tree.

One may also deﬁne the notion of covering map for graphs (see e.g. [FK19]), and one has that any connected cover of is covered by ( ). G T G

2 Motivation and related work

2.1 Density of States of Operators on Universal Covers Studying the density of states of Jacobi matrices on universal covers is a hard problem and only in speciﬁc cases explicit results can be obtained. Here we provide a quick overview of what we consider are the most relevant results in the context of this paper. We refer the reader to [ABS20] for a broader and more detailed discussion. The density of states of the adjacency operator of the universal cover of d-regular graphs (i.e. the regular d-regular tree) was ﬁrst computed by Kesten [Kes59] in the context of Cayley graphs, then revisited by McKay [McK81] in the context of random graphs. Godsil derived a formula for the density of states of the adjacency matrix of the universal cover of bipartite biregular graphs

10 [GM88]. In the case of d-regular graphs, but now allowing arbitrary coefficients, Figa-Talamanca` and Steger [FTS94] provided a complete description of the spectrum of the lift of Jacobi matrices to the infinite d-regular tree. In the more general setting of arbitrary universal covers, Aomoto [A+88] (see Theorem 1.4 above) derived a system of coupled equations (with square roots) for the Cauchy transforms of the spectral measures of lifts of Jacobi matrices. In a similar spirit, polynomial coupled equations for certain transforms were then obtained by Avni, Breuer and Simon [ABS20] in the context of periodic Jacobi matrices on trees, and by Keller, Lenz and Warzel [KLW13] in the context of infinite trees of finite cone type (which are a family of infinite trees that contains universal covers [KLW15]). In [Aom91] Aomoto used the coupled equations for the Cauchy transforms that he obtained in [A+88], to show that lifts to the universal cover of arbitrary Jacobi matrices on d-regular graphs have no point spectrum. A result in a similar direction was later obtained in [KLW13], where it was shown that under some conditions on the base graph (that allow non-constant degree graphs), the density of states of of the adjacency matrix of the universal cover is absolutely continuous. Certainly, this result does not apply to all universal covers, since the density of states may contain atoms; see Table1. A general result was obtained in [ ABS20], where it was shown that periodic Jacobi matrices on trees (i.e. any lift of a Jacobi matrix on a graph to its universal covers) have no singular continuous spectrum. Finally, we point out the work of Sunada [Sun92], which was discussed above (see Theorem 1.7). The reasons for studying the spectrum of operators on universal covers has varied from author to author. Our main motivation is that, as shown in [BC19], the spectra of these infinite objects govern the behaviour of large random lifts of finite graphs, and on the other hand, random lifts have been instrumental for obtaining groundbreaking results on the existence of optimal (Ramanujan) expanders [MSS13, HPS18]. We refer the reader to Section 2.2 for a definition and discussion of random lifts. That said, the complexity and variety of features that one can observe when looking at the spectrum of different universal covers is appealing in its own right, and it is the source of many interesting spectral problems. To give the reader some intuition on the behaviour of the density of states of these Jacobi matrices below we include some simulations and observations.

Observation 2.1. For simplicity let us consider only the adjacency operator (so bv = 0 and ae = 1 for all vertices v and edges e.) Table1 shows that even if the base graphs are similar in some sense, the spectra of their universal cover may be quite diﬀerent. In particular, we make the following remarks:

i) (Topologicalequivalence) and 2 are homeomorphic and, in particular, they have the same fundamental G1 G group. Yet the spectra of the universal covers diﬀer.

ii) (Eigenvalues of base graph) The graphs 3 and are cospectral (i.e. their non-weighted adjacency G G4 matrices have the same multiset of eigenvalues); however, the spectra of their universal covers possess very diﬀerent features. iii) (Perturbations) 3 is obtained by adding a leaf to 2. In this case a gap in the spectrum around zero G G is created. From experiments, it seems adding leaves or edges can often cause major changes in the spectrum. iv) (Degrees) Graphs 5 and 6 have the same degree sequence. G G

11 100 1400

90 1200

1000 70

60 800

50 600 40

400 30

20 200

0 0 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3 4 2 µ 2 1 µ 1 150 G 1200 T G T

1000

100 800

600

50 400

200

0 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 4 µ 3 µ 3 4 G 1200 T G 80 T 70 1000

800 50

600 40

30 400

200 10

0 0 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

5 µ 6 µ 6 G T5 G T

Table 1: Six graphs i alongside the respective approximation to µ i , the density of states of the adjacency G T operator on A i (that is, ae = 1 and bv = 0 for all e and v). The approximations presented are (scaled) histograms ofT the eigenvalues of a large random lift (taken as in Section 2.2) of the respective graph. See Table2 for the spectral radii of the i. T

2.2 Random Lifts A random d-lift of a graph is a random d-fold cover of the graph with a particular structure. Recently, random d-lifts have been studied in the context of expander graphs. For example, random 2-lifts were used by Marcus, Spielman and Srivastava to show the existence of Ramanujan graphs of every degree [MSS13]; these techniques were later generalized by Hall, Puder and Sawin [HPS18] to n-lifts for n 2. ≥ Definition 2.2 (Random lift). Let be a finite graph with vertices labeled 1,..., n, and let d 1. For G ≥ every edge e in the multiset E( ), let Ue be an independent uniform random d d permutation matrix. A G × random d-lift of is a random graph with nd vertices, whose adjacency matrix is given by G X ∆ij Ue + ∆ji U∗ ⊗ ⊗ e e= i,j E( ) { }∈ G where ∆ij denotes the n n matrix with a 1 in the (i, j) entry and 0 everywhere else. × It is a well known fact that as d goes to infinity, random d-lifts of a fixed graph converge in the G Benjamini-Schramm sense [BS11] to the universal covering graph of . In particular, this implies T G that the density of states of A is the weak limit of the mean eigenvalue distribution of the random d-lifts: T

12 Lemma 2.3 (Limits of random lifts). Let A be a Jacobi matrix on a ﬁnite graph and let A be its G G T pullback to the universal cover ( ). Let µA be its density of states. For every d, denote by Ad the pullback T G T of A to a random d-lift of . Then, for every k we have that G G Z 1 k k lim E[Tr(Ad)] = x dµA (x). d d T →∞ R Recently, using tools from free probability, Bordenave and Collins showed that the convergence above holds in a much stronger sense, implying convergence of the edges of the spectrum [BC19]. Their theorem also handles half-loops e, for which Ue is deﬁned to be a random matching (see [FK19] for a discussion of related models of random lifts). Bordenave and Collins viewed the limiting operator A as an element of a certain group C∗-algebra. We expand this discussion in Section4. T

2.3 Spectral Radii of Operators on Universal Covers

In this discussion we fix a finite graph and let A denote is adjacency matrix (that is ae = 1 and G G bv = 0 for all e and v), and we denote the universal cover of by . We denote the spectral radius G T of A and A by ρ( ) and ρ( ) respectively. ρ( ), ρ( ) G T G T G T The quantity ρ( ) is fundamental in the theory of Ramanujan graphs [LPS88]. Indeed, in T general, a graph is said to be Ramanujan if σ( ) is contained in [ ρ( ), ρ( )] ρ( ), ρ( ) G G − T T ∪ {− G G } [Gre95]; and many of the results in the area are stated in terms of ρ( ). For example, the main result T in [MSS13] states that any graph has a 2-lift 0 such that the new eigenvalues of A are bounded G G G0 above by ρ( ), while its generalization in [HPS18] shows that the same is true for n-lifts for every T n 2. Similar techniques have been used in [MO20] to obtain analogous results for quotients of ≥ a class of infinite graphs that goes beyond universal covers, where the results are also given in terms of the spectral radius of the given infinite graph. We refer the reader to [SS92, HR19] for discussions on the relation between ρ( ) and ρ( ). G T In most cases ρ( ) is mentioned only as an implicit quantity and no quantitative bounds on it T are provided. When is d-regular, by the work of Kesten [Kes59] we know that ρ( ) = 2 √d 1. If G T − is a (c, d)-biregular bipartite graph, by Godsil [GM88] we know that ρ( ) = √c 1 + √d 1. In G G − − the case of general graphs, Hoory [Hoo05] proved that if davg( ) is the average degree of then G G q ρ( ) 2 davg( ) 1. (6) T ≥ G −

On the other hand, an upper bound can be obtained trivially by noting that if dmax( ) is the G maximum degree of , then can be embedded in the infinite dmax( )-regular tree and hence G T G p 2 dmax( ) 1 ρ( ). (7) G − ≥ T Therefore, if is a regular graph we have davg( ) = dmax( ), so by putting together (6) and (7) the G G G formula of Kesten is recovered. It is hard to find any explicit formula for ρ( ) that only depends on the adjacencies in and T G not, for example, on the paths in or the powers of A . This is due in part to the fact that two G G similar base graphs may have universal covering trees with fairly different spectral radii. In the table below we used the system of equations in Corollary 6.7 to compute the spectral radii of the respective universal covering trees in Table1.

13 Base graph ρ( ( )) G T G 3.0368 G1 q ≈ 1 2 (7 + √33) 2.5243 G 2 ≈ s q ! 1 3 6 + √5 + 27 + 6 √5 2.7012 G 2 ≈ 2.6589 G4 ≈ 5 1 + √2 2.4142 G ≈ 6 2.4461 G ≈ Table 2: Using Mathematica, the system of equations in Corollary 6.7 was solved for the graphs in Table 1. In some cases explicit solutions in radicals were output. Previous results on the biregular bipartite case mentioned above imply the result for . G5

2.4 Graph Products and Noncommutative Probability In recent years different problems in spectral graph theory have been approached from the perspective of non-commutative probability theory; we refer the reader to the book of Hora and Obata [HO07] for a unified exposition. Of particular interest for the present work are a sequence of results [Oba04, ABGO04, ALS07] which establish a correspondence between different combinatorial graph products and different notions of stochastic independence in non-commutative probability (see Example 2.4 below). We summarize this correspondence in the dictionary presented in the table below. As mentioned before, part of the motivation of this project is to extend this dictionary to include the notion of freeness with amalgamation5.

Graph product Notion of independence Cartesian Classical (tensor) Free Free Comb Monotone Star Boolean

Table 3: On the left, graph products are mentioned (see [ABGO04] for deﬁnitions). On the right, the corresponding notions of stochastic independence are given (see [SW97, Mur01] for deﬁnitions of Boolean and Monotone independence, respectively.)

Example 2.4 (Cartesian product). Given two graphs and 2, one may form their Cartesian product G1 G 2, with vertex set and edge set as follows: G1 G

V( 2) = V( ) V( 2) G1 G G1 × G

E( 2) = (v, w ), (v, w2) : w , w2 E( 2) (v , w), (v2, w) : v , v2 E( ) . G1 G {{ 1 } { 1 } ∈ G } ∪ {{ 1 } { 1 } ∈ G1 } 5Graph products related to the notion of amalgamation in group theory, such as the one appearing in [Moh06], have appeared in the past. However, these unrelated products were introduced with the purpose of studying symmetries in graphs and it is not clear if they relate to spectral theory.

14 In other words, we have the relation of adjacency matrices A = A I + I A . Using this G1 G2 G1 ⊗ ⊗ G2 representation, one sees that the spectrum of 2 is the Minkowski sum λ +λ2 : λ σ( ), λ2 σ( 2) . G1 G { 1 1 ∈ G1 ∈ G } In the language of spectral measures, this says that the density of states µ of A 1 2 is the (classical) convolution of the density of states for the constituent graphs; that is, µ is theG lawG of the sum of two independent random variables distributed according to the density of states of A and of A , respectively. G1 G2 In this work, the free graph product is of particular interest. The connections of this product to free probability were developed by Accardi, Lenczewski and Sałapata [ALS07]. The authors pointed out that Voiculescu’s notion of free independence introduced in [Voi85] could be used when analysing the spectrum of free products of graphs. As an example provided in their paper, one can recover the spectral measure of the d-regular infinite tree, as the free convolution of d signed Bernoulli distributions (also known as Rademacher distributions). However, the roots of the theory of free graph products go back to Kesten [Kes59], who studied the spectra of random walks on free groups. We survey this area in the next subsection. Finally, we draw attention to the additive graph product recently defined by Mohanty and O’Donnell [MO20], who showed that X-Ramanujan graphs (a generalized notion of Ramanujan graph) can always be obtained by taking certain quotients of any graph constructed via their product. In the aforementioned work the features of the spectrum of the resulting infinite graphs are left as implicit quantities. Hence, a natural complementary line of research would then be that of understanding the spectrum of the infinite graphs arising from the additive graph product. Understanding the intersection between the class of graphs obtained via the amalgamated free graph product and the additive graph product would shed some light on this question.

2.5 Random Walks on Groups and Free Probability The ’80s and ’90s saw a ﬂurry of activity on the spectra of random walks on Cayley graphs of free products of groups; see [Woe00] for a detailed exposition. The analytic formula relating the spectral measure of the product graph to the spectral measures of its factors (essentially, Voiculescu’s R-transform) was discovered independently by McLaughlin [McL88], Soardi [Soa86], and Woess [Woe86], but it was Voiculescu’s work [Voi85] that put it into the more general context of non-commutative probability. Interestingly, also in [Voi85], Voiculescu laid out a more general theory of freeness with amalgamation, which extends the scalars C to an arbitrary unital algebra . On the other hand, in a parallel B way, some progress was also made by the graph theory community in the study of random walks on amalgams [PW85]. In particular Cartwright and Soardi [CS86] developed the combinatorial tools to obtain the Green function of the Cayley graph of an amalgamated free product of ﬁnite groups, in the particular case in which the subgroup over which amalgamation is performed is a normal subgroup of the groups in the product. In Section 3.3 we explain why the tools of free probability allow one to approach this problem even if the normality assumption is dropped.

3 Preliminaries on Free Probability

In this section we describe the tools from the theory of free probability that will be used throughout this preprint. A basic background on C∗-algebras is recommended, but not necessary for all of the following discussion. We refer the reader to [Dav96] for an introduction to C∗-algebras.

15 3.1 Free Probability Free probability was introduced by Dan-Virgil Voiculescu in his seminal papers [Voi85, Voi86]. We refer the reader to [VDN92] for a detailed introduction. This theory is developed in the context of non-commutative probability, in which random variables are viewed as elements of a non-commutative algebra and the notion of expectation from classical probability is substituted by a linear functional on the algebra. Deﬁnition 3.1 (Non-commutative probability space). A non-commutative probability space is a pair ( , τ) where is a unital C-algebra and τ : C is a unital linear map. A A A → In this work the following type of non-commutative probability space is of primary importance. Example 3.2. Let G be a discrete group and denote the reduced C -algebra of G by C (G).6 Then the C ( ) ∗ red∗ red∗ · C∗-algebra Cred∗ (G) has a canonical state given by τe

def τe(x) = δe, xδe , x C∗ (G), (8) h i ∀ ∈ red 2 where δe ` (G) denotes the indicator of the identity element e G. ∈ ∈ As Voiculescu showed, in non-commutative probability there is not a unique notion of stochastic independence.

Deﬁnition 3.3 (Free independence). Let ( , τ) be a non-commutative probability space, and let i i I A {A } ∈ be a family of unital subalgebras of . We say that the algebras i are freely independent (or just free) if A 7 A τ(a an) = 0 whenever ai j , j , j2 , , jn and τ(ai) = 0. 1 ··· ∈ A i 1 ··· Sets of random variables are said to be freely independent if the algebras they generate are free. In the proof of Theorem 1.7 we will use the following.

8 Proposition 3.4 (Voiculescu). Let 11,..., 1m be the m canonical generators of Fm and λ be the left regular Fm 2 representation of Fm on ` (Fm). Then, the random variables λ(11), . . . , λ(1m) are free in the non-commutative F probability space (Cred∗ ( m), τ), where τ is as in (8).

Note that in Proposition 3.4 the random variables λ(1i) are unitaries and satisfy that

k τ(λ(1i) ) = 0 k Z 0 . (9) ∀ ∈ \{ } In non-commutative probability, random variables satisfying (9) are called Haar unitaries.

3.2 Operator-valued Probability Spaces We will require the following generalization of the notion of non-commutative probability space. Deﬁnition 3.5 (Operator-valued probability space). A triple ( , E, ) is called an operator-valued A B probability space if is a unital algebra, with 1 , and E : is a conditional expectation, A B ⊂ A A ∈ B A → B i.e. E is a linear map satisfying E = Id and E[b1ab2] = b1E[a]b2 for every a and b1, b2 . Often, B B ∈ A ∈ B to stress the choice of we will often say that ( , E, ) is a -valued probability space. B A B B 6 2 2 In other words, Cred∗ (G) is the norm closure in B(` (G)) of the left regular representation of G on ` (G). See [Dav96, §7]. 7 Here and throughout, we use this shorthand to mean ji , ji+1 for all 1 i < n. 8 ≤ Here, Fm denotes the free group on m generators.

16 In the present we are interested in the following situations.

Example 3.6 (Matrices with entries in the algebra). Let ( , τ) be a non-commutative probability space. A Then, the algebra Mn(C) (i.e. n n matrices with entries in ) has a copy of Mn(C) as a subalgebra, ⊗ A × A namely Mn(C) 1 . Hence, Mn(C) can be viewed as an Mn(C)-valued probability space with the ⊗ A ⊗ A n conditional expectation Id C τ. In other words, if X Mn(C) is given by X = (xij) then E Mn( ) ⊗ ∈ ⊗ A i,j=1 deﬁned by def n E(X) = (τ(xij)) Mn(C), (10) i,j=1 ∈ is a Mn(C)-valued conditional expectation.

Example 3.7 (Group C -algebras). Let H be a subgroup of G, and denote by the canonical copy of C (H) ∗ B red∗ inside C (G). Then, we can view C (G) as a -valued probability space, by considering the canonical red∗ red∗ B conditional expectation E : C (G) , namely the projection of C (G) onto that is orthogonal with red∗ → B red∗ B respect to the GNS inner product induced by the canonical trace of Cred∗ (G).

3.3 Freeness with Amalgamation One of the main technical ingredients of this preprint is Voiculescu’s notion of freeness with amalagamation [Voi85, §5]. See [VDN92, §3.8] and [MS17, §9] for an introduction to the subject, and [Jek18] for a detailed development.

Definition 3.8 (Freeness with amalgamation). Consider an operator-valued probability space ( , E, ). A B Let i i I be a family of subalgebras of such that i for every i I. We say that the algebras i are {A } ∈ A B ⊂ A ∈ A free with amalgamation over if E(a an) = 0 whenever ai j , j , j2 , , jn and E(ai) = 0. B 1 ··· ∈ A i 1 ··· Note in the above definition that in the particular case in which C, we recover the usual B definition of free independence. In this work we exploit the following relation between freeness and freeness with amalgamation.

Observation 3.9. Let ( , τ) be a non-commutative probability space and 1, 2 be freely independent A n nA A ⊂ A subalgebras. Let X, Y Mn(C) with X = (xij) and Y = (yij) . If xij and yij 2 for ∈ ⊗ A i,j=1 i,j=1 ∈ A1 ∈ A every i, j = 1,..., n, then X and Y are free with amalgamation over Mn(C) with respect to the conditional expectation deﬁned in (10).

The connection with the amalgamated free product of groups is the following.

Proposition 3.10 (Voiculescu). Let G1, G2 be discrete groups with a common subgroup H. Let G be def the group free product with amalgamation of G , G2 over H, i.e. G = G H G2. Consider the inclusions 1 1 ∗ ρ0 : Cred∗ (H) Cred∗ (G) and ρi : Cred∗ (Gi) Cred∗ (G) for i = 1, 2. Now, as in Example 3.7 put def → → = ρ0(C (H)) and view C (G) as a -valued probability space. Then ρ (C (G )) and ρ2(C (G2)) B red∗ red∗ B 1 red∗ 1 red∗ are free with amalgamation over . B We are now ready to explain in precise terms the way in which free probability appears in the study of the spectra of Cayley graphs. Consider a ﬁnitely generated group G and let S G be a ⊂ ﬁnite generating set. Denote by Γ(G, S) the (left) Cayley graph of G with respect to S. When it is clear from the context, we will simply write Γ. Let λ be the left regular representation of G, and note that the operator TΓ

17 def X TΓ = λ(s) C∗ (G) ∈ red s S ∈ is the adjacency operator of Γ. Indeed, if G is identiﬁed with the vertices of Γ, then for every 1 G ∈ X TΓ(δ1) = δs1. s S ∈ 1 Moreover, if S is symmetric, meaning S = S− , then clearly Γ is undirected and TΓ is self-adjoint.

Observation 3.11. Let G , G2 be two finitely generated groups with a common subgroup H. Let S G 1 1 ⊂ 1 and S2 G2 be respective finite generating sets. For j = 1, 2, let ιj : Gj G1 H G2 be the canonical ⊂ → ∗ 2 inclusion and let S = ι1(S1) ι2(S2). Let λ be the left regular representation of G1 H G2 on ` (G1 H G2). def def∪ def ∗ ∗ Let Γ = Γ(G H G2, S), Γ = Γ(G , S ) and Γ2 = Γ(G2, S2). Then we have the following decomposition 1 ∗ 1 1 1 X X X TΓ = λ(s) = λ(s) + λ(r) = TfΓ1 + TfΓ2 , s S s ι (S ) r ι (S ) ∈ ∈ 1 1 ∈ 2 2 2 where TfΓ denotes the natural lift of TΓ C (Gj) into C (` (G H G2)). From Proposition 3.10 we j j ∈ red∗ red∗ 1 ∗ f f know that TΓ1 and TΓ2 are free with amalgamation over Cred∗ (H) with respect to the conditional expectation E : C (G H G2) C (H) defined as in Example 3.7. red∗ 1 ∗ → red∗ 3.4 The Amalgamated Free Product of Hilbert -modules B Note that if ( , E, ) is an operator-valued probability space then naturally can be viewed as a A B A -bimodule, in which case E being a conditional expectation is equivalent to E being a -bimodule B B homorphism. In the remaining of this section we will assume that is a unital C -algebra and we B ∗ will focus on the following type of -valued probability space. B Example 3.12 (Operators on Hilbert modules). This setup is based on [Voi85, 5.1]. Assume that B is a unital C -algebra and that is a Hilbert -bimodule.9 Furthermore assume that ξ satisfies ∗ H B ∈ H def ξ, ξ = 1 . Let ξ = ξb : b and consider the decomposition = ξ ◦ , where ◦ denotes the h iH B B { ∈ B} H B ⊕ H H orthogonal complement of ξ with respect to the -valued inner product of . Denote the -algebra of B B H ∗ bounded, adjointable, right -linear operators on by eB( ) and assume that there is a -algebra homorphism eB( ) B H H ∗ H ◦ χ : eB( ), satisfying that χ(b )(ξb2) = ξb b2 for every b , b2 and that is invariant under the B → H 1 1 1 ∈ B H action of χ( ) . Then, via χ we can view as a subalgebra of eB( ) and it is easy to verify that under the B B H aformentioned conditions the triple (eB( ), E, ) is a -valued probability space, if we define H B B def E(X) = ξ, Xξ , X eB( ). h iH ∀ ∈ H

Now consider a family of Hilbert -bimodules i i I. For every i I suppose that ξi i, i = B {H } ∈ ∈ ∈ H H ◦ ξi i, eB( i) and χi : eB( i) are as in Example (3.12) and that the respective assumptions B ⊕ H H B → H def are satisﬁed. Then, as mentioned above, by taking Ei(X) = ξi, Xξi we can view each eB( i) as a h iHi H 9Roughly speaking, a right Hilbert -module is a - -bimodule with a -valued inner product for which is B B B B H complete in some sense. We refer the reader to [Spe98, §4.6], [Bla06, §2.7] or [Jek18, §1.2] for a development of this and other related concepts.

18 -valued probability space. We will now explain how to construct a bigger -valued probability B B space in which we can ﬁnd copies of the eB( i) that are free with amalgamation over . We refer H B the reader to [Voi85, §5] for the original source and [Jek18, §4] for an extended exposition. With this end deﬁne the amalgamated free product over of the i as B H M def def ◦ ◦ ( , ξ) = ( i, ξi) = ξ i1 in H ∗i BI H B ⊕ H ⊗B · · · ⊗B H i ,i , ,in ∈ 1 2 ··· Next, for every i I we consider the subspace of ∈ H M def ◦ ◦ (i) = ξ i in . H B ⊕ H 1 ⊗B · · · ⊗B H i,i ,i ,i , ,in 1 1 2 ···

Now define the identifications Vi : i (i) by sending ξi ξ to ξ, and in the obvious H ⊗B H → H ⊗ ◦ ◦ ◦ ◦ ◦ ◦ way identifying i ξ with i, each ξi ( i in ) with i in and each H ⊗ H ⊗ H 1 ⊗B · · · ⊗B H H 1 ⊗B · · · ⊗B H ◦ ◦ ◦ ◦ ◦ ◦ i ( i in ) with i i in . Then, for every i I, define ρi : eB( j) eB( ) H ⊗B H 1 ⊗B · · ·⊗B H H ⊗B H 1 ⊗B · · ·⊗B H ∈ H → H by def 1 ρi(X) = Vi(X Id (i))Vi− . ⊗ H Note that also ξ defines a -valued conditional expectation on eB( ) given by ∈ H B H def E(X) = ξ, Xξ . h i Theorem 3.13 (Voiculescu). The above construction satisfies the following :

i) For every i I, ρi : eB( i) eB( ) is an injective -algebra homomorphism. ∈ H → H ∗

ii) For every i I and every X eB( i) it holds that E(ρi(X)) = Ei(X). ∈ ∈ H iii) The family of subalgebras ρi(eB( i)) i I of eB( ) are free with amalgamation over with respect to E. { H } ∈ H B 3.5 The Operator-Valued R-Transform In what follows we consider an operator valued space ( , E, ) with and being unital A B A B C -algebras. Moreover, we assume that E is positive, meaning that E[XX ] 0 for all X . ∗ ∗ ≥ ∈ A In the scalar case, namely when C, Voiculescu’s R-transform is used to compute the B distribution of sums of free random variables. See [VDN92, §3] for an introductory reference. The same machinery extends without many modifications to the operator-valued context. A good introductory reference for this subject is [MS17, §9]. The first step towards defining the operator-valued R-transform is to define the operator-valued version of the Cauchy transform, which essentially is the conditional expectation applied to the resolvent.

def Deﬁnition 3.14 (Operator-valued Cauchy transform). Let H+ = b : Im(b) > ε1 , for some ε > 0 A def + { ∈ B } and H− = H . Then, for any self-adjoint X , the operator-valued Cauchy transform of X is the − + ∈ A function GX : H H− given by GX( ) → def 1 · GX(b) = E[(b1 X)− ]. (11) A −

19 Observation 3.15. We can recover the scalar Cauchy transform from the operator-valued Cauchy transform. More specifically, consider a non-commutative probability space ( , τ) and a unital subalgebra A B ⊂ A together with a conditional expectation E : . Furthermore, assume that τ and E are compatible in the A → B sense that τ(X) = τ(E(X)) for every X . ∈ A Let X be self adjoint and let µ be the probability measure on R given by the distribution of X with ∈ A respect to τ, i.e., µ is the unique probability measure whose k-th moment equals τ(Xk). Then, for z C with ∈ z we have that Im( ) > 0 Z 1 τ(GX(z1 )) = dµ(t), B z t R − where GX is as in (11). Without going into much detail we recall that the operator-valued Cauchy transform has a left inverse when restricted to a proper neighborhood of infinity, so the following definition makes sense.

Deﬁnition 3.16 (Operator valued R-transform). Let X be self-adjoint. In a suitable neighborhood of ∈ A 0 , the operator valued R-transform of X is well deﬁned by the following equation RX( ) B ∈ B ·

bGX(b) = 1 + RX(GX(b))GX(b), (12) where GX(b) is as in (11). Many things are known about the R-transform (both the scalar and operator-valued). Here we will only need the following fact.

Theorem 3.17 (Voiculescu). Let X, Y be self-adjoint random variables that are free with amalgamation ∈ A over , then B RX+Y = RX + RY, where RX and RY are as in Deﬁnition 3.16.

4 Band Structure of Universal Covers

4.1 Sunada’s Theorem via Asymptotic Freeness In this section, we give an alternate route to a theorem of Sunada [Sun92, ABS20] upper-bounding number of connected components of the spectrum of the universal cover. We also point out a generalization to the case where the base graph is allowed to contain half-loops. We state the theorem below:

Restatement of Theorem 1.7. Let A be a Jacobi matrix on a graph with n vertices, and let A be its G G T lift to the universal cover of . Then the density of states of A assigns a positive integer multiple of 1/n to G T each connected component of the spectrum of A . Consequently, the spectrum of A contains at most n connected components. T T

Let A be a Jacobi matrix on a ﬁnite graph with n vertices (and no half-loops). Recalling DeﬁnitionG 2.2, the pullback of A to a random d-lift of is given by the random nd nd Hermitian G G × matrix

20 X X bi∆ii I + ae(∆ij Ue + ∆ji U∗) ⊗ ⊗ ⊗ e i V( ) e= i,j E( ) ∈ G { }∈ G where Ue e E( ) are independent uniform random permutation matrices conditioned on Uı(e) = Ue∗ { } ∈ G for all e, and ∆ij denotes the n n matrix with a 1 in the entry (i, j) and a 0 everywhere else. × As usual, we can view the Ue in the above deﬁnition as elements of the non-commutative 1 probability space of d d random matrices with state E Tr. It easy to see that each Ui converges × d ◦ in distribution to a Haar unitary random variable. The asymptotic joint distribution is given by the following classical result of Nica.

Theorem 4.1 (Nica [Nic93]). Fix m > 0. For every d let U(d),..., U(d) be a family of independent uniform { 1 m } d d permutation matrices. Then, with respect to the state 1 Tr ( ), the family U(d),..., U(d) converges in × d d · { 1 m } distribution to a family u ,..., um of free Haar unitaries as d goes to infinity. { 1 } This result of Nica says that independent random permutation matrices converge to free Haar unitaries. On the other hand, the result in Proposition 3.4 gives a construction for a family of free F Haar unitaries in Cred∗ ( m), while Lemma 2.3 above says that random lifts converge in distribution to A . These arguments put together yield the following lemma, which appeared implicitly in [BC19T ]. Lemma 4.2. Let A be a Jacobi matrix on a graph with n vertices (with no half-loops), and let A be its G G T pullback to its universal cover. Let γ : E( ) [m] be a numbering of the edges (i.e. γ is a bijection) and let 1 1 G →F F 1,..., m be the canonical free generators of m. Let τ be the canonical trace on Cred∗ ( m), and let λ denote 2 the left regular representation of Fm on ` (Fm). Then the density of states of A is the same as the spectral measure of T X X 1 bi 1 + ae((∆ij λ(1 ) + ∆ji λ(1− )) (13) ⊗ ⊗ γ(e) ⊗ γ(e) i V( ) e E( ) ∈ G e∈= i,Gj { } 1 in the non-commutative probability space (Mn(C) C (Fm), Tr τ). ⊗ red∗ n ⊗ Now the problem has been reduced to studying the spectral measure of a particular operator 10 that lives in a well studied C∗-algebra. In order to study the band structure of the spectrum of this random variable a standard trick in operator K-theory will be used. In the context of graph theory, this technique was first used by Aomoto [A+88] and has been used in the study of different infinite graphs by different authors (e.g. [Sun92, KFSH19]). The main technical result that one needs from the theory of C∗-algebras is the following. F Proposition 4.3. Let m and n be positive integers and τ be the canonical trace in Cred∗ ( m). Then, if P Mn(C) C (Fm) is a projection, (Tr τ)(P) is a non-negative integer. ∈ ⊗ red∗ ⊗ Using a standard K-theory argument it can be seen that the above proposition follows from a deep and technical result of Pimsner and Voiculescu [PV82]. A self-contained proof appears in [ABS20]. Sunada’s theorem now follows from a simple argument:

10 A similar approach is used by [ABS20] using a diﬀerent C∗-algebra. Their approach does not use asymptotic freeness or any other free probability concept.

21 Proof of Theorem 1.7. We use Lemma 4.2; let X denote the operator in (13). Let I be a connected def T component of σ(A ) = σ(X ). Define f (x) = χI(x) to be the indicator function of the set I. Since I is a connected componentT T of the spectrum, f is a continuous function on σ(X ) and hence T2 f (X ) Mn(C) C∗ (Fm). Moreover, since the range of f is contained in R and f = f , by the T ∈ ⊗ red continuous functional calculus f (X ) is a projection. T On the other hand, if µA is the density of states of A , we have that T T Z 1 µA (I) = µX (I) = χI(t)dµX (t) = Tr τ ( f (T)). T T σ(X ) T n ⊗ T So by Proposition 4.3, the quantity above is k/n for some integer k 1, as desired. ≥ If the base graph has half-loops, a similar result can be proven. In this case the random G permutations in the lift associated to half-loops should be taken to be random matchings. By the work in [Nic93], asymptotic freeness also holds for independent random matchings and independent random uniform permutations. Hence, for some m1 and m2, the analog of Lemma 4.2 still holds if one replaces the C -algebra with C (Fm Γm ), where Γm is the free product of m2 ∗ red∗ 1 ∗ 2 2 copies of Z2. The K-theory of this C∗-algebra is also well understood (see for example [Sun92]), and hence one can get an analog of Proposition 4.3. In this case, one obtains that the spectrum of A can have at most 2n bands. T The 2n result is again tight since can be a finite tree with a half-loop added to some vertex, in G which case will be a finite tree with 2n vertices. If one allows the coefficients bv to be nonzero, T then one can obtain many more examples with 2n bands. In particular we have the following example, which was pointed out to us by Barry Simon. Roughly speaking, one may take n disjoint copies ,..., n of a fixed finite graph (and its graph Jacobi matrix), add different large multiples G1 G G of the identity to the Jacobi matrix on each i, and then connect the i by some edges with very G G small weights ae. This produces a connected graph whose universal cover has n times as many G0 spectral bands as A . If is a single vertex with a whole-loop and a half-loop with coefficients G G chosen so that the spectrum of A ( ) has 2 bands by Theorem 4.4, then A ( 0) has 2n bands in its spectrum. T G T G

4.2 Spectral Splitting Theorem 1.7 provides an upper bound to the number of bands in the spectrum of the universal cover, and we have provided examples of when this bound is tight above. A natural further question is to determine what properties of a graph result in one band, two bands, and so on, as the number of bands has some relevance in physics (see [ABS20]). In this section, we discuss some interesting examples yielding various numbers of bands. One way to vary the number of bands in the spectrum of the universal cover is to fix the base graph and vary the coefficients ae. For the specific case of regular trees, Figa-Talamanca´ and G Steger [FTS94] gave an explicit description of this phenomenon. Here, to make it compatible with our context, the theorem below paraphrases Lemma 1.4 in Chapter 2 of [FTS94].

Theorem 4.4 (Figa-Talamanca,´ Steger). Let be the graph with two vertices u, v and d parallel edges G e ,..., e connecting them. Assume that ae ae > 0 and bu = bv = 0. Then, zero is in the spectrum 1 d 1 ≥ · · · ≥ d

22 of A ( ) if and only if T G Xd a2 a2 . (14) e1 ≤ ei i=2 Let be as in the above theorem. Note that from Theorem 1.7 it follows that the spectrum of G A ( ) has at most two bands. Combining this with the fact that the spectrum is symmetric about zeroT G (as is bipartite), we have the following: T Observation 4.5. Using the notation of Theorem 4.4, A ( ) has a connected spectrum if and only inequality T G (14) is satisﬁed. Moreover, this remains true if we add a constant to bu and bv.

We pause to mention some examples. If d = 2k + 1 and the aei are chosen to be distinct and in such a way that (14) holds, we get a (2k + 1)-regular tree with non-constant coeﬃcients and connected spectrum. On the other hand, if d = 2k and the aei are chosen with the same characteristics 11 as above, we get that A ( ) has connected spectrum . More can be said aboutT G the graph from Theorem 4.4: G Proposition 4.6. Using the notation of Theorem 4.4, if bu and bv are instead taken to be arbitrary distinct reals, then the spectrum of A ( ) has two bands. T G Proof. Applying Lemma 4.2, in particular we obtain that the spectrum of A ( ) is the spectrum of an operator-valued matrix of the form T G ! b1 x M2(C) Cred∗ (Fd). x∗ b2 ∈ ⊗

Since b1 , b2 by assumption, let us subtract a suitable multiple of the identity to obtain ! t x X := x t ∗ − for some real t; this merely translates the spectrum. Note that the spectrum of X is symmetric ! ! 1 0 1 12 2 y 0 2 about zero, as UXU− = X for the unitary U = − . Since X = where y = xx∗ + t is − 1 0 0 y∗ invertible, 0 is not in the spectrum of X. Thus the spectrum has a gap, as desired.

Now we show that non-constant-degree universal covers with similar characteristics can be constructed. To this end, we will now use to denote the graph consisting of two vertices u, v, G with a loop e1 on u and two parallel edges e2, e3 connecting u and v. Put bu = bv = 0 and assume aei > 0. As in the example of regular trees, since has two vertices, we also have that the spectrum of G A ( ) is connected if and only if it contains zero. By Lemma 4.2, determining the invertibility of T G A ( ) is equivalent to deciding if the following operator-valued matrix is invertible T G ! def ae1 (λ(11) + λ(11)∗) ae2 λ(12) + ae3 λ(13) X = M2(C) Cred∗ (F3), ae2 λ(12)∗ + ae3 λ(13)∗ 0 ∈ ⊗

11These two examples answer Conjectures 9.6 and 9.7 in [ABS20] in the negative, and we thank Barry Simon for pointing out that the case d = 3 negatively answers their Conjecture 10.5. 12We thank Barry Simon for this observation.

23 Figure 1: The graph and a ﬁnite portion of its universal cover ( ), along with an approximate eigenvector for zero. G T G

where 11, 12 and 13 are the canonical generators of F3.

We separate our analysis into two cases. First assume that ae2 , ae3 . We can further assume without loss of generality that ae2 > ae3 . Note that ! a 1 1 1 e3 1 11 ae2 λ( 2) + ae3 λ( 3) = ae2 λ( 2) 1 + λ( 2− 1) . (15) ae2 aer 1 11 aer 1 11 From ae2 > ae3 we have a λ( 2− 1) < 1, which implies that 1 + a λ( 2− 1) is invertible, and in k e2 k e2 def 1 view of (15) this implies that ae2 λ(12) + ae3 λ(13) is also invertible. Set x = (ae2 λ(12) + ae3 λ(1e3 ))− and ! 0 x Y = ∗ . x ae x(λ(1 ) + λ(1 ) )x − 1 1 1 ∗ ∗

It is easy to see that XY = I2 1 and hence that zero is not in the spectrum of A ( ) when ae2 , ae3 . ⊗ T G Now assume that ae2 = ae3 . In this case one can show that A ( ) is not invertible. Indeed, let xn be the vector whose entries alternate between 1 and 1 along Tn Gconsecutive degree-two vertices − on the y-axis of ( ) as depicted in Figure1. Then xn = √n while A ( )xn = √2, so xn is a T G k k k T G k { } sequence of approximate eigenvectors for zero. Alternatively, one can show that X is not invertible by noting that X(1, 2) is a scalar multiple of λ(12) + λ(13), which is not invertible since its spectrum is z C : z √2 (see Example 5.5 in [HL00]). This means that both X(1, 2) and X(2, 1) are not { ∈ | | ≤ } invertible, and hence X is not invertible. This discussion can be summarized as follows. Example 4.7. Let be the graph consisting of two vertices u, v, with a loop e on u and two parallel G 1 edges e2, e3 connecting u and v. Put bu = bv = 0 and assume that aei > 0 for i = 1, 2, 3. Then, if ae1 = ae2 , the spectrum of A ( ) is connected; otherwise it has two bands. T G This example disproves Conjecture 9.5 in [ABS20], since it provides a graph of non-constant G degree and speciﬁc coeﬃcients for which A ( ) has a connected spectrum. T G

24 Finally, in [ABS20] an interesting conjecture was made regarding the possibility of generalizing the Borg-Hochstadt theorem. Roughly speaking, in the language of our work, it was conjectured that for an arbitrary universal cover , if the cumulative distribution function of the density of T states of A is of the form j/p inside every gap, then there exists a quotient of , say , such that T T G V( ) is a divisor of p.13 In some sense, Example 4.7 is already a counterexample of this conjecture, | G | since in this case when ae1 = ae2 there is only one band in the spectrum, while the smallest quotient of the universal cover has two vertices. However, it is still of interest to ﬁnd an example with an interior spectral gap where the extension of the Borg-Hochstadt theorem does not hold. We do this below14. Let m 4 and let 1 ,..., 1m be the canonical generators of Fm. Take x C (Fm) self-adjoint. If ≥ 1 ∈ red∗ x is invertible then     x λ(1 ) 0 x 1 0 x 1λ(1 1 )  1   − − 1 2     −   λ(11)∗ 0 λ(12)   0 0 λ(12)  = I3 1. (16)    1 1 1 1 1 1 1  ⊗ 0 λ(12) 0 λ(1 1 )x λ(1 ) λ(1 1 )x λ(1 12) ∗ − 2− 1− − 2− 2− 1− − 1

Now consider a graph with three vertices u, v, w and edges e , e2 connecting u with v, and v with w, G 1 respectively. Assume that bu = bv = bw = 0. If we add whole-loops on the vertex u then by Lemma 4.2 A ( ) will have the form of the ﬁrst matrix in the left side of (16). Moreover, x will correspond to theT loopsG on u and it can be made invertible by putting at least two loops on u and varying their Jacobi coeﬃcients as shown in Example 4.7. So, for the cases where x is invertible, zero will not be in the spectrum of A , but by Theorem 1.7 and since the density of states is symmetric about zero, this means that A hasT exactly two bands, each with mass 1/2. This provides an example of a universal cover whose smallestT quotient has three vertices, and where the mass of the bands has the form j/2.

5 The Amalgamated Free Product for Graphs

In Section 5.1 we give a combinatorial description of our product and show that it extends both the free product of graphs given in [Que94] and the notion of universal cover of a graph. At the end of Section 5.1 we state Theorem 5.8, the main result of this section, which explains the use of having such a product. In Section 5.2 we delve into the technicalities of the product, showing the connection between the combinatorial description given in Section 5.1 and the construction of freeness with amalgamation described in Section 3.4. To lighten notation, in the remaining of this section for a graph we will write = ( , , e) to G G V E denote that is ( ), is E( ), and e is the root of .( , , e) V V G E G G V E 5.1 Combinatorial Description First, following the presentation in [ALS07] we recall Quenell’s free product of rooted graphs. Let = ( , , e ),..., n = ( n, n, en) be rooted graphs and for every i [n] denote G1 V1 E1 1 G V E ∈ ◦ def n ◦ i = i ei . The free product of rooted graphs will be denoted by i i=1. i V V \{ } n ∗{G } V n The vertex set of i will be the following set of words: i ∗{G }i=1 ∗{V }i=1 13Actually, the conjecture was stated in terms of the notion of period discussed in [ABS20], where an ultimate deﬁnition of period was left open. However, there does not seem to be a sensible deﬁnition of period that rules out the

25 n def ◦ i = e vm v : v i and i , i for 1 k m 1, m 1 . (17) ∗ {V }i=1 { } ∪ { ··· 1 k ∈ V k k k+1 ≤ ≤ − ≥ } As in [ALS07] we think of e as the empty word and we allow the roots ei to appear in words also n playing the role of an empty word, that is, if w i i=1 then we have the convention wei = eiw = w. ∈ ∗{V } n The set of edges in the free product is deﬁned as follows i ∗{E }i=1 n n n o i i = vu, v0u : v, v0 i [n] i and u, vu, v0u Vi i . (18) ∗ {E } =1 { } { } ∈ ∪ ∈ E ∈ ∗{ } =1 n n n n In summary i = ( i , i , e). See Figure2 for an example. i ∗{G }i=1 ∗{V }i=1 ∗{E }i=1 ∗{G }i=1

Figure 2: On the left side of the equality we show two graphs 1, 2 rooted at their red vertices. On the right side we show the free product graph , also rooted at itsG redG vertex. G1 ∗ G2 We now expand the notion of free product of graphs by introducing the concept of relator graph. Definition 5.1 (Relator graph). Let be a finite set. A relator graph with colors in is a finite rooted C C graph = ( , , r) together with an edge coloring c : . G V E E → C Our amalgamated free product of rooted graphs = ( , , e ),..., n = ( n, n, en) will be G1 V1 E1 1 G V E defined relative to a relator graph ( , c) and a coloring of the i compatible with the coloring of the G G relator graph. The colorings together with the graph structure of the relator graph indicate how the i will be combined. Thus, one may obtain different results even if the i are fixed, if the relator G G ( , c) is modified. G Definition 5.2 (Amalgamated free product for graphs). Let = ( , , e ),..., n = ( n, n, en) G1 V1 E1 1 G V E be finite rooted graphs. Assume that each i comes equipped with an edge coloring ci : i i such that def Sn G E → C i j = for every i , j. Let = i and let ( , c) be a relator graph with colors in and k vertices. C ∩ C ∅ C i=1 C G C Denote = ( , , r) and set = [k]. G V E V n Then, the free product of the ( i, ci) with amalgamation over ( , c), which we denote by ( ,c) ( i, ci) i=1, G G ∗ G { G } is the rooted graph with vertex set

n def ◦ ( ,c) i i = (r, e) ([k] ), ∗ G {V } =1 { } ∪ × W ◦ def n where = ( i ) e , and whose edge set is W ∗{V }i=1 \{ } n def ( ,c) ( i, ci) i = (i, vu), (i0, v0u) : v, v0 j for some j [n], i, i0 and c(i, i0) = cj(v, v0) (19) ∗ G { E } =1 {{ } { } ∈ E ∈ { } ∈ E } counterexamples presented here. 14We should mention that our example consists of a graph with a leaf, while in [ABS20] only leaﬂess graphs were considered.

26 n where, as in (18), we are assuming that u, uv, uv0 ( ,c) i and allowing the ei to play the role of empty ∈ ∗ G {V }i=1 words. This product generalizes diﬀerent constructions.

Example 5.3 (Free products of graphs). Using the notation from 5.2, if each i has only one element and C is a graph with a single vertex and n loops, each colored with a diﬀerent color, we have G n n ( ,c) ( i, ci) i = i i . ∗ G { G } =1 ∗{G } =1 Hence, this construction generalizes Quenell’s free product of graphs.

Below in Section 5.3 we prove a stronger result: roughly, we show that if G1,..., Gn are discrete groups with a common subgroup H, and ,..., n are their Cayley graphs, then the Cayley graph G1 G of the amalgamated free product H Gi can be constructed as an amalgamated graph product of the ∗ i for a suitable choice of colorings and relator graph. G Example 5.4 (Universal cover). Let = ( , ) be a finite graph with n vertices and let m be the number G V E of edges in where every whole loop is counted twice. E First we consider the case where has no loops. Let c : [m] be a coloring that assigns a different G E → color to each edge. Then, by rooting at an arbitrary vertex r we turn ( , c) into a relator graph. G ∈ V G Now, for every i [m] take i to be a rooted graph consisting of two vertices (one of them being the root ei) ∈ G m connected by a single edge with color c(i). Then the graph ( ,c) i may be disconnected; however, the ∗ G {G }i=1 connected component containing the vertex (r, e) is isomorphic to the universal cover of . Moreover, the G root (r, e) is an element of the fiber of r. See Figure3 for an example. If has loops an extra step is needed. Define to be the set of edges obtained by splitting every whole G Ehalf loop in into two loops and let = ( , ). In this case we consider the coloring c : [m]. E Ghalf V Ehalf Ehalf → Then by rooting at r, we turn ( , c) into a relator graph. The rest is done as in the no-loop case. Ghalf Ghalf

Figure 3: Example of the construction of a universal cover using the amalgamated free product of graphs.

As Example 5.4 shows, even if the i and the are connected, the amalgamated graph product G G may be a disconnected graph. Typically, we will only care about the connected component containing the root, this motivates the following deﬁnition.

27 n Definition 5.5 (Core of the product). Using the notation of Definition 5.2 we define the core of ( ,c) i ∗ G {G }i=1 to be the connected component containing the vertex (r, e).

So far all the discussion in this section has been at the level of combinatorics, however, our main result here is about Jacobi operators. It is then necessary to clarify how the coefficients associated to a graph are incorporated to this framework. Note that conceptually weights on vertices are treated differently than loops, but Jacobi operators do not distinguish between these two objects. It is for this reason that working only with weighted adjacency matrices (i.e. assuming bv = 0 for all vertices) heavily simplifies our analysis. For the sake of clarity, and because we are introducing a new concept, we have decided to only discuss the simpler case in this work.

Definition 5.6 (Lifting coefficients). Using the notation of Definition 5.2, assume that for i = 1,..., n the (i) (0) Jacobi operators A i and A are weighted adjacency matrices. Let ae be the coefficients of A i , and ae be G G n G the coefficients of A . Then, these coefficients lift to ( ,c) as follows G ∗ G {G}i=1

def a = a(0) a(i) . (i,vu),(i0,v0u) i,i v,v { } { 0} { 0} When working with the amalgamated free product of graphs it is convenient to have the following notation.

Deﬁnition 5.7 (Notation). Fix = ( , ) a graph and c : a coloring of the edges. For each α G V E E → C ∈ C we denote by [α] the graph with the same vertex set as but that only includes those edges of color α, i.e. [α] G G G = ( , c 1(α)). G V − We are now ready to state the main theorem.

Theorem 5.8. Use the notation in Deﬁnition 5.2 and Deﬁnition 5.7. For i [n] and every α i let A ∈ ∈ C α and X be weighted adjacency matrices of i[α] and [α] respectively. Then, there exists a non-commutative α G G probability space ( , τ) and random variables T ,..., Tn with the following properties: A 1 ∈ A i) The sum T1 + + Tn distributes with respect to τ as the Jacobi operator corresponding to the core of n ··· ( ,c) i with respect to the state induced by its root. ∗ G {G }i=1 def ii) If k = , there exists a subalgebra with M (C) and a conditional expectation E : |V| B ⊂ A B k A → B compatible with τ, such that the Ti are free with amalgamation over with respect to E. B def iii) Fix i [n], let li = i and note that X A M (C) M (C) for every α i. Then, if we view ∈ |V | α ⊗ α ∈ k ⊗ li ∈ C M (C) as a non-commutative probability space with the functional induced by the root of i, and from li G there construct the M (C)-valued probability space (M (C) M (C), Ei, M (C)) as in Example 3.6, we k k ⊗ li k have that X X A α ⊗ α α ∈Ci distributes with respect to Ei as Ti with respect to E.

28 5.2 Proof of Theorem 5.8

Since we care about the explicit action of the operators in question we think of the Xα as elements 2 2 in B(` ( )) and of the A as elements of B(` ( i)) if α i. To lighten notation let V α V ∈ C def = B(`2( )). B V

In what follows, the notion of tensor will be used in diﬀerent ways, we will use , C and to ⊗B ⊗ ⊗ denote the tensor of -modules, C-algebras and C-vector spaces respectively. Although, to avoid B overloading, the subscript in will be dropped when using it to denote pure tensor elements in a ⊗B -module tensor product. B We begin by constructing the building blocks i to which we apply the construction of the H amalgamated free product for Hilbert modules described in Section 3.4.

Step 1: Construction of the operator-valued probability spaces. For every i [n] let ∈ def 2 i = ` ( i), H B ⊗ V and note that this is a - -bimodule with the right and left actions of given by B B B def 2 (X v)Y = XY v and Y(X v) = YX v X, Y and v ` ( i). (20) ⊗ ⊗ ⊗ ⊗ ∀ ∈ B ∀ ∈ V

Moreover, we can turn i into a Hilbert -bimodule by using the -valued inner product H B B def 2 2 X u, Y v i = u, v ` ( )X∗Y, X, Y and u, v ` ( i). (21) h ⊗ ⊗ iH h i Vi ∀ ∈ B ∀ ∈ V

Recall that for every i [n], ei i denotes the root of i, so δei is the distinguished vector in 2 ∈ ∈ V G def the Hilbert space ` ( i). Following the construction outlined in Section 3.4, we set ξi = I δe to be V k ⊗ i the distinguished vector of i, and we note that, by (20), ξi = δe . Hence, the decomposition H B B ⊗ i ◦ i = ξi i satisfies H B ⊕ H ◦ 2 ◦ i = ` ( i), H B ⊗ V ◦ def where as before i = i ei . V V \{ } As mentioned in Example 3.12, the -algebra of bounded, adjointable, right -linear operators ∗ B def eB( i) is a -valued probability space with the conditional expectation defined by Ei(S) = ξi, Sξi i H B 2 h iH for every S eB( i). But since and B(` ( i)) are finite dimensional, we can easily do the ∈ H B V identification 2 eB( i) C B(` ( i)), H B ⊗ V 2 by letting each X A C B(` ( i)) act on i by ⊗ ∈ B ⊗ V H 2 X A(X0 v) = XX0 Av, X0 , v ` ( i). (22) ⊗ ⊗ ⊗ ∀ ∈ B ∀ ∈ V

Now note that under this identiﬁcation, the Ei we just deﬁned also coincide with the construction 2 given in Example 3.6, since for a pure tensor X A C B(` ( i)) we have, by (21) and (22), that ⊗ ∈ B ⊗ V

2 Ei(X A) = ξi, X A(ξi) i = Ik δei , X A(Ik δei ) i = Ik δei , X Aδei i = δei , Aδei ` ( )X. ⊗ h ⊗ iH h ⊗ ⊗ ⊗ iH h ⊗ ⊗ iH h i Vi

29 We can now use Voiculescu’s construction of the free product of Hilbert -modules with amalga- B mation over . B

Step 2: Deﬁning , τ, and the Ti. Note that and the Hilbert -bimodules i satisfy all the conditions A B B H required in Section 3.4, so we can go ahead with the construction of the amalgamated free product. Deﬁne M def ◦ ◦ = ξ i im , H B ⊕ H 1 ⊗B · · · ⊗B H i ,i , ,in 1 2 ···

def where ξ = In δe and δe is the vector resulting from identifying the δe . Also deﬁne = eB( ). Now, ⊗ i A H for every i [n], let ρi : eB( i) eB( ) denote the inclusions described in Section 3.4 and set ∈ H → H def X Ti = ρi(X A ). α ⊗ α α ∈Ci

We will denote by E the natural -valued conditional expectation on eB( ), and τr will denote the def B H state on given by τr(b) = δr, bδr `2( ). We can then lift τr to a state τ on eB( ) by deﬁning B h i V H

def τ = τr E. ◦

With these deﬁnitions, from Theorem 3.13 it is clear that the Ti are free with amalgamation over B with respect to the conditional expectation E, and that each Ti distributes as X X A . α ⊗ α α ∈Ci

It remains to show that T + Tn distributes as the Jacobi operator on the infinite graph constructed 1 ··· in Definition 5.2 when the lift of Jacobi coefficients is done as in Definition 5.6.

Step 3: Relating T + + Tn to the Jacobi operator. For every i, j [n] let ∆ij denote the operator in 1 ··· ∈ B(`2( )) corresponding to the matrix in M (C) with a 1 in the entry (i, j) and 0 everywhere else. We V k will use that   ∆im if j = l, ∆ij∆lm =  (23) 0 otherwise.

def Deﬁne T = T + + Tn. The main idea of Step 3 is to ﬁnd a C-vector subspace V of (containing 1 ··· H ∆rr δe) that is invariant under the action of T and such that a C-base of V can be bijected with ⊗ V( ( ,c) i ) so that it is clear that T acts on V in the same way that the Jacobi operator acts on 2 {∗ G G } ` (V( ( ,c) i )). First note that {∗ G G } M def ◦ ◦ = ξ i im H B ⊕ H 1 ⊗B · · · ⊗B H i1,i2, ,in ···M 2 ◦ 2 ◦ = δe ( ` ( i )) ( ` ( im )) B ⊗ ⊕ B ⊗ V 1 ⊗B · · · ⊗B B ⊗ V i ,i , ,in 1 2 ···

30 and that as -bimodules we have the identification B 2 ◦ 2 ◦ 2 ◦ 2 ◦ ( ` ( i )) ( ` ( im )) ` ( i ) ` ( im ), B ⊗ V 1 ⊗B · · · ⊗B B ⊗ V B ⊗ V 1 ⊗ · · · ⊗ V for every i , i2 , , im. Using this identification, for every l [k] define the subset of Θ 1 ··· ∈ H l

def ◦ ◦ Θ = ∆ δe ∆ δv δv : i [k], v i ,..., vm i , i , i2 , , im . l { ll ⊗ } ∪ { il ⊗ 1 ⊗ · · · ⊗ m ∈ 1 ∈ V 1 ∈ V m 1 ··· }

Now, by deﬁnition of τr and , we have h· ·iH Xk p p p p τ(T ) = τr( Ik δe, T (Ik δe) ) = τr( ∆ll δe, T (Ik δe) ) = τr( ∆rr δe, T (Ik δe) ) h ⊗ ⊗ iH h ⊗ ⊗ iH h ⊗ ⊗ iH l=1

On the other hand, from (23) and the deﬁnition of T it follows that the C-span of each Θl is is p p T-invariant, which implies that τr( ∆rr δe, T (Ik δe) ) = τr( ∆rr δe, T (∆rr δe) ) and hence h ⊗ ⊗ iH h ⊗ ⊗ iH p p τ(T ) = τr( ∆rr δe, T (∆rr δe) ). h ⊗ ⊗ iH

It follows that the distribution of T with respect to τ is determine by the action of T on Θr. Then n we note that Θr is in bijection with ( ,c) i i=1, and under this correspondence T acts on Θr in the ∗ G {V }n 2 n same way as the Jacobi operator of ( ,c) i acts on ` ( ( ,c) i ). This concludes the proof. ∗ G {G }i=1 ∗ G {V }i=1 5.3 Amalgamated Free Product of Groups as Amalgamated Free Product of Graphs Here we show that Cayley graphs of amalgamated free products of finite groups can be realized as an amalgamated free product of finite graphs. Note that this is not an obvious fact and requires a proof. In the setup of Observation 3.11, freeness with amalgamation appears because we are working in a group C∗-algebra of an amalgamated free group product. In contrast, in the case of the amalgamated free graph product, the freeness with amalgamation comes from the fact that we are working in a tensor algebra. The following discussion uses a technique sketched in [KFSH19, Appendix C] following Sunada [Sun92], and allows us to go from the former setup to the latter. Let G1,..., Gn be finite groups with a common subgroup H and finite symmetric generating Gi, H 1 sets Si = S−i . For each i fix a set of representatives Ri for the set of right cosets of H in Gi, and Si stipulate that eG Ri for each i. The following structure theorem for the amalgamated free product R i ∈ i of groups is well known; see e.g. [Ser80]:

Theorem 5.9 (Structure of amalgamated free products of groups). Each element 1 of the amalgamated n free product of groups H Gi can uniquely be represented in the normal form ∗ { }i=1

1 = hr rn, 1 ··· where h H, r Ri for all 1 k n and i , i for all 1 k n 1. ∈ k ∈ k ≤ ≤ k k+1 ≤ ≤ − The above theorem is useful even in the case n = 1 corresponding to a single group G amalgamated over a subgroup H, in which case one obtains the induced representation. To be precise, the theorem gives a bijection G H R, so the left regular representation of G induces a left group ↔ × action φ of G on H R, which yields an inclusion of algebras χ : C[G] B(`2(H)) B(`2(H G)). φ × → ⊗ \

31 We are now ready to express Cayley graphs of amalgamated free group products via our amalgamated free graph product, following along with the combinatorial description of our graph product in Definition 5.2. The reader may find it helpful to first look at Example 5.11.

n Construction. We produce a family of colored graphs ( i, ci) and a relator graph ( , c) as { G }i=1 G follows.

(i) Vertices: Set V( ) = H and for every i set V( i) = Ri. Then the vertex set of the amalgamated G n G n product graph is H Ri , which is the desired vertex set for the Cayley graph of H Gi × ∗{ }i=1 ∗ { }i=1 by Theorem 5.9.

(ii) Roots: Set the root of to be eH, and the root of i to be eG for all i. G G i Sn (iii) Color sets: For every i let i = Si H Ri and = i. C × × C i=1 C (iv) Edges: For each element 1 = hr ... rn HGi (as in Theorem 5.9), for each i [n] and for 1 ∈ ∗ ∈ each generator s Si, we may uniquely write sh = h r for some h H and r Ri, where ∈ 0 0 ∈ ∈ (h0, r) = φ(s, (h, e)). We have one of two cases:

(a) i , i1, in which case s1 has the normal form

s1 = h0rr1 ... rn

and we create directed edges (h, h ) in the relator graph and (e, r) in the graph i, both of 0 G the same color (s, h, e).

(b) i = i , in which case h rr = h r for r Ri and s1 has the normal form 1 0 1 00 10 10 ∈ 1 1 s = h00r10 r2 ... rn,

and we create directed edges (h, h ) in the relator graph and (r , r ) in i , both of the 00 1 10 G 1 same color (s, h, r1). As a consequence, in our amalgamated product graph, we will have the edge (1, s1) as desired, 1 with the same color. Because Si = Si− and φ is a group action, for any directed edge constructed its reverse edge will also be constructed, so if desired we may identify the colors associated to these edges and obtain colored undirected graphs.

Thus, by construction we have the following theorem.

Sn Theorem 5.10. Let S = Si and let ( , c), ( i, ci) be deﬁned as above. Then i=1 G G n n ( ,c) ( i, ci) i Γ( H Gi i , S). ∗ G { G } =1 ∗ { } =1 Example 5.11. Here we provide the explicit construction for the Cayley graph of SL(2, Z), which can be viewed as the free product of Z4 and Z6 with amalgamation over Z2, and has the ﬁnite presentation

SL(2, Z) x, y x4 = y6 = 1, x2 = y3 . h | i

32 Let λZ4 , λZ6 denote the left regular representations of Z4 and Z6, respectively. We start by evaluating the 2 2 inclusion χ : C[Z ] B(` (Z2)) B(` ( 0, 2 , 1, 3 )) at the element λZ (1) + λZ ( 1) where we use the 1 4 → ⊗ { } { } 4 4 − coset representatives R = 0, 1 . It is easily seen that { } ! ! ! ! 1 0 0 1 0 1 0 1 χ (λZ (1) + λZ ( 1)) = + . 1 4 4 − 0 1 ⊗ 1 0 1 0 ⊗ 1 0

2 2 Now we consider the inclusion χ2 : C[Z6] B(` (Z2)) B(` ( 0, 3 , 1, 4 , 2, 5 )) where the set of → ⊗ { } { } { } coset representatives R = 0, 1, 2 is used. We then have { }     !  0 1 0  !  0 0 1  1 0   0 1   χ2(λZ (1) + λZ ( 1)) =  1 0 1  +  0 0 0  . 6 6 − 0 1 ⊗   1 0 ⊗    0 1 0   1 0 0 

Due to the simple form of the above equations, ultimately arising from the abelian nature of the groups, we may get away with using fewer colors than in the fully general construction above. Associating a color to each pure tensor above, we obtain the colored graphs depicted in Figure4.

a = 1, 3 u = 1, 4 { } { }

v = 2, 5 (1, au) { } (0, au) ··· 1 (0, v) ∗ (0, u) (0, ua) e2 = 0, 3 e = 0, 2 { } 1 { } 0 (0, a) (1, va) = (0, e) (1, e) (1, a) ··· (0, va)

(1, ua) (1, u) (1, v) (1, av) (0, av) ···

Figure 4: (A ﬁnite portion of) the Cayley graph of SL(2, Z), represented as an amalgamated product of graphs.

5.4 Implications for A ( ) T G Here we consider a graph with n vertices and let m be the number of edges in where each G G whole loop is counted twice. We denote the universal cover of by . To lighten notation we G T assume that V( ) = [n] and let ∆ij be the n n matrix with a 1 in the entry (i, j) and 0 elsewhere. G × Let E ( ) be the set obtained by taking E( ) and splitting every whole loop into two loops. Fix a E ( ) half G G half G numbering γ : E ( ) [m]. γ half G →

33 def Let Γm denote the discrete group Γm = Z2 Z2 and let 11,..., 1m be its canonical generators. Γm, 1i ∗ · · · ∗ 2 As usual λ will denote the left regular representation of Γm on ` (Γm). λ Example 5.4 together with Theorem 5.8 indicate that we can decompose A as a sum of random T variables that are free with amalgamation over Mn(C). If we look closer into the construction of the amalgamated graph product, we actually see that we can represent A as an n n operator-valued T × matrix with entries in Cred∗ (Γm). This yields a representation resembling that of Lemma 4.2, but note the diﬀerent C∗-algebra used here. We formalize this representation below. Proposition 5.12. Let r V( ) and r˜ V( ) be a representative in the ﬁber of r. Let A be a Jacobi ∈ G ∈ T G operator on and A be its lift to . Then, the spectral measure of A with respect to the root r˜ coincides G T T T with that of X G

def X X X X = bi∆ii 1 + ae∆ii λ(1γ(e)) + ae(∆ij + ∆ji) λ(1γ(e)) (24) G ⊗ ⊗ ⊗ i V( ) e= i,i Ehalf( ) e Ehalf( ) ∈ G { }∈ G e=∈ i,j , iG,j { } viewed as an element of the non-commutative probability space (Mn(C) C (Γm), τr τe), where ⊗ red∗ ⊗

def def τr(X) = Xrr X Mn(C), and τe(T) = δe, Tδe T C∗ (Γm). ∀ ∈ h i ∀ ∈ red

Proof. We follow the construction from Example 5.4. Let r be the root of . For every i [m] let i G ∈ G be a graph with two vertices (one of them being the root ei) connected by an edge and let Ai be the 15 adjacency matrix of i. Note that Ai distributes as a Rademacher random variable, which is also G 1 the distribution of any λ( i) in the probability space (Cred∗ (Γm), τe). def As in Example 5.4, we will use = (V( ), E ( )) to build the relator graph. Color the Ghalf G half G elements in E ( ) with colors in [m], assigning a different color to every element in E ( ). Since half G half G we only defined the our graph product for weighted adjacency matrices, for now we will ignore the bv and only assign the Jacobi coefficients ae to . So, for every i, j = e E ( ) with i , j, the Ghalf { } ∈ half G subgraph [γ(e)] whose vertex set is and whose only edge is e has ae(∆ij + ∆ji) as its adjacency Ghalf V matrix. While for every i, i = e E ( ), the subgraph [γ(e)] whose vertex set is and { } ∈ half G Ghalf V whose only edge is e has ae∆ii as its adjacency matrix. For i [m], denote the adjacency matrix of [i] by Xi. Then, here the operators from Theorem ∈ Ghalf 5.8 are given by Xi Ai. ⊗ Note that for any e = i, j E ( ), the operator X A distributes as the random variable { } ∈ half G γ(e) ⊗ γ(e) ae(∆ij + ∆ji) λ(1 ) if i , j and as ae∆ii λ(1 ) otherwise. But Theorem 5.8 produces copies in ⊗ γ(e) ⊗ γ(e) Mn(C)-distribution of the Xi Ai, namely Ti, that are free with amalgamation over Mn(C), and says ⊗ that T = T + + Tm distributes as the weighted adjacency matrix on . At this point it is clear 1 ··· T that T is equal in distribution to the right side of (24) if bv = 0 for all v . ∈ G In some way, our graph product has brought conceptual understanding of why we can represent C A as an element of Mn( ) Cred∗ (Γm) when the bv = 0. Once this is clear, one way to handle the T ⊗ n 2 case when bv , 0 is to identify V( ) with a certain orthonormal set of C ` (Γm) in the obvious T ⊗ way, and compare action the of the diagonal part of A on `2(V( )) with the diagonal part of X n 2 T T G on C ` (Γm). We skip the details of how this is done, since it is lengthy to make this more specific, ⊗ but not very enlightening.

15 1 By Rademacher random variable, we mean a random variable with distribution (δ 1 + δ1). 2 −

34 6 Universal Covers: Algebraic Description of the Spectrum

In this section we will use the setup from Section 5.4. That is, will be a graph with n vertices G and m edges, where each loop is counted as two edges. The vertex set V( ) will be identiﬁed with G the set [n]. We will use X to denote the random variable deﬁned in (24), and Γm will denote the G free product of m copies of Z2. We will also be working in the Mn(C)-valued probability space Mn(C) C (Γm) with conditional expectation E = Id τe. ⊗ red∗ ⊗ The starting point of this subsection is the result from Proposition 5.12, which implies that the problem of computing the density of states of A is equivalent to that of computing the spectral measure of an operator-valued matrix with free RademacherT random variables in the entries. The latter problem has already been addressed by Lehner in [Leh99].

6.1 Aomoto’s Equations via the R-transform The goal of this subsection is to prove Theorem1 .4 via free probability. We begin by noting that if we apply [Leh99, Proposition 3.1], and specialize from positive deﬁnite matrices to diagonal positive deﬁnite matrices, we get the following result:

def Proposition 6.1 (Lehner). Let W Mn(C) be a diagonal matrix with positive entries and let wi = Wii. ∈ Then the Mn(C)-valued R-transform of X at W, i.e. RX (W), is also diagonal, and the diagonal entries are given by G G       1  X q X q  R (W)(i, i) = b +  1 + 4a2w w 1 + 2 1 + 4a2w2 1  , i [n]. X i  e i j l i  G 2wi  − −  ∀ ∈  e E( ) l E( )  e=∈i,j G, j,i l∈= i,Gi { } { } In order to use the above proposition we will need the following lemma.

Lemma 6.2. Let GX denote the Mn(C)-valued Cauchy transform of X . Then for z C in a neighborhood G G ∈ of infinity, GX (zIn) is diagonal. G Proof. We will use the following power series expansion (in a neighborhood of infinity) for the Cauchy transform X 1 1 k GX (B) = E[B− (X B− ) ]. G G k 0 ≥ (k+1) k For B = zIn, the terms in the right-hand side of the above equation will be of the form z− E[X ]. It k G is then enough to show that E[X ] is a diagonal matrix for every k. To this end note that XG(i, j) = 0 G whenever i, j < . By expanding out the matrix product, we see that Xk (i, j) is a linear combination { } E of terms of the form G def π(e ,..., e ) = λ(1 )λ(1 ) λ(1 ), 1 k γ(e1) γ(e2) ··· γ(ek) dir where e1,..., ek is a sequence in E forming a (possibly backtracking) path in from i to j. k k G Fix i, j [n] with i , j. To see that E[X ](i, j) = τ(X (i, j)) = 0, we show that each τ(π(e1,..., ek)) ∈ G G 2 is 0. If the path e ,..., en is backtracking, then repeatedly applying λ(1 ) = 1 , we can reduce 1 γ(e) Cred∗ (Γm) π(e1,..., ek) to a product π0 that corresponds to a nonbacktracking path. Now since τ(1γ(e)) = 0 for all e, the definition of free independence implies directly that τ(π0) = 0, as desired.

35 We can now recover Theorem 1.4.

Restatement of Theorem 1.4 (Aomoto). Let A be a Jacobi matrix on a ﬁnite graph with parameters G G ae , bv as in (1) and let wu(z) be the Cauchy transforms for A as in (4). Then, the following system of { } { } T equations holds for all z C in a neighborhood of inﬁnity and for all real z outside the convex hull of the ∈ spectrum σ(A ): T       q   1  X q X  w z  u a2 w z w z a2w z 2 u V  u( ) = 2 deg( ) + 1 + 4 u,v u( ) v( ) + 2 1 + 4 e u( )  ( ).  2(z bu)  − { }  ∀ ∈ G  −  e E( ) e= u,u E( )   e= u∈,v G, v,u { }∈ G { } Proof of Theorem 1.4. As in the statement, for every i [n] let wi(z) be the Cauchy transform of the ∈def spectral measure of A corresponding to i. Let W(z) = GX (zIn). From Lemma 6.2 we know that W(z) is diagonal. Moreover,T from Observation 3.15 we haveG that

wi(z) = W(z)(i, i), and wi(z) > 0 for suﬃciently large real z. Plugging into the deﬁnition of R-transform (12) we obtain

zwi(z) = 1 + RX (W)(i, i)wi(z) G for all 1 i n. After substituting in the expression for RX from Proposition 6.1 and simplifying, ≤ ≤ G we get that the system of equations in the theorem statement holds for sufficiently large positive real z. Since wu(z) is holomorphic and wu(z) 0 as z , both sides of the equations are → | | → ∞ holomorphic in a neighborhood of infinity, so by analytic continuation the system of equations holds in a neighborhood of infinity. Since wu(z)wv(z) > 0 for all u, v when z is real and outside the convex hull of the spectrum σ(A ), the system of equations holds for these z as well, as the singularity of the square root is alwaysT avoided.

Remark 6.3. If is allowed to have half-loops (see Section 2.1) then Theorem 1.4 holds where each half-loop p G 2 2 e = u, u contributes 1 + 4a wu(z) to the summation, without the factor of 2 that would appear if e were { } e a whole-loop.

6.2 Behavior of the Density of States at the Right Edge of the Spectrum Using Aomoto’s systems of equations for the Cauchy transforms, we may prove the following condition on when the spectral radii of a graph and its universal cover match. The proof will use Theorem 6.6 of [ABS20] which states that the Cauchy transforms defined in Theorem 1.4 are algebraic. Among other things, this result implies that if µ is the density of states of A , and ρ is T the right edge of σ(A ), then the (possibly infinite) limit lim 0+ µ((ρ , ρ])/ exists. T → − Proposition 6.4. Let A be Jacobi matrix on a finite graph with coefficients ae > 0 for all e E( ). Let G G ∈ G A be its pullback to the universal cover ( ). Let µ and ρ denote the density of states of A and maximum T T G T element of the spectrum of A respectively. Suppose that T µ((ρ , ρ]) lim − > 0. (25) 0+ → Then the maximum eigenvalue of A is equal to ρ. G

36 Proof. Let w(z) denote the Cauchy transform of µ. We begin by proving that the condition in (25) implies that limt ρ+ w(t) = . There are two cases. If µ has an atom at ρ the statement is clearly true. → ∞ If there is no atom at ρ, we note that Theorem 6.6 in [ABS20] implies that w(t) is algebraic (since the sum of algebraic functions is algebraic), and hence µ is absolutely continuous in a neighborhood of ρ and has a power-like behavior near ρ (see Theorem 2.9 in [AZ08] for a formal statement). In particular, (25) implies that there is some C > 0 such that for all > 0 small enough we have dµ (s) > C for all s (ρ , ρ). Hence, for any > 0 dx ∈ − Z Z ρ 1 ρ 1 1 w 2 d s C ds C (ρ + ) 2 µ( ) 2 = log 1 + . ≥ ρ ρ + s ≥ ρ ρ + s − − − − So taking 0+ we get → lim w(t) = . (26) t ρ+ ∞ → 1 P Now, for all u V( ), let wu(z) be deﬁned as in (4). Recall that w(z) = V( ) v V( ) wv(z). ∈ G | G | ∈ G Note that wu(t) is analytic, positive and strictly decreasing for t > ρ, so limt ρ+ wu(t) exists and → lies in (0, ]. Thus, for all u, v V( ), we have that limt ρ+ wv(t)/wu(t) exists and lies in [0, ]. ∞ ∈ G → ∞ In fact, we claim limt ρ+ wu(t)/wv(t) < when u is a neighbor of v. If not, then setting z = t in → ∞ Theorem 1.4 and rearranging, we would have s s 2 deg(u) X 1 w (t) X 1 t b − a2 v a2 = u + + 2 + e + 2 2 + e , (27) 2wu(t) 4wu(t) wu(t) 4wu(t) e E( ) e E( ) e= u∈,v G, v,u e=∈ u,Gu { } { } and the right hand side would diverge as t ρ+ while the left hand side would converge, a → contradiction. Taking the reciprocal and switching the roles of u and v, we in fact obtain

0 < lim wv(t)/wu(t) < (28) t ρ+ ∞ → whenever u and v are neighbors. By the connectedness of ,(28) actually holds for arbitrary vertices G u, v. Together with (26), this implies limt ρ+ wu(t) = for all u V( ). → ∞ ∈ G By (28), there exist positive real numbers wfu u V( ) with the property { } ∈ G wfu wu(t) = lim . w t ρ+ wv(t) fv → Taking the limit as t ρ+ of (27), we get → p X p X p ρ wfu = bu + ae wfv + 2 al wfu. e E( ) l E( ) e= u∈,v G,v,u l=∈ u,Gu { } { } Recalling the deﬁnition (1) of A , the above equation explicitly shows that ρ is an eigenvalue of G p A with an eigenvector with positive entries we . Now take λ > 0 large enough so that e e E( ) G ∈ G λ + bu > 0 for all u V( ). Then the entries of A + λ are nonnegative, the eigenvector for ρ + λ has ∈ G G positive entries, and A is irreducible (since as a directed graph is strongly connected), so by G G the Perron-Frobenius theorem ρ + λ is in fact the maximum eigenvalue of A + λ. Thus ρ is the maximum eigenvalue of A , as desired. G G

37 The following theorem of Sy and Sunada will be useful. We rephrase it in the language of Jacobi matrices on graphs. Their original formulation is in terms of what they call a discrete Schrodinger operator, but every graph Jacobi matrix with positive edge weights ae can be represented in their framework.

Theorem 6.5 ([SS92]). Let 1 be a ﬁnite connected graph with no loops or multi-edges, and let A be a G G1 Jacobi matrix on 1 with ae > 0 for all e E( 1). Let 2 be a graph covering 1, and let A be the lift of G ∈ G G G G2 A . Then ρr(A ) ρr(A ) with equality if and only if the deck transformation group of the covering is G1 G1 ≥ G2 amenable, where ρr( ) denotes the maximum element of the spectrum. · We may now prove our main result on the edge of the spectrum:

Restatement of Theorem 1.6. Let A be a Jacobi matrix on a ﬁnite graph with at least two cycles, and G G let A be its pullback to the universal cover ( ). Furthermore assume that ae > 0 for all e E( ). Let T T G ∈ G µ and ρ be the density of states and the maximum element of the spectrum of A , respectively. Then µ is dµ T absolutely continuous in a neighborhood of ρ and limx ρ (x) = 0. → dx Proof. Since has at least two cycles, the deck transformation group of the universal cover contains G the free group on two generators as a subgroup, and is therefore nonamenable. If has no loops G and no multi-edges, Theorem 6.5 then implies that the maximum eigenvalue of A is strictly greater than ρ. Thus, applying the contrapositive of Proposition 6.4 we have G

µ((ρ , ρ]) lim − = 0. 0+ → In particular µ has no atoms in (ρ , ρ] for sufficiently small. Also, µ has no singular continuous − part ([ABS20]) due to algebraicity of the Cauchy transforms. Thus, the conclusion follows. If, on the other hand, has loops or multi-edges, we use the following workaround. One may G first take a finite cover of that does not contain loops or multi-edges. (If j is the maximum G0 G number of loops at any vertex and k is the maximum number of edges in any multi-edge, a cover with max 2j + 1, k sheets suffices.) Since ( ) = ( ), the only thing left to check is { } T G T G0 that ρr(A ) ρr(A ), where ρr( ) denotes the maximum eigenvalue. This holds because the top G ≥ G0 · eigenvector v of A can be projected down to an eigenvector v0 of A for the same eigenvalue by G0 G summing the entries of v in each fiber. Note that v0 is nonzero because v has positive entries, by Perron-Frobenius applied to A + λ for λ > 0 sufficiently large (as the bu may be negative). Thus G ρr(A ) ρr(A ) as desired. G ≥ G0 6.3 Formula for the Spectral Radius Let A be a Jacobi matrix on a graph with n vertices, and let A be its pullback to the universal G G T covering tree ( ). Let ρ( ) be the spectral radius of A . The main objective of this subsection is T G T T to prove Theorem 1.1. We start by stating some intermediate results. We will build on one of the main results of [Leh99], namely Theorem 1.1. As above we will denote the free product of m copies of Z2 by Γm and we will denote the canonical generators by 11,..., 1m. The left regular representation of Γm will be denoted by λ.

38 Theorem 6.6 (Lehner). Assume that m 2, and let A0,..., Am be n n Hermitian matrices, with A0 ≥ × positive semideﬁnite. Then

m m X X 1 1 1 1 1 2 2 2 2 2 2 A0 1C (Γm) + Ai λ(1i) = inf 2Z + A0 + Z (In + (Z− AiZ− ) ) In Z (29) ⊗ red∗ ⊗ Z>0 − i=1 i=1 where the infimum is taken over all positive definite invertible n n matrices Z. Moreover, the infimum can × be restricted to those Z for which the expression inside the norm sign equals a positive scalar multiple of the identity matrix In. We are now ready to prove Theorem 1.1. In short, the proof uses Aomoto’s equations (Theorem 1.4) to reduce the expression (29).

Restatement of Theorem 1.1. Let A be a Jacobi matrix on a graph with vertex set [n], and let A be G G T its pullback to the universal cover ( ). Let ρr(A ) denote the right edge of the spectrum of A . Then ρr T G T T     1  X q    2  ρr(A ) = inf max bi + 2 deg(i) + 1 + 4a i,j yi yj . (30) T y1,...,yn>0 i [n]  2yi  −  ∈ j: i,j E( ) { } { }∈ G where deg(i) is the degree of the vertex i in , and where loops each contribute 2 to the degree, and each loop G is understood to appear twice in the summation. Moreover, the inﬁmum can be restricted to those n-tuples (y1,..., yn) for which the n expressions inside the max symbol are equal to each other.

Proof. For i = 1,..., n deﬁne 1i(y1,..., yn) to be the expression inside the max symbol in the theorem statement. Let wi(z) denote the Cauchy transform of the spectral measure of A rooted at an T element in the ﬁber of i. Fix t > ρr(T). On the one hand, > wi(t) > 0 for every i [n]. On the ∞ ∈ other hand, from Theorem 1.4 we have t = 1i(w1(t),..., wn(t)) for every i. Together, this implies that

t inf max 1i(y1,..., yn). ≥ y1,...,yn>0 i [n] ∈

Since the above inequality holds for any t > ρr(A ), it holds for t = ρr(A ). It remains to show the opposite inequality. T T From Proposition 5.12 we have that ρ(A ) = X , where X is as in (24). We would like to T || G|| G apply Theorem 6.6 on X , but the theorem requires A0 to be positive semideﬁnite. To remedy this, G take λ 0 large enough so that λ + bi 0 for all i; we may now apply Theorem 6.6 on X to obtain ≥ ≥ G

X 1 2 1 1 2 1 1 ρ(A + λ) = X + λIn = inf 2Z + B + λIn + Z 2 ((In + ae (Z− 2 AeZ− 2 ) ) 2 In)Z 2 (31) T || G || Z>0 − e E( ) ∈ G where B = diag(b ,..., bn), and where for each edge e = i, j with i , j we have Ae = ∆ij + ∆ji, and 1 { } each loop e = i, i is understood to appear twice in the summation, both times with Ae = ∆ii. We { } will now see that in this case the inﬁmum is achieved by diagonal matrices. Let y1,..., yn > 0 and take Y := diag(1/2y1,..., 1/2yn). Simple computations yield that upon setting Z = Y in (31), the quantity inside the norm on the right hand side becomes

λIn + diag(11(y1,..., yn),..., 1n(y1,..., yn)). (32)

39 Since λ + 1i(y ,..., yn) 0 for all i, we have 1 ≥

λIn + diag[11(y1,..., yn),..., 1n(y1,..., yn)] = λ + max 1i(y1,..., ym). || || i [n] ∈ Then, (31) and (32) yield

ρ(A + λ) λ + inf max 1i(y1,..., yn). T ≤ y1,...,yn>0 i [n] ∈

Since ρr(A ) + λ = ρr(A + λ) ρ(A + λ), we have T T ≤ T

ρr(A ) inf max 1i(y1,..., yn) T ≤ y1,...,yn>0 i [n] ∈ as desired.

Applying Lagrange multipliers to the optimization problem (3), using the constraint that all n expressions inside the max symbol are equal, we have the following corollary:

Corollary 6.7. Let A be a Jacobi matrix on a ﬁnite graph with vertex set [n] and Jacobi coeﬃcients G G ai i , be e E( ), and let A be its pullback to the universal cover ( ). Let ρr denote the right edge of the { } ∈G { } ∈ G T T G spectrum of A . T Then t = ρr is the only real number such that, under the constraint y1,..., yn > 0, the following system of 2n + 1 equations in the variables t, y ,..., yn, λ , . . . , λn R has a solution: 1 1 ∈  n  X 1 = λ ,  i  i=1   X yjλi + yiλj  t b a2 i n λi( i) = e q [ ],  − 2 ∀ ∈  e E( ) 1 + 4ae yi yj  e∈= i,Gj   { }         1  X q  t b  i a2 y y  i n  = i + 2 deg( ) + 1 + 4 e i j [ ],  2yi  −  ∀ ∈   e E( )   e∈= i,Gj { } where each loop of is understood to appear twice in each summation, and loops count twice towards the G degree deg(i).

7 Future Research

We ﬁnd that there are many potentially fruitful directions to pursue going forward.

Amalgamated Free Product for Graphs. For certain operators on diﬀerent classes of inﬁnite graphs, the respective Green functions (or other related functions) have been shown to be algebraic [Woe87, Lal01, NW02, KLW13, ABS20]. On the other hand, algebraicity is relevant in the context of spectral theory since it provides means to show that the operators in question have no singular continuous spectrum [ABS20] or in some cases that the spectrum is purely absolutely continuous

40 [KLW13]. We believe that Jacobi operators on any graph defined via our product also have no singular continuous spectrum, and it is possible that the work of Anderson [And14] might lead to showing algebraicity of their Green functions. Proving this would provide a generalization of Theorem 6.7 in [ABS20] and other results in the literature of spectral analysis of Cayley graphs. We state this as a conjecture: Conjecture 7.1. The Cauchy transforms of the spectral measures of any Jacobi operator on a graph constructed via the amalgamated free product of graphs (where the Jacobi coefficients are lifted from the graphs in the product in a sensible way) are algebraic, and the Jacobi operator has no singular continuous spectrum. There are some natural extensions of our graph product. In the present work we have only considered the case of finite graphs; one could hope to move beyond the finite realm. More concretely, it may be possible to extend the amalgamated free graph product to consider (possibly infinite) locally finite graphs with bounded degree. This would allow, for example, to decompose the Cayley graph of a surface group using an analog of Theorem 5.8. On a different direction, our product can be extended to directed graphs without much work. Another direction is to obtain a more complete characterization of which graphs can be obtained using our graph product. We have shown that Cayley graphs of amalgamated free products of groups can be obtained, as well as universal covers of graphs. One specific question is whether the “additive product” graphs defined by Mohanty and O’Donnell [MO20] can be obtained using our graph product.

Universal Covers. Many fundamental questions regarding Jacobi operators on universal covers remain open. For example, even if Sunada’s theorem (Theorem 1.7 above) is tight, determining the exact number of bands in the spectrum of A ( ) in terms of and the ae, bv, is currently out of reach. T G G Here we pose some more modest, but still challenging goals that would bring some understanding to this problem.

Problem 7.2. Find a characterization of the base graphs and coeﬃcients ae, bv for which A ( ) has G T G connected spectrum.

Problem 7.3. Find an inﬁnite sequence of Jacobi matrices A n on graphs n (without half-loops) with (n) G G V( n) as n , and with bv = 0 for every v V( n), such that for every n, n has at least two | G | → ∞ → ∞ 16 ∈ G G cycles and the spectrum of A ( n) has V( n) bands. T G | G | It is also not well-understood for which base graphs Jacobi operators on the universal cover have atoms in their density of states. Moreover, it is not known where the atoms may be located. In this direction, there is the recent work of Belinschi, Bercovici and Liu [BBL19], which analyzes the atom structure of operator-valued free convolutions. Their work could prove useful, in light of the fact that Jacobi operators on universal covers are essentially operator-valued free convolutions.

Acknowledgements

We thank Nikhil Srivastava and Dan-Virgil Voiculescu for many helpful discussions and suggestions of research directions. We also thank Peter Sarnak for bringing us up to speed with the literature in

16 (n) In an earlier version of this paper we did not impose the condition bv = 0, but this simpler problem is solved in [ABS20]; see Remark 1.8 in this paper.

41 the area and for his encouragement. We thank Barry Simon for his helpful feedback on a previous version of this paper, for pointing out key references and for helpful discussions. We are also grateful to Nalini Anantharaman, Octavio Arizmendi, Benson Au, Hari Bercovici, Shai Evra and Sidhanth Mohanty for helpful discussions.

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