Zeta functions of infinite graph bundles

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Zeta functions of infinite graph bundlesSamuel Cooper a & Stratos Prassidis aa Department of Mathematics , Vanderbilt University , Nashville,TN, 37240b Department of Mathematics , Canisius College , Buffalo, NY,14208Published online: 22 Jan 2009.

To cite this article: Samuel Cooper & Stratos Prassidis (2010) Zeta functions of infinite graphbundles, Linear and Multilinear Algebra, 58:2, 185-201, DOI: 10.1080/03081080801928084

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Linear and Multilinear AlgebraVol. 58, No. 2, February 2010, 185–201

Zeta functions of infinite graph bundles

Samuel Coopera and Stratos Prassidisb*

aDepartment of Mathematics, Vanderbilt University, Nashville, TN 37240; bDepartment ofMathematics, Canisius College, Buffalo, NY 14208

Communicated by B. Mohar

(Received 11 May 2007; final version received 16 January 2008)

We compute the equivariant zeta function for bundles over infinite graphs andfor infinite covers. In particular, we give a ‘transfer formula’ for the zeta functionof infinite graph covers. Also, when the infinite cover is given as a limit of finitecovers, we give a formula for the limit of the zeta functions.

Keywords: zeta functions; graph bundles; von Neumann algebra; limits of graphs

2000 Mathematics Subject Classifications: primary 05C50; secondary 11M36;11M41

1. Introduction

The Ihara zeta function of a finite graph reflects combinatorial and spectral propertiesof that graph [2,12,16]. Originally, Ihara defined the zeta function on finite graphsimitating the classical definition of the zeta function:

�XðzÞ ¼Y½C�

ð1� z‘ðCÞÞ�1;

where the product is over all equivalence classes of primitive closed loops C in X and ‘(C)denotes the length of C. In [2], it was shown that, for a finite graph X:

�XðzÞ�1¼ ð1� z2Þ��detðI� zAþ z2QÞ;

where � is the number of edges, � is the number of vertices, A is the adjacency matrix of Xand Q is the diagonal matrix with entries deg(v)� 1, for each v2V(X ). In [8], the definitionof Ihara zeta function was extended to infinite graphs that are limits of sequences of finitegraphs. In particular, it was shown in [8], using the results in [15], that the sequence of thezeta functions of the finite graphs converges. In [3,4,9–11], the expression of the zetafunction as a rational function was extended to infinite graphs that admit an action ofa discrete group � with finite quotient. The determinant in the finite case is replaced by thedeterminant in a von Neumann algebra N 0(X,�) of all the bounded operators onL2(V(X )). In [6], the zeta function of finite graph bundles over finite was computed

*Corresponding author. Email: [email protected]

ISSN 0308–1087 print/ISSN 1563–5139 online

� 2010 Taylor & Francis

DOI: 10.1080/03081080801928084

http://www.informaworld.com

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generalizing the results on graph coverings that appear [16–18]. Their results can be

described as transfer results for the Ihara zeta function.We combine the results on infinite graphs and bundles to derive a transfer formula for

infinite bundles and coverings. Let � be an Aut(F )-assignment on X. Let (�,�) be a pair

of groups that act on X and F in such a way that the actions are �-compatible and by finite

co-volume.

THEOREM (Main Theorem 1) With the above assumptions, the equivariant zeta function is

given by:

�X� F,��ðzÞ�1¼ ð1� z2Þ��

ð2ÞðX� F Þ det�� I�X

�2AutðF Þ

ðAX!

ð ,�Þ

�P� þ IX�AF ÞzþQz2

0@ 1A,where A

X!

ð�,�Þ

is the adjacency matrix of the directed graph spanned by the edges in � 1(�), P�

is the permutation matrix induced by the action of � on V(F ), �(2) is the Euler characteristicof the quotient X1 �

�1 F, det�� is the determinant defined on the von Neumann algebra of

��, and Q is the diagonal operator such that Q(y, i)¼deg(y)þdeg(i)� 1.

Using similar methods, we prove a decomposition formula of the Ihara zeta function

for infinite covers. Let p : Y!X be a cover with X finite. Let �¼Cov( p).

THEOREM (Main Theorem 2) With the above notation,

�Y,�ðzÞ�1¼ ð1� z2Þ��

ð2ÞðYÞ det� I�X�2�

AðY!

ð , �Þ Þ� P�

!zþQz2

!,

where Q is the diagonal operator with (x, �)-entry deg(x)� 1.

We apply the above calculations to sequences of strongly convergent graphs.

In particular, a sequence {(Xn,wn)}n2N is strongly convergent to (X,w) if it is a covering

sequence of regular graphs converging to X in such a way that X covers compatibly each

element of the sequence. Such sequences appear when we consider the Cayley graphs of

finite quotients of groups converging to the Cayley graph of the group.

2. Preliminaries

We now define a number of terms that we will use later on.

Definition 2.1 Let G be any locally finite graph. Then we define the adjacency operator

AG as follows: for any u, v2V(G),

AGðu, vÞ ¼1, if u � v0, otherwise.

�The definition makes sense even if the graph is directed. If G is undirected, then AG is

symmetric.

186 S. Cooper and S. Prassidis

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Definition 2.2 Let ~G and G be locally finite graphs. We say that

p : ~G! G

is a graph covering if the following two conditions hold:

(1) If x � ~G y, then p(x)�G p(y).(2) For any x 2 ~G, p : N(x)!N( p(x)) is a bijection.

The first condition in the definition means that p is a graph map. The second condition

is a local triviality condition.Graph bundles are defined in [14]. They generalize the graph coverings in the sense

that the ‘fibre graph’ is allowed to have a non-empty set of edges. We will concentrate

on bundles with finite fibres. For a graph X, we denote by EðX!Þ the set of ordered edges –

i.e. each edge of X appears twice, each with opposite orientation.

Definition 2.3 Let G be any locally finite graph (possibly infinite), let F be a finite graph.

We define an Aut(F )-voltage assignment on G by

� : EðG!Þ ! AutðF Þ, �ðuvÞ ¼ �ðvuÞ�1:

Definition 2.4 Let G be a locally finite graph, F a finite graph and � an Aut(F )-voltage

assignment on G. We define a graph bundle G� �F to be the graph with vertex set

V(G)�V(F ), with two vertices (u, i), (v, j)2G� �F adjacent, if either one of the following

two conditions hold:

(1) u� v and j¼ i�(uv)

(2) u¼ v and i� j.

Let � be a Aut(F )-voltage assignment on G. Let � 2Aut(F ).

(1) Let G!ð�,�Þ denote the spanning sub-graph of the digraph G

!whose directed edge set

is �� 1(�).(2) We define the permutation operator P� by the following formula: for any two

vertices i, j in V(F ),

P�ði, jÞ ¼1, if j ¼ i�

0, otherwise.

�Remark 2.5 When the graphs are infinite, the matrices defined above are operators on the

Hilbert space with basis on the vertex set of the graph. More precisely, if G is any locally

finite graph, set L2(G) to be the Hilbert space:

L2ðGÞ ¼ f : VðGÞ ! C :X

u2VðGÞ

jfðuÞj2 <1

( ):

Then the adjacency operator is given by

Að f ÞðuÞ ¼Xu�v

fðvÞ:

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With the above notation,

P�ð f ÞðiÞ ¼ fð�ðiÞÞ:

The following combines covering maps and bundles.

THEOREM 2.6 Let F and X be locally finite graphs. Let X be equipped with an

Aut(F )-voltage assignment �. Let p : Y!X be a covering map and the Aut(F )-voltage

assignment

: EðY!Þ ! AutðF Þ, ðxyÞ ¼ �ðpðxÞpðyÞÞ:

Define a graph map

~p : Y� F! X��F, ~pðx, iÞ ¼ ðpðxÞ, iÞ:

Then ~p is a covering map.

Proof First we will prove that ~p is a graph map i.e. preserves adjacency. Let

(x, i)� (y, j) in Y� F. There are two cases to consider:

(1) Suppose x� y in Y. Then p(x)� p(y), and j¼ i (xy)¼ i�( p(x)p(y)). Thus, by definition,

~pðx, iÞ ¼ ðpðxÞ, iÞ � ðpðyÞ, jÞ ¼ ~pðy, jÞ, in X��F:

(2) Suppose x¼ y. Then i� j in F, and clearly p(x)¼ p(y). Thus, by definition,~pðx, iÞ � ~pðy, jÞ.

Thus ~p preserves adjacency.Now we must show that ~pjNðx, iÞ is a bijection.~pjNðx, iÞ is an injection. Let (y1, j1), (y2, j2)2N(x, i) with ~pðy1, j1Þ ¼ ~pðy2, j2Þ: Then we

know that p(y1)¼ p(y2) and j1¼ j2. Now there are two cases to consider:

(1) Suppose y1¼x and i� j1. Then i� j2, and since (x, i)� (y2, j2), we must have x¼ y2.

Thus, y1¼ y2, so (y1, j1)¼ (y2, j2). The same argument works if y2¼ x.(2) Suppose y1� x and i ¼ j1

ðy1xÞ ¼ j1�ðpðy1ÞpðxÞÞ. Since p(y1)¼ p(y2), we see that

i ¼ j2�ðpðy2ÞpðxÞÞ. Thus, since (y2, j2)� (x, i), we must have y2� x. Now, since p is a

graph covering map, pjN(x) is a bijection. But y1, y22N(x) and p(y1)¼ p(y2); thus,

y2¼ y1, so (y1, j1)¼ (y2, j2).

~pjNðx, iÞis a surjection. Let (u, k)2N(( p(x), i). Again, there are two cases to consider:

(1) Suppose u¼ p(x) and i� k. Then ~pðx, kÞ ¼ ðu, kÞ, and by definition, (x, k)2N(x, i).(2) Suppose u� p(x) and i¼ k�(up(x)). Since pjN(x) is a surjection, there exists some

y2N(x) such that p(y)¼ u. Then y� x and i¼ k�( p(y)p(x))¼ k (yx). Thus, (y, k)2

N(x, i) and ~pðy, kÞ ¼ ðu, kÞ.

Therefore, ~pjNðx,iÞ is a bijection. This completes the proof. g


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The vertex set of a bundle over G is V(G)�V(F ). Then

L2ðG��F Þ ¼ L2ðGÞ�L2ðF Þ

where the tensor product takes place in the category of Hilbert spaces. More precisely,

it is the completion of the algebraic tensor product. The following theorem (proved in [14]

for the finite case) provides a decomposition for the adjacency operator of any graph

bundle.

THEOREM 2.7 Let � be an Aut(F )-voltage assignment on a locally finite graph G, with F

locally finite. Then

AG��F ¼M

�2AutðF Þ

AG!

ð�,�Þ

� P� þ IG � AF:

Proof It is enough to prove the result for functions of the form f� g, where f2L2(G) and

g2L2(F ). Let (u, i)2V(G� �F ). Then

AG��Fðf�gÞðu, iÞ ¼X

ðu, iÞ�ðv, jÞ

fðvÞgðjÞ:

The right-hand side is given by:M�2AutðF Þ

AG!

ð�,�Þ

� P�ðf�gÞðu, iÞ þ IG � AFðf�gÞðu, iÞ

¼M

�2AutðF Þ, u2G!

ð�,�Þ

AG!

ð�,�Þ

ðfjÞðuÞP�ðgÞðiÞ þ fðuÞAFðgÞðiÞ

There are two possibilities for (u, i)� (v, j):

(1) u� v and i¼ j�(uv). Then the right-hand side becomes:

AG!

ð�,�ðuvÞÞ

ðfjÞðuÞP�ðgÞðiÞ ¼ fðvÞgðjÞ:

(2) u¼ v and i� j. In the right-hand side, only the last summand is non-zero and it is

equal to f(u)g(j).

Finally, it is clear that if neither u� v nor u¼ v, then the sum on the right-hand side is

zero. This completes the proof. g

By a marked graph, we mean a pair (X,w) with X, a graph, and w, a distinguished

vertex.

Definition 2.8 On the space of marked graphs, there is a metric dist defined as follows:

dist�ðX1,w1Þ, ðX2,w2Þ

�¼ inf

1

nþ 1;BX1ðw1, nÞ is isometric to BX2

ðw2, nÞ

� �,

where BX(w, n) is the combinatorial ball of radius n in X centred on w.


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For a sequence of marked graphs {(Xn,wn)}n2N, we say that (X,w) is the limit graph if

limn!1

dist ðX,wÞ, ðXn,wnÞð Þ ¼ 0:

For a finite graph X, let cr(X ) denote the number of closed paths in X of length r. Let

ðX,wÞ ¼ limn!1ðXn,wnÞ,

where {(Xn,wn)}n2N is a covering sequence of k-regular marked graph. In [8], the definition

of the number cr is extended for the graph X as follows:

~cr ¼ limn!1

crðXnÞ

jXnj:

In [8], it was shown that the limit exists. The zeta function �(X,w) of the marked graph X,

with respect to the sequence {(Xn,wn)}n2N, is defined by

ln�ðX,wÞðzÞ ¼ limn!1

1

jXnjln �Xn

ðzÞ ¼X1r¼1

~crzr

r, jzj <

1

k� 1:

The proof that the series has a non-trivial radius of convergence is given in [8] and depends

on results from [15].Let X be a graph such that the degrees of vertices are bounded. Let � be a group of

graph automorphisms of the graph X that acts on X without inversions and satisfying the

following properties:

(1) For each v2V(X ), the stabilizer �v¼ {� 2� : �v¼ v} is finite.(2) If F 0�V(X ) is a complete set of orbit representatives of the action of � on V(X ),

then

volðX=�Þ ¼Xv2F 0

1

j�vj<1:

In particular, if the action of � on V(X ) is free, the second condition is equivalent to the

condition that the orbit space V(X )/� is finite. In this case, the Ihara zeta function is

defined as

�X,�ðzÞ ¼Y

C2P=�

1� z‘ðCÞ� �j�Cj

:

where:

. P/� are equivalence classes of closed, primitive, tailless edge-paths without

backtracking. Two such circuits are equivalent if they differ only by a shift. P/�

denotes the orbit space of P under the � action.. For each class C2P/�, ‘(C) denotes the length of C i.e. the number of edges in C.. �C denotes the isotropy group of C.

This formula generalizes the classical zeta function on finite graphs.


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We will describe the analogue of Bass’ formula for �X,�(z). Let L2(X ) be the Hilbert

space of functions on V(X ). A unitary representation is given by:

�0 : �! UðL2ðXÞÞ, ð�0ð�ÞfÞðvÞ ¼ fð��1vÞ, � 2 �, f 2 L2ðXÞ, v 2 VðXÞ:

Then the von Neumann algebra of all bounded operators on L2(X ) that commute with the

� action is defined as:

N 0ðX,�Þ ¼ f�0ð�Þ : � 2 �g0:

The algebra N 0(X, �) inherits a trace given by:

Tr�ðAÞ ¼Xv2F 0

1

j�vjAðv, vÞ, A 2 N 0ðX,�Þ:

With this setting, the Bass formula for the Ihara zeta function has the form [3,4,9–11]:

��1X,�ðzÞ ¼ ð1� z2Þ��ð2ÞðXÞ det�ð�X, zÞ,

where

. det�¼ exp �Tr� � ln is the determinant in the von Neumann algebra N 0(X, �).

. �X,z¼ I�AzþQz2, with A the adjacency operator on X and Q is the operator on

L2(X ) given by:

Qð f ÞðvÞ ¼ ð degðvÞ � 1ÞfðvÞ, for each v 2 VðXÞ:

Remark 2.9

(1) In [3,4,9–11], it was shown that the function �X,� is defined for sufficiently small juj.

More precisely, if k is the maximum degree of X, �X,�(u) is a holomorphic function

for all juj < ð1=k� 1Þ.(2) �(2)(X ) is the Euler characteristic defined in [3]. In most applications, it is equal to

�(X/�), the Euler characteristic of the orbit space.(3) Let X be a k-regular graph and q¼ k� 1. Using the determinant formula, the zeta

function can be extended to a holomorphic function in the open set [3,9]:

�q ¼ R2n ðx, yÞ 2 R

2 : x2 þ y2 ¼1

q

� �[ðx, 0Þ 2 R

2 :1

q jxj 1

� �� :

(4) In the above references there is an interpretation of the Bass formula

over the determinant on N 1(X, �), the von Neumann algebra on the set of edges

of X.

Notation. There are three types of zeta functions used in this article.

(1) We write �X(z) for the classical zeta function defined for a finite graph X.(2) We write �X,�(z) for the equivariant zeta function defined on an infinite graph X

equipped with an action of a group � with finite co-volume.(3) We write �(X,w)(u) for the zeta function that it is the limit of �Xn

ðzÞ1=jVðXnÞj, where

{(Xn,wn)}n2N is a covering sequence of finite regular graphs converging to (X,w).


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Definition 2.10 The sequence {Xn,wn}n2N strongly converges to (X,w) if:

(1) {Xn,wn}n2N is a covering sequence of marked k-regular graphs with

pm�1 : Xm! Xm�1

the covering map.(2) X is k-regular.(3) There are covering maps

n : X! Xn

such that:

(a) 1(u)¼ p1, p2, . . . , pn� 1(n(u)), for all n.(b) For each n, the isometry between BX(w, sn) and BXn

ðwn, snÞ is given by therestriction of n.

Remark 2.11 Cayley graph of groups give sequences of graphs that strongly converge.Let � be a group, S a symmetric set of generators and {Kn}n2N a sequence of normalsubgroups of finite index such that:

K1 K2 K3 . . . , and\n2N

Kn ¼ K:

Then the sequence of the marked Schreier graphs {(S(�,Kn,S), 1)}n2N strongly convergesto (S(�,K,S), 1).

Let {Xn,wn}n2N strongly converge to (X,w) and �n¼Cov(X,Xn). The next results give aconnection between the different types of zeta functions.

THEOREM 2.12 Assume that all the graphs in the sequence are k-regular finite graphsconverging to a k-regular graph (X,w). Then

limn!1

�XnðzÞðjVðX1Þj=jVðXnÞjÞ ¼ �X,�1

ðzÞ ¼ �ðX,wÞðzÞjVðX1Þj:

Proof By [4], Theorem 2.1,

limn!1

�XnðzÞð1=jNnjÞ ¼ �X,�1

ðzÞ

where Nn¼ [�n :�1]¼ jV(Xn)j/jV(X1)j. The result follows from the definition of�(X,w)(z). g

The following is the main part of the proof of Theorem 2.1 in [4].

COROLLARY 2.13 With the above notation,

det�1ð�X, zÞ ¼ lim

n!1detð�Xn, zÞ� �ðjVðX1Þj=jVðXnÞjÞ

Let F be a locally finite graph and �1 an Aut(F )-voltage assignment on X1. Inductively,define an Aut(F )-voltage assignment on Xn by:

�nðuvÞ ¼ �n�1ðpn�1ðuÞpn�1ðvÞÞ:


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Also, define an Aut(F )-voltage assignment on X by:

�ðuvÞ ¼ �1ð1ðuÞ1ðvÞÞ:

The details are presented in the following diagram.

Now, by Theorem 2.6, we know that for any finite, d-regular graph F, the sequence

fXn ��n Fgn2N is a kþ d-regular covering sequence; thus, by [8], it converges. We will show

that in fact it converges to the graph X� �F. To do this, we will need the following:

LEMMA 2.14 Assume that {(Xn, wn)}n2N strongly converges to (X, w). Then

en : BX��Fððw, iÞ, snÞ ! BXn��n Fððwn, iÞ, snÞ, enðu, iÞ ¼ ððnðuÞ, iÞ,

is an isometry, for any i2V(F ) and for all n2N.

Proof Since n is a bijection, it is clear that en is a bijection; thus, we must show that enpreserves adjacency. To this end, assume (u, i)� (v, j), for ðu, iÞ, ðv, jÞ 2 BX��Fðw, iÞ.

Then there are two cases:

(1) u� v and j¼ i�(uv)

(2) u¼ v and i� j.

In the case where u¼ v, since clearly n(u)¼ n(v), we must have enðu, iÞ � enðv, jÞ. In the

case where u� v, we must have n(u)� n(v). Thus we must show that

j ¼ i�nðnðuÞnðvÞÞ:


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Now, by the definition of �,

�ðuvÞ ¼ �1ð1ðuÞ1ðvÞÞ,

and by assumption,

�1ð1ðuÞ1ðvÞÞ ¼ �1ðp1 � � � pn�1ðnðuÞÞ, p1 � � � pn�1ðnðvÞÞÞ:

But by the definition of �n,

�nðnðuÞnðvÞÞ ¼ �1ðp1 . . . pn�1ðnðuÞÞ, p1 . . . pn�1ðnðvÞÞÞ:

This shows that

�nðnðuÞnðvÞÞ ¼ �ðuvÞ,

and thus

j ¼ i�ðuvÞ¼)j ¼ i�nðnðuÞnðvÞÞ:

This shows that n preserves adjacency, and thus, is an isometry. This completes the

proof. gAs a corollary, we have the following theorem.

THEOREM 2.15 For each i2F, the covering sequence fðXn ��n F, ðwn, iÞÞgn2N strongly

converges to (X� �F, (w, i)).

Proof Theorem 2.6 implies that the covering conditions of the strong convergence are

satisfied. The rest of the proof follows from Lemma 2.14 and [8]. g

3. Zeta functions for bundles and coverings

In this section we will use our previous result and [3,4,9–11], to generalize the results of [6]

to infinite graph bundles.

Definition 3.1 Let X be a graph equipped with an Aut(F )-voltage assignment �.

(1) An action of a group � on X without edge inversions is called (F,�)-compatible if

�ð�ðuÞ�ðvÞÞ ¼ �ðuvÞ, for all u, v 2 VðXÞ, � 2 �:

(2) An action of a group � without inversions on F is called (X,�)-compatible if

Im(�)�CAut(F )(�) i.e. the image of � centralizes �.(3) The pair of groups (�, �), as before, is called �-compatible if � is (F, �)-compatible

and � is (X, �)-compatible.

LEMMA 3.1 With the above notation, if the pair (�,�) is �-compatible, then the product

action:

ð�, Þðx, iÞ ¼ ð�x, iÞ, ð�, Þ 2 ��, ðx, iÞ 2 VðX��F Þ,

is an action by graph automorphisms on X� �F. Furthermore, if the action of � and � are of

finite co-volume, so is the action of �� on X� �F.

Proof The proof follows from the definitions. g


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THEOREM 3.3 Assume that (�,�) is a pair of �-compatible actions. Also, assume that the

actions are of finite co-volume. Then

�X��F,��ðzÞ�1¼ ð1� z2Þ��

ð2ÞðX��F Þdet�� I�X

�2AutðF Þ

ðAX!

ð�,�Þ

�P� þ IX�AF ÞzþQz2

0@ 1A,�(2) is the Euler characteristic and Q is the diagonal operator with (x, i)-entry

deg(x)þ deg(i)� 1. Furthermore, the zeta function is holomorphic for

jzj < ð1=ðkþ d� 1ÞÞ. If X is k-regular and F d-regular, then �X��F,�ðzÞ can be extended to

a holomorphic function on �kþ d� 1.

Proof From [9,10], we have

�X��F,��ðzÞ�1¼ ð1� z2Þ��

ð2ÞðX��F Þ det�� I� zAX��F þ z2QX��F

� �:

The theorem now follows immediately from Theorem 2.7. g

We will now use Theorem 3.3 to provide a decomposition for the zeta function of any

infinite cover. Let p : Y!X be a cover with X finite and Y locally finite. Let Cov( p)¼�.

Now we define the function

� : EðX!Þ ! �:

For this we write X¼ {x1, . . . , xn}. For each i, choose vi2Y such that p(vi)¼ xi.

Now, since p : N(vi)!N(xi) is a bijection, for each xj2N(xi) there exists a unique

uj2N(vi) such that p(uj)¼xj. So, since p(vj)¼ xj¼ p(uj), there exists some � 2� such that

�vj¼ uj. Thus, define

� : EðX!Þ ! �, �ðxixjÞ ¼ �:

We then have the following:

LEMMA 3.4 Let � be defined as above. Then

(1) The � action on Y is of finite co-volume and it is �-compatible.(2) The map

� : Y! X�� , �ðuÞ ¼�pðuÞ,�

�is an isomorphism, where �u¼ vi, p(u)¼ xi¼ p(vi).

Proof The proof is folklore. g

Now, in order to prove an analogue of 2.7 for Y’X� ��1, we need to define the

following operator: for �1, �2, � 2�1,

P�ð�1, �2Þ ¼1, if �2 ¼ �1�

0, otherwise:

(


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LEMMA 3.5 Let � be defined as above. Then

AX�� ¼X�2�

AX!

ð�,�Þ

� P� :

Proof The proof is analogous to that of 2.7. g

The following theorem provides a decomposition for the zeta function of any infinite

cover.

THEOREM 3.6 With the above notation,

�Y,�ðzÞ�1¼ ð1� z2Þ��

ð2ÞðXÞ det� I�X�2�

AðX!

ð�,�ÞÞ� P�

!zþQz2

!,

where Q is the diagonal operator with (x, �)-entry deg(x)� 1. The function is holomorphic for

jzj < ð1=ðk� 1ÞÞ. If Y is k-regular, then �Y,�(z) is holomorphic on �k� 1.

Proof This follows immediately from Theorem 3.3 and Lemma 3.4. g

Let {(Xn,wn)}n2N be a sequence of finite k-regular marked graphs that strongly

converges to the k-regular marked graph (X,w). Let F be a finite d-regular graph. With the

notation as in Theorem 2.15, we know thatWe write an¼ jV(Xn)j and f¼ jFj.

COROLLARY 3.7 With the above notation, for jzj < 1=ðkþ d� 1Þ,

�ðX��F, ðw, iÞÞðzÞ�1

¼ limn!1

ð1� z2Þ��ðXn��n F Þ det I�

X�2AutðF Þ

ðAXn

!

ð�n , �Þ

� P� þ IXn� AF ÞzþQnz

2

!" #1=fan

¼ ð1� z2Þ��ð2ÞðX��F Þ det� I�

X�2AutðF Þ

ðAX!

ð�,�Þ

� P� þ IX � AF ÞzþQz2

!" #1=fa1

Proof The first identity follows because of Theorem 2.15:

�ðX��F, ðw, iÞÞðzÞ�1

¼ limn!1

�Xn��nFðzÞ

�1=fan

¼ limn!1

ð1� z2Þ��ðXn��n F Þ det I�

X�2AutðF Þ

ðAXn

!

ð�n , �Þ

� P� þ IXn� AF ÞzþQnz

2

0@ 1A24 351=fan

The second identity follows from Theorem 3.3 and Theorem 2.12. g

4. Application

Let p : Y!X be a cover with Cov( p)¼� and X a finite graph. Let F be a finite d-regular

graph with n such that Aut(F ) contains a dihedral group D2n of order 2n. Let � be an


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Aut(F )-voltage assignment on X whose image is contained into D2n and the induced

Aut(F )-voltage assignment on Y (Theorem 2.6). By Theorem 2.6, the induced map

~p : Y� F! X��F, ~pðx, iÞ ¼ ðpðxÞ, iÞ

is a covering map. Also, Covð ~pÞ ¼ �.The following is the setup (for the finite case this is the same as in [6] and [13]):

set V(F )¼ {1, 2, . . . , n} and Sn the symmetric group on V(F ). Let a¼ (1, 2, . . . , n� 1 n) be

an n-cycle and let

b ¼

ð1 nÞð2 n� 1Þ . . .n� 1

2

nþ 3

2

� nþ 1

2

� if n is odd,

ð1 nÞð2 n� 1Þ . . .n

2

nþ 2

2

� if n is even

8>>><>>>:be a permutation in Sn. The permutations a and b generate the dihedral subgroup Dn of Sn:

Dn ¼ ha, bj an ¼ b2 ¼ 1, bab ¼ a�1i:

Let ¼ exp(2�i/n) and xk¼ (1, k, 2k, . . . , (n� 1)k)T be the column vector in Cn. Then 1,

, . . . , n� 1 are the distinct eigenvalues of the permutation matrix P(a) and xk is

the eigenvector corresponding to the eigenvalue k. Let P(b) be the permutation matrix

of b and

M ¼½x0 PðbÞx1 x2 PðbÞx2 . . . xðn�1Þ=2 PðbÞxðn�1Þ=2� if n is odd

½x0 PðbÞx1 x2 PðbÞx2 . . . xðn�2Þ=2 PðbÞxðn�2Þ=2 xn=2� if n is even

(

In [13] (also [6]), it was shown that P(b)xk is an eigenvector of P(a) associated with the

eigenvalue n� k. Thus M is invertible. Also, P(a) and AF commute, and thus, they are

simultaneously diagonalizable. Also, xk and P(b)xk(1 k (n� 1)/2 when n is odd and

1 k (n� 2)/2 when n is even) are eigenvectors of AF with the same eigenvalue of P(b),

denoted �(F,k). Also, x0 is the eigenvector of AF corresponding to the eigenvalue d and,

for n even, �(F,n/2) is the eigenvalue associated to the eigenvector x2. Then as in [13], using

Theorem 2.7, we get that

ðIY�MÞ�1AY� FðIY�MÞ

¼

ðAY þ dIYÞ�Lðn�1Þ=2

i¼1 ðAt þ �ðF, tÞðIY�IYÞ� �

if n is odd

ðAY þ dIYÞ�Lðn�2Þ=2

i¼1 ðAt þ �ðF, tÞðIY�IYÞ�ðBþ �ðF, n=2ÞIY

� �if n is even

8><>:where

B ¼Xn�1k¼0

ð�1ÞkAðY!ð , akÞÞ þ ð�1Þ

kþ1AðY!ð , akbÞÞ

� �,


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and

At ¼Xn�1k¼0

tkAðY!ð , akÞÞ tkAðY

!ð , akbÞÞ

ðn�tÞkAðY!ð , akbÞÞ ðn�tÞkAðY

!ð , akÞÞ

24 35Also, let LY¼ (QYþ dIY)� I2. Then the calculation in Section 4 in [6] can be carried

through in our setting and we get the following:

THEOREM 4.1 Let p : Y!X be as above. Then

�Y� F,�ðzÞ�1¼ ð1� z2Þ��

ð2ÞðY� F ÞfY,FðzÞYðn�1Þ=2t¼1

gY,F, tðzÞ

when n is odd, and

�Y� F,�ðzÞ�1¼ ð1� z2Þ��

ð2ÞðY� F ÞfX,FðzÞhY,FðzÞYðn�2Þ=2t¼1

gY,F, tðzÞ

when n is even, where

(1) gY,F,t(z)¼ det�(IY� IY� (Atþ �(F,t) (IY� IY))zþLYz2)

(2) hY,F(z)¼det�(IY� (Bþ �(F,n/2) IY)zþ (QYþdIY)z2)

(3) fX,F(z)¼ det�(IY� (AYþ dIY)zþ (QYþ dIY)z2)

Proof This follows by simple calculation from Theorem 3.3 with �¼ {1}, and

Theorem 9 of [6]. g

Let {(Xm,wm)}m2N be a sequence of finite regular graphs that strongly converges

to (X,w). Let F be a finite d-regular with n vertices such that Aut(F ) contains D2n.

Let � be an Aut(F )-voltage assignment on X1 whose image is contained in D2n. Let �nbe the induced Aut(F )-voltage on Xm and be the induced Aut(F )-voltage assignment

on X. Set

� ¼ CovðX! X1Þ, �m ¼ CovðXm ! X1Þ, m 2 N:

COROLLARY 4.2 Let am¼ jV(Xm)j. With the above notation,

(1) If n is odd:

�X� F,�ðzÞ�1¼ �ðX� F, ðw, iÞÞðzÞ

�a1 ¼ limm!1

ð1� z2Þ��ðXm��mF ÞfXm,FðzÞ

Yðn�1Þ=2t¼1

gXm,F, tðzÞ

!a1=am

:

(2) If n is even

�X� F,�ðzÞ�1¼ �ðX� F,ðw, iÞÞðzÞ

�a1 ¼ limm!1

ð1� z2Þ��ðXm��mF ÞfXm,FðzÞhXm,FðzÞ

Yðn�1Þ=2t¼1

gXm,F,tðzÞ

!a1=am

:

Proof It follows from Theorems 2.12, 2.15 and 4.1. g


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In some cases, we can get a better description of the functions appearing in the

expression for the zeta function of the limit. Assume that all the graphs Xm, m2N, and Xare p-regular. Following [8], for each m2N set:

�m ¼X �iðXmÞ

am

where �i(Xm) are the eigenvalues of the Markov operator ð1=kÞAXmon Xm and x is the

Dirac function. The sequence {�m}m2N weakly converges to the spectral measure �associated to the Markov operator (1/k)AX. Using the calculation of Section 5 in [8] andCorollary 2.13, we get:

ln fX,FðzÞ ¼ ln det� IY � AY þ dIYð Þzþ QY þ dIYð Þz2� �

¼ limm!1

1

amln det IXm

� AXmþ dIXm

� �zþ QXm

þ dIXm

� �z2

� �¼ lim

m!1

Z 1

�1

ln 1� ðp�þ dÞzþ ðp� 1þ dÞz2� �

d�mð�Þ

¼

Z 1

�1


d�ð�Þ

where � is the spectral measure associated to (1/p)AX.Summarizing:

COROLLARY 4.3 With the above notation,

ln fX,FðzÞ ¼

Z 1

�1


d�ð�Þ, for jzj <1

pþ d� 1,

where � is the spectral measure associated to the regular random walk on X.

We give a specific example. The same method works for any group for which thespectral measure is known. Let � be the Grigorchuk group [1,5,7,8]. Then � can be

represented as a subgroup of automorphisms of the rooted binary tree. Let P¼ St(11) bethe stabilizer of the infinite sequence of 1s. Let Pm be the stabilizer of all the elements that

start with m 1s and it has finite index in �. Then

P ¼[1m¼1

Pm:

If S¼ {a, b, c, d} be the standard set of generators of �, then the Schreier graphs

{Sm¼S(�,Pm,S)}m2N converge to S¼S(�,P,S). All the graphs have as the base point theidentity coset. Then in [8], Corollary 9.2, we have that

ln �S,PðzÞ ¼ �3 lnð1� z2Þ �

Z 0

�1=2

1� 8xzþ 7z2� �

j1� 4xj

2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixð2x� 1Þð2xþ 1Þð1� xÞ

p dx

�

Z 1

1=2

1� 8xzþ 7z2� �

j1� 4xj


p dx

Let F be a d-regular graph as in the beginning of the section and � an Aut(F )-voltageassignment whose image lies into D2n. Combining Theorem 4.1 and Corollary 4.3 we get.


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THEOREM 4.4 With the above notation, let be the Aut(F )-voltage assignment on S and

R¼P1/P. Then

ln �ðS� F, ðP, iÞÞðzÞ ¼1

da1

"� �ð2ÞðS�� F Þ lnð1� z2Þ

þ ln detR I�X

�2AutðF Þ

ðAðS!ð , �ÞÞ � P� þ IS � AF ÞzþQz2

0@ 1A35Furthermore,

�ðS� F, ðP, iÞðzÞ�1¼ ð1� z2Þ��

ð2ÞðS� F ÞfS,FðzÞYðn�1Þ=2t¼1

gS,F, tðzÞ

when n is odd, and

�ðS� F, ðP, iÞðzÞ�1¼ ð1� z2Þ��

ð2ÞðS� F ÞfX,FðzÞhS,FðzÞYðn�2Þ=2t¼1

gS,F, tðzÞ

when n is even, where g and h are as in Theorem 4.1 and

ln fS,FðzÞ ¼

Z 0

�1=2

lnð1� ð8xþ dÞzþ ð7þ dÞz2Þj1� 4xj


p dx

þ

Z 1

1=2

lnð1� ð8xþ dÞzþ ð7þ dÞz2Þj1� 4xj


p dx

Acknowledgements

The research was partially supported by an NSF REU grant and Canisius College Summer Grant.

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Zeta functions of infinite graph bundles

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