Weighted prime geodesic theorems
Anton Deitmar

TL;DR
This paper extends prime geodesic theorems to weighted infinite graphs and building quotients, relating growth rates to spectral data of translation operators, inspired by Bass's work.
Contribution
It introduces weighted prime geodesic theorems for complex structures, connecting spectral analysis with geometric growth in new weighted settings.
Findings
Prime geodesic theorems are established for weighted graphs.
Growth rates are expressed via spectral data of translation operators.
The approach generalizes classical results to weighted and more complex structures.
Abstract
Prime geodesic theorems for weighted infinite graphs and weighted building quotients are given. The growth rates are expressed in terms of the spectral data of suitable translation operators inspired by a paper of Bass.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGraph theory and applications · Advanced Topics in Algebra · Matrix Theory and Algorithms
Weighted prime geodesic theorems
Anton Deitmar
Abstract: Prime geodesic theorems for weighted infinite graphs and weighted building quotients are given. The growth rates are expressed in terms of the spectral data of suitable translation operators inspired by a paper of Bass.
MSC: 11N80, 11F72, 11N05, 20E08, 20F65, 51E24, 53C22
[TABLE]
Contents
- 1 -positive operators
- 2 The Ihara Zeta function
- 3 The prime geodesic theorem for a weighted graph
- 4 Affine buildings
- 5 Discrete groups
- 6 The prime geodesic theorem for a weighted building quotient
Introduction
The first Prime Geodesic Theorem was given by Huber in [Huber]. It states that for a compact hyperbolic surface the number of prime closed geodesics of lenght satisfies
[TABLE]
It has been sharpened by giving estimates on the error term and it has been extended to other manifolds [Hejhal, Iwaniec, Katsunada, Koyama, ADhigherRank, Sound]. Applications to class numbers are in [Sarnak], and, extending this result, in [class, classNC]. It was extended to more general dynamical systems, culminating in Margulis’s celebrated result on Anosov flows, stating that the number of closed orbits of length satisfies , where is the entropy of the system [Margulis].
An extension to the graph case was formulated in [Hashi, ADCont]. In [weightedIhara], zeta functions of graphs with weights were introduced. Here weights are representing resistance to a flow along the edges or a individual distribution of the flow at the nodes. In this paper we present a notion flexible enough to take into account these different requirements.
The paper [PGTbuild] gives Prime Geodesic Theorems for compact quotients of buildings, generalizing the graph case. This generalization is natural, as in number theoretical situations, graphs and building quotients both turn up in the -adic setting. In the present paper the latter two ideas are combined in stating Prime Geodesic Theorems for weighted infinite graphs and weighted building quotients. In the latter case, a full expansion of the numbers of closed geodesics of a given length is presented.
The first section treats an extension of the Perron-Frobenius Theorem suitable for our purposes. The next two sections deal with the graph case and the last three with building quotients.
1 -positive operators
Definition 1.1**.**
Let be a Hilbert space and an orthonormal basis. Let denote the cone of all with for every . A bounded linear operator on is called -positive, if . This is equivalent to
[TABLE]
for all . Note that the adjoint is -positive if is. Further, if and are -positive, then so is .
Definition 1.2**.**
We say that a bounded operator is -reducible, if there exists a proper subset such that
[TABLE]
where by we mean the closure of the span of . This is in accordance with the isomorphism . In this case, the closed subspace is -stable.
If is not -reducible, we say that is -irreducible.
Definition 1.3**.**
For a compact operator and an eigenvalue the geometric multiplicity is the dimension of the eigenspace . In this situation, the sequence , is eventually stationary. The algebraic multiplicity is the limit
[TABLE]
Theorem 1.4**.**
Let be a positive compact operator on with positive spectral radius . Then the spectral radius is an eigenvalue of . Assume that is -irreducible and trace class. Then the algebraic multiplicity of is 1. The eigenvalues with distribute evenly over the unit circle times . More precisely, if are all eigenvalues of with , then we can order them in a way that . Finally, every has algebraic multiplicity 1.
Proof.
If , this is the Theorem of Frobenius, see [Gant]. So we assume to be infinite-dimensional. We first show that if is -irreducible, then so is the adjoint operator . For this assume -irreducible and let be such that maps to itself. Then for every and every we have
[TABLE]
This implies that maps to itself, hence is either empty or equals , which means that is -irreducible.
The fact that is an eigenvalue is known as the Krein-Rutman Theorem, see for instance Theorem 7.10 in [Abram]. This theorem also says that there exists an eigenvector for the eigenvalue which is -positive in the sense that for every . Let be the set of all such that . For we write . have
[TABLE]
Since is positive, and so we have if . This means that is -stable, hence, since , we get . So we conclude that for every . This means that is a totally -positive vector.
We now assume that is -irreducible, -positive and trace class. Then the Fredholm determinant is an entire function. For such that we set
[TABLE]
For each let be the largest -stable subspace on which has spectrum . If , then is finite-dimensional and
[TABLE]
Let . Then and are closed, -stable subspaces and
[TABLE]
Then , where the indicates the adjugate matrix. This implies that extends continuously to . Applying to a vector in we see that . For and we have
[TABLE]
Letting tend to we conclude that there is a sign such that is -positive. Since is totally positive, we have
[TABLE]
This implies that is a simple zero of , which is to say that the algebraic multiplicity is 1. Therefore the condition (G) of Definition 4.7 in [Schae] is satisfied, and hence the remaining points of the theorem follow from Theorem 5.2 in [Schae]. ∎
We finally consider the situation without the condition of irreducibility.
Proposition 1.5**.**
Let be a compact operator with positive spectral radius . Then for a given orthonormal basis there exists a disjoint decomposition with the following properties:
- (a)
The space is -stable for each . 2. (b)
For each let denote the orthogonal projection onto then the operator
[TABLE]
either has a spectral radius or is -irreducible.
If is -positive, then is -positive for each . If is trace class, then each is and for the Fredholm determinant one has
[TABLE]
Proof.
If is -irreducible, we are done. Otherwise there is a such that is -stable. We consider and , where is the orthogonal projection to . We have . If either of the operators or has spectral radius or ist irreducible, we leave this factor in peace and continue with the other, which we then decompose further. This process will stop, as there are only finitely many spectral values with and these are eigenvalues of finite multiplicity. By the fact that the Fredholm determinant distributes it follows that the spactral values distribute and by the finiteness, this process will stop, yielding the proposition. ∎
2 The Ihara Zeta function
Definition 2.1**.**
Let denote an oriented graph. This means that consists of the following data: a set of nodes and a set of oriented edges. We call the source of the oriented edge and the target and we write this as , . The nodes are called the endpoints of the edge .
Definition 2.2**.**
The valency, of a node is the number of edges having for one of their endpoints. Throughout, we will assume that has bounded valency, i.e., that there exist such that
[TABLE]
holds for all nodes .
We introduce the notion of a weight. Usually, a weight is put on an edge representing a length or a resistance or the reciprocal of that. In order to be more flexible and also consider paths with partial backtracking, it is more convenient for us to consider a transition weight, which may be interpreted as the likelihood of a particle or a current, of choosing a certain edge.
Definition 2.3**.**
A transition weight on , henceforth simply called a weight, is a map
[TABLE]
such that
[TABLE]
and
[TABLE]
Example 2.4**.**
A natural example is given in the case of being a quotient , where is a tree of bounded valency and is a tree lattice [treelat]. In this case the weight
[TABLE]
is a natural choice which fits the approach of Bass [Bass] to the Ihara zeta function in case of ramified quotients, see also [treelattice].
Definition 2.5**.**
An oriented path of length in is an -tuple of oriented edges such that holds for all . We write the length as . The path is called a closed path, if . For a closed path we define its weight as
[TABLE]
Definition 2.6**.**
(Shifting the starting point) On the set of all closed paths we instal an equivalence relation generated by
[TABLE]
An equivalence class of closed paths is called a cycle. The length and weight functions factor through the quotient of this equivalence, so and are well-defined for a cycle .
Definition 2.7**.**
For a cycle and a natural number we define to be the cycle one gets by iterating the cycle for -times. One has
[TABLE]
A cycle is called primitive, if is not a power of some shorter cycle . For every cycle there is a uniquely determined primitive and a uniquely determined number such that
[TABLE]
The cycle is called the underlying primitive and is called the multiplicity of the cycle .
Definition 2.8**.**
The Ihara zeta function of the weighted oriented graph is defined by the product
[TABLE]
where the product runs over all primitive cycles in .
Definition 2.9**.**
Let be the Hilbert space of all -functions on the elements of which we write as formal series with satisfying .
Definition 2.10**.**
Inspired by [Bass], we consider the Bass operator given by
[TABLE]
If for two edges we have , we write , so we have .
The following theorem is a straightforward generalization of Theorem 1.6 of [weightedIhara].
Theorem 2.11**.**
The operator is of trace class. The product converges for sufficiently small. The function extends to a meromorphic function, more precisely, is entire and satisfies
[TABLE]
where is the Fredholm-determinant.
Proof.
The proof is essentially the same as the proof of Theorem 1.6 of [weightedIhara]. We repeat it here for the convenience of the reader. We show that the operator is of trace class, and for every we have
[TABLE]
where the sum runs over all cycles of length and is the underlying primitive cycle to .
For this we consider the natural orthonormal basis of given by . Using this orthonormal basis, one easily sees that has the claimed trace, once we know that is of trace class. For this we estimate
[TABLE]
This implies that is of trace class. For small values of we have
[TABLE]
This a fortiori also proves the convergence of the product. ∎
3 The prime geodesic theorem for a weighted graph
Theorem 3.1**.**
Let be a weighted oriented graph. For let
[TABLE]
Then there are and natural numbers such that, as ,
[TABLE]
for some .
Proof.
Let the Ihara zeta function of . By the definition of we get
[TABLE]
The formula on the other hand yields
[TABLE]
where is the spectral radius of and the last sum runs over all eigenvalues of with . The number is the algebraic multiplicity of . Finally, the numbers is the number of components as in Proposition 1.5 with and is the number of eigenvalues of that component, which satisfy . Since the sum is zero unless is a multiple of , the theorem follows. ∎
Let
[TABLE]
Proposition 3.2**.**
Assume that . Let the numbers of Theorem 3.1, let be their least common multiple and let . Then, as one has
[TABLE]
and
[TABLE]
Proof.
Let
[TABLE]
We compute
[TABLE]
and this implies as well as . Next we let and we get
[TABLE]
So that
[TABLE]
Since tends to zero, this implies and this finally implies and . As was arbitrary it follows and , so and . The claim follows if we finally show that . We have
[TABLE]
and so
[TABLE]
Proposition 3.3**.**
- (a)
For the spectral radius one has
[TABLE] 2. (b)
If there exist two different primitive cycles with a common oriented edge and , then .
Proof.
(a) We have
[TABLE]
Hence it follows that equals the supremum of all for which . Let . We show that . So suppose that . Then, as power series may be differentiated element-wise and on the other hand they converge locally uniformly and may be integrated, it follows that
[TABLE]
As , the claim follows.
(b) The cycles and are primitive, distinct, and of the same length. Replacing , with these, we assume that . Next we fix representing closed paths of and which start with the edge . For we have
[TABLE]
where the sum runs over all closed paths starting with and being of the form for some . This implies that
[TABLE]
where is the Bass operator of the oriented graph with constant weight
\bullet$$\bullet$$\bullet$$\bullet$$f
One sees that Therefore and so the operator norm satisfies . The spectral radius therefore satisfies
[TABLE]
4 Affine buildings
For background on this section, the reader may consult [bldglat]. Let be a locally finite affine building. By this we understand a polysimplicial complex which is the union of a given family of affine Coxeter complexes, called apartments, such that any two chambers (=cells of maximal dimension, which is fixed) are contained in a common apartment and for any two apartments containing chambers there is a unique isomorphism fixing and point-wise. A chamber is called thin if at every wall it has a unique neighbor chamber, it is called thick, if at each wall it neighbors at least two other chambers. The building is called thin or thick if all its chambers are. For the ease of presentation, we will always assume that the building is simplicial instead of polysimplicial.
Note that our definition includes buildings which are not Bruhat-Tits. In higher dimensions, buildings tend to be of Bruhat-Tits type [BruhatTits]. For buildings of dimension at most two the situation is drastically different. Indeed, Ballmann and Brin proved that every 2-dimensional simplicial complex in which the links of vertices are isomorphic to the flag complex of a finite projective plane has the structure of a building [BallmannBrin].
When speaking of “points” in , we identify the complex with its geometric realization. Note that the latter carries a topology as a CW-complex. In this topology, a set is compact if and only if it is closed and contained in a finite union of chambers. Note that an affine building is always contractible, see Section 14.4 of [Garrett].
Definition 4.1**.**
Generally, there are different families of apartments which make a building, but there is a unique maximal family (Theorem 4.54 of [Bldgs]). In this paper, we will always choose the maximal family. Let be the automorphism group of the building , that is, the set of all automorphisms of the complex which map apartments to apartments. In the geometric realisation these are cellular maps which are affine on each cell.
Definition 4.2**.**
A choice of types is a labelling that attaches to each vertex a label , or type in such that for each chamber the set of vertices of is mapped bijectively to .
Restricting the labelling gives a bijection between the set of all choices of types and the set of all bijections , where is any given chamber. Therefore the number of different choices of types is . We fix a choice of types such that each vertex of type zero is a special vertex. This means that the set of reflection hyperplanes containing it, meets every parallelity class of reflection hyperplanes of the ambient apartment, see Definition 1.2.3 of [bldglat].
Definition 4.3**.**
Pick a chamber and an apartment containing . Let be the vertices of with . Pick as origin to give the affine space the structure of a vector space. Let denote the open cone in spanned by the interior of the chamber , i.e.
[TABLE]
Further let
[TABLE]
For two chambers we finally write
[TABLE]
if and only if
[TABLE]
Definition 4.4**.**
Let be the other vertices of and let
[TABLE]
where and for each is the largest rational number such that where is the lattice of vertices of type zero.
Let
[TABLE]
denote the set of all such that lies in . Analogously define
[TABLE]
For given the element is contained in a unique chamber such that as in the picture.
v_{0}$$C$$C(k)$$k_{1}$$k_{2}
We say that the chamber is in relative position to and we write this as
[TABLE]
5 Discrete groups
The automorphism group of the building carries a natural topology, the compact-open topology. It is a locally-compact group which is totally disconnected. A subgroup is discrete if and only if for each chamber the stabilizer group
[TABLE]
is finite. A discrete group is a lattice in , i.e., there exists a finite -invariant Radon measure on if and only if
[TABLE]
We fix a discrete group . The subgroup of all which preserve a given labelling, is normal and of finite index in . Replacing by we will henceforth assume that preserves labellings.
Definition 5.1**.**
Fix a discrete, label preserving subgroup . A -weight is a map
[TABLE]
such that
- •
for all and all ,
- •
for some ,
- •
,
- •
holds for all which satisfy .
Definition 5.2**.**
Let be a -weight. For we define an operator
[TABLE]
by
[TABLE]
where the sum runs over all chambers in relative position to . Note that for given in each apartment containing there is at most one in position , but the same can lie in infinitely many apartments containing . As we assume the building to be locally finite, the sum defining is actually finite.
Lemma 5.3**.**
For we have
[TABLE]
In particular, the operators and commute.
Proof.
On the one hand, the chamber is in relative position and on the other, for any chamber in relative position to there exist uniquely determined chambers and in relative positions and such that each apartment containing and contains and . This and the transitivity of proves the claim. ∎
Definition 5.4**.**
A quasicharacter on is a map with . For a given quasicharacter let
[TABLE]
be the generalized eigenspace, i.e, the set of all such that for every one has
[TABLE]
for some . For every non-zero the space is finite-dimensional. Let denote its dimension. We then get
[TABLE]
where the sum runs over the set of all quasi-characters .
Proposition 5.5**.**
For let be bounded, i.e., there exists such that holds for all . Then the operator
[TABLE]
is well defined and maps to itself. On this Hilbert space, is a trace class operator.
Proof.
We compute
[TABLE]
This implies that is trace class. ∎
Definition 5.6**.**
Let denote the unital subring of generated by the translation operators with . This is a commutative integral domain. Let denote its quotient field.
For indeterminates we define the formal power series
[TABLE]
where . Note that the summation only runs over the set of all such that .
Theorem 5.7**.**
* is a rational function in . More precisely, there exists a finite set and for every as well as such that*
[TABLE]
Proof.
The case of trivial weight is Theorem 3.1.4 of [bldglat]. The proof given there extends without problems to the case of general weight. ∎
Definition 5.8**.**
We say that is singular if there exists such that is an eigenvalue of . Otherwise, is regular. The singular set is a countable union of complex submanifolds of codimension 1 in , so the regular set is connected, open and dense.
Proposition 5.9**.**
The family is a meromorphic family of trace class operators on . It is holomorphic on the regular set. The map
[TABLE]
is meromorphic on and holomorphic on the regular set. We have
[TABLE]
Proof.
This is clear from the theorem and the fact that all are trace class. ∎
6 The prime geodesic theorem for a weighted building quotient
Let .
Theorem 6.1** (Prime Geodesic Theorem).**
For let
[TABLE]
There are such that
[TABLE]
as independently. Moreover, there exists a sequence with such that for every we have, with absolute convergence of the sum,
[TABLE]
Proof.
We show the last asertion first. By definition we get and so
[TABLE]
As is a quasi-character, there exists with
[TABLE]
Now let be the sequence that runs through all with , where each is repeated with multiplicity . The claim follows. The first claim follows from the fact that in . ∎
Writing we write the statement of the theorem as
[TABLE]
References
Mathematisches Institut
Auf der Morgenstelle 10
72076 Tübingen
Germany
