Weighted prime geodesic theorems

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.

1 Introduction

The first prime geodesic theorem was given by Huber in [19]. It states that for a compact hyperbolic surface X the number N(T) of prime closed geodesics of length $\leqslant T$ satisfies

It has been sharpened by giving estimates on the error term and it has been extended to other manifolds [8, 18, 20,21,22, 26]. Applications to class numbers are in [24], and, extending this result, in [7, 11]. It was extended to more general dynamical systems, culminating in Margulis’s celebrated result on Anosov flows, stating that the number N(T) of closed orbits of length $\leqslant T$ satisfies $N(T)\sim \frac{e^hT}{hT}$, where h is the entropy of the system [23].

An extension to the graph case was formulated in [9, 17]. In [10], zeta functions of graphs with weights were introduced. Here weights are representing resistance to a flow along the edges or an individual distribution of the flow at the nodes. The paper [14] 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 p-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 to the infinite-dimensional case, which is suitable for our purposes. The next two sections deal with the graph case and the last three with building quotients.

2 E-Positive operators

Definition 2.1

Let H be a Hilbert space and E an orthonormal basis. Let $H^+_E$ denote the cone of all $\sum _{e\in E}c_ee\in H$ with $c_e\geqslant 0$ for every $e\in E$. A bounded linear operator T on H is called E-positive if $T(H^+_E)\subset H^+_E$. This is equivalent to

$$\begin{aligned} \langle Te,f\rangle \geqslant 0 \end{aligned}$$

for all $e,f\in E$. Note that the adjoint $T^*$ is E-positive if T is. Further, if S and T are E-positive, then so are $S+T$ and ST.

Definition 2.2

We say that a bounded operator $T:H\rightarrow H$ is E-reducible if there exists a proper subset $\varnothing \ne F\subsetneq E$ such that

$$\begin{aligned} T(F)\subset \ell ^2(F), \end{aligned}$$

where by $\ell ^2(F)$ we mean the closure of the span of F. This is in accordance with the isomorphism $H\cong \ell ^2(E)$. In this case, the closed subspace $\ell ^2(F)$ is T-stable.

If T is not E-reducible, we say that T is E-irreducible.

Definition 2.3

For a compact operator T and an eigenvalue $\lambda$ the geometric multiplicity is the dimension of the eigenspace . In this situation, the sequence , $n\in {{\mathbb {N}}}$, is eventually stationary. The algebraic multiplicity is the limit

Theorem 2.4

Let T be an E-positive compact operator on H with positive spectral radius $r=r(T)>0$. Then the spectral radius r is an eigenvalue of T. Assume that T is E-irreducible and trace class. Then the algebraic multiplicity of r is 1. The eigenvalues $\lambda$ with $|\lambda |=r$ distribute evenly over the unit circle times r. More precisely, if $r=\lambda _0,\lambda _1,\dots ,\lambda _{n-1}$ are all eigenvalues of T with $|\lambda _j|=r$, then we can order them in a way that $\lambda _j=re^{2\pi ij/n}$. Finally, every $\lambda _j$ has algebraic multiplicity 1.

Proof

If $\dim H<\infty$, this is the Theorem of Frobenius, see [15]. So we assume H to be infinite-dimensional. We first show that if T is E-irreducible, then so is the adjoint operator $T^*$. For this assume T is E-irreducible and let $F\subset E$ be such that $T^*$ maps $\ell ^2(F)$ to itself. Then for every $f\in F$ and every $e\in E{{{\smallsetminus }}} F$ we have

This implies that T maps $\ell ^2(E{{{\smallsetminus }}} F)$ to itself, hence F is either empty or equals E, which means that $T^*$ is E-irreducible.

The fact that $r=r(A)$ is an eigenvalue is known as the Krein–Rutman Theorem, see for instance [2, Theorem 7.10]. This theorem also says that there exists an eigenvector $v_0$ for the eigenvalue $\lambda _0=r$ which is E-positive in the sense that $\langle v_0,e\rangle \geqslant 0$ for every $e\in E$. Let F be the set of all $f\in E$ such that $\langle v_0,f\rangle =0$. For $f\in F$ we write . We have

Since $T^*$ is positive, $c_e\geqslant 0$ and so we have $e\in F$ if $c_e\ne 0$. This means that $\ell ^2(F)$ is $T^*$-stable, hence, since $v_0\ne 0$, we get $F=\varnothing$. So we conclude that $\langle v_0,e\rangle >0$ for every $e\in E$. This means that $v_0$ is a totally E-positive vector.

We now assume that T is E-irreducible, E-positive and trace class. Then the Fredholm determinant is an entire function. For $u\in {{\mathbb {C}}}$ such that $\frac{1}{u}\notin \sigma (T)$ we set

For each $\lambda \in {{\mathbb {C}}}$ let $H(\lambda )$ be the largest T-stable subspace on which T has spectrum $\{\lambda \}$. If $\lambda \ne 0$, then $H(\lambda )$ is finite-dimensional and

Let $H^r=\bigoplus _{\lambda \ne r}H(\lambda )$. Then H(r) and $H^r$ are closed, T-stable subspaces and

Then , where the $\#$ indicates the adjugate matrix. This implies that B(u) extends continuously to $u=1/r$. Applying B(1/r) to a vector in $H_{<r}$ we see that $B(1/r)\ne 0$. For $0<u<1/r$ and $e,f\in E$ we have

Letting u tend to 1/r we conclude that there is a sign $\sigma \in \{\pm 1\}$ such that $\sigma B(1/r)$ is E-positive. Since $v_0$ is totally positive, we have

This implies that 1/r is a simple zero of , which is to say that the algebraic multiplicity is 1. Therefore the condition (G) of [25, Definition 4.7] is satisfied, and hence the remaining points of the theorem follow from [25, Theorem 5.2].$\square$

We finally consider the situation without the condition of irreducibility.

Proposition 2.5

Let T be a compact operator with positive spectral radius $r(T)>0$. Then for a given orthonormal basis E there exists a disjoint decomposition $E=E_1\cup \dots \cup E_n$ with the following properties:

• (a) The space $\ell ^2(E_1\cup \cdots \cup E_k)$ is T-stable for each k.

• (b) For each k let $P_k$ denote the orthogonal projection onto $\ell ^2(E_k)$ then the operator

  $$\begin{aligned} T_k=P_kT:\ell ^2(E_k)\rightarrow \ell ^2(E_k) \end{aligned}$$

  either has a spectral radius $r(T_j)<r(T)$ or is E-irreducible.

If T is E-positive, then $T_k$ is $E_k$-positive for each k. If T is trace class, then each $T_k$ is as well and for the Fredholm determinant one has

Proof

If T is E-irreducible, we are done. Otherwise there is a $\varnothing \ne F\subsetneq E$ such that $\ell ^2(F)$ is T-stable. We consider $T_F=T|_{\ell ^2(F)}$ and $T_{E{{{\smallsetminus }}} F}=PT:\ell ^2(E{{{\smallsetminus }}} F)\rightarrow \ell ^2(E{{{\smallsetminus }}} F)$, where P is the orthogonal projection to $\ell ^2(E{{{\smallsetminus }}} F)$. We have . If either of the operators $T_F$ or $T_{E{{{\smallsetminus }}} F}$ has spectral radius $<r(T)$ or is 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 $\lambda$ with $|\lambda |=r(T)$ and these are eigenvalues of finite multiplicity. By the fact that the Fredholm determinant distributes it follows that the spectral values distribute and by the finiteness, this process will terminate, yielding the proposition.$\square$

3 The Ihara zeta function

Definition 3.1

Let X denote an oriented graph. This means that X consists of the following data: a set N(X) of nodes and a set of oriented edges. We call x the source of the oriented edge $e=(x,y)$ and y the target and we write this as $x=s(e)$, $y=t(e)$. The nodes x, y are called the endpoints of the edge (x, y).

Definition 3.2

The valency, of a node x is the number of edges having x for one of their endpoints. Throughout, we will assume that X has bounded valency, i.e., that there exists $M>0$ such that

holds for all nodes x.

We introduce the notion of a weight. Usually, a weight $w(e)>0$ is put on an edge representing a length or a resistance or the reciprocal of that. In order to be more flexible, 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 3.3

A transition weight on X, henceforth simply called a weight, is a map

such that

$$\begin{aligned} w(e,f)\ne 0\;\;\Rightarrow \;\; t(e)=s(f) \end{aligned}$$

and

Example 3.4

A natural example is given in the case of X being a quotient $X=\Gamma \backslash Y$, where Y is a tree of bounded valency and $\Gamma$ is a tree lattice [5]. We let $\Gamma _f$ denote the stabiliser of f in the group $\Gamma$. In this case the weight

$$\begin{aligned} w(e,f)=\frac{1}{|\Gamma _f|} \end{aligned}$$

is a natural choice which fits the approach of Bass [4] to the Ihara zeta function in case of ramified quotients, see also [12].

Definition 3.5

An oriented path of length n in X is an n-tuple $p=(e_1,e_2,\dots ,e_n)$ of oriented edges such that $t(e_j)=s(e_{j+1})$ holds for all $j=1,\dots ,n-1$. We write the length as $\ell (p)=n$. The path p is called a closed path if $t(e_n)=s(e_1)$. For a closed path $p=(e_1,\dots ,e_n)$ we define its weight as

Definition 3.6

(Shifting the starting point) On the set CP(X) of all closed paths we install an equivalence relation $\sim$ generated by

$$\begin{aligned} (e_1,e_2,\dots ,e_n) \sim (e_2,e_3,\dots ,e_n,e_1). \end{aligned}$$

An equivalence class $c=[p]$ of closed paths is called a cycle. The length and weight functions factor through the quotient of this equivalence, so $\ell (c)$ and w(c) are well-defined for a cycle c.

Definition 3.7

For a cycle c and a natural number k we define $c^k$ to be the cycle one gets by iterating the cycle c for k times. One has

$$\begin{aligned} \ell (c^k)=k\ell (c)\quad \text {and}\quad w(c^k)=w(c)^k. \end{aligned}$$

A cycle c is called primitive if c is not a power $c_1^k$ of some shorter cycle $c_1$. For every cycle c there is a uniquely determined primitive $c_0$ and a uniquely determined number $\mu (c)\in {{\mathbb {N}}}$ such that

$$\begin{aligned} c=c_0^{\mu (c)}. \end{aligned}$$

The cycle $c_0$ is called the underlying primitive and $\mu (c)$ is called the multiplicity of the cycle c.

Definition 3.8

The Ihara zeta function of the weighted oriented graph (X, w) is defined by the product

where the product runs over all primitive cycles in X.

Definition 3.9

Let $H=\ell ^2(E)$ be the Hilbert space of all $\ell ^2$-functions on E(X) the elements of which we write as formal series $\sum _{e\in E(X)}c_e e$ with $c_e\in {{\mathbb {C}}}$ satisfying $\sum _{e\in E(X)}|c_e|^2<\infty$.

Definition 3.10

Inspired by [4], we consider the Bass operator $T:H\rightarrow H$ given by

If for two edges we have $t(e)=s(f)$, we write $e\rightarrow f$, so we have $T(e)=\sum _{e\rightarrow f}w(e,f)f$.

The following theorem is a straightforward generalization of [10, Theorem 1.6].

Theorem 3.11

The operator T is of trace class. The product Z(u) converges for |u| sufficiently small. The function Z(u) extends to a meromorphic function, more precisely, $Z(u)^{-1}$ is entire and satisfies

where $\det$ is the Fredholm-determinant.

Proof

The proof is essentially the same as the proof of [10, Theorem 1.6]. We repeat it here for the convenience of the reader. We show that the operator T is of trace class, and for every $n\in {{\mathbb {N}}}$ we have

where the sum runs over all cycles c of length n and $c_0$ is the underlying primitive cycle to c.

For this we consider the natural orthonormal basis of $\ell ^2(E)$ given by E. Using this orthonormal basis, one easily sees that $T^n$ has the claimed trace, once we know that T is of trace class. For this we estimate

$$\begin{aligned} \sum _{e\in E}\left\| Te\right\|&=\sum _e\left( \langle Te,Te\rangle \right) ^{\frac{1}{2}} =\sum _e\,\Biggl (\sum _{f\in E}\langle Te,f\rangle \langle f,Te\rangle \Biggr )^{\frac{1}{2}}\\&\leqslant \sum _e\sum _f|\langle Te,f\rangle |^2=\sum _{e,f}w(e,f)^2<\infty . \end{aligned}$$

This implies that T is of trace class. For small values of u we have

This a fortiori also proves the convergence of the product.$\square$

4 The prime geodesic theorem for a weighted graph

Theorem 4.1

Let (X, w) be a weighted oriented graph. For $m\in {{\mathbb {N}}}$ let

Then there are $r>0$ and natural numbers $n_1,\dots , n_s$ such that, as $m\rightarrow \infty$,

$$\begin{aligned} N_m=r^m\sum _{k=1}^s\,n_k\textbf{1}_{n_k{{\mathbb {N}}}}(m)+O((r-\varepsilon )^m) \end{aligned}$$

for some $\varepsilon >0$.

Proof

Let Z(u) be the Ihara zeta function of X. By the definition of Z we get

$$\begin{aligned} u\,\frac{Z'}{Z}(u)=\sum _{m=1}^\infty N_mu^m. \end{aligned}$$

The formula on the other hand yields

where $r=r(T)>0$ is the spectral radius of T and the last sum runs over all eigenvalues $\lambda$ of T with $|\lambda |<r$. The number $m(\lambda )$ is the algebraic multiplicity of $\lambda$. Finally, the numbers s is the number of components as in Proposition 2.5 with $r(T_j)=r(T)$ and $n_j$ is the number of eigenvalues $\lambda$ of that component, satisfying $|\lambda |=r(T)$. Since the sum $\sum _{j=0}^{n_k-1}e^{\frac{2\pi ijm}{n_k}}$ is zero unless m is a multiple of $n_k$, the theorem follows.$\square$

Let

Proposition 4.2

Assume that $r=r(T)>1$. Let $n_1,n_2,\dots ,n_s$ be the numbers of Theorem 4.1, let K be their least common multiple and let $C=\sum _{k=1}^s\frac{n_kr^{n_k}}{r^{n_k}-1}$. Then, as $n\rightarrow \infty$ one has

$$\begin{aligned} \vartheta (nK)\sim \psi (nK)\sim r^{nK}C \end{aligned}$$

and

$$\begin{aligned} \pi (nK)\sim \frac{r^{nK}}{nK}\,C. \end{aligned}$$

Proof

Let

We compute

and this implies $l_1\leqslant l_2\leqslant l_3$ as well as $L_1\leqslant L_2\leqslant L_3$. Next we let $0<\alpha <1$ and we get

So that

$$\begin{aligned} \frac{\vartheta (n)}{r^n}\geqslant \alpha \,\frac{n\pi (n)}{r^n}-\frac{\alpha n\pi (\alpha n)}{r^{\alpha n}}\, r^{(\alpha -1)n}. \end{aligned}$$

Since $r^{(\alpha -1)n}$ tends to zero, this implies and this finally implies $\alpha l_3\leqslant l_1$ and $\alpha L_3\leqslant L_1$. As $\alpha$ was arbitrary it follows $l_3\leqslant l_1$ and $L_3\leqslant L_1$, so $l_1=l_2=l_3$ and $L_1=L_2=L_3$. The claim follows if we finally show that $l_2=L_2=C$. We have

and so

$$\begin{aligned} \frac{\psi (nK)}{r^{nK}}=\sum _{k=1}^s\,n_kr^{n_k}\frac{1-r^{-nK}}{r^{n_k}-1}+O((r-\varepsilon )^{nK}) =\sum _{k=1}^s\,n_k\frac{r^{n_k}}{r^{n_k}-1}+o(1). \end{aligned}$$

$\square$

Proposition 4.3

• (a) For the spectral radius $r=r(T)$ one has

  

• (b) If there exist two different primitive cycles $c_0, d_0$ with a common oriented edge e and $w(c_0)=w(d_0)\geqslant 1$, then $r>1$.

Proof

(a) We have

Hence it follows that $\frac{1}{r}$ equals the supremum of all $u>0$ for which . Let . We show that $l=\frac{1}{r}$. 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

As $1\leqslant l(c_0)\leqslant l(c)$, the claim follows.

(b) The cycles $c_0d_0$ and $d_0c_0$ are primitive, distinct, and of the same length. Replacing $c_0$, $d_0$ with these, we assume that $l(c_0)=l(d_0)=l$. Next we fix representing closed paths of $c_0$ and $d_0$ which start with the edge e. For $n\in {{\mathbb {N}}}$ we have

where the sum runs over all closed paths starting with e and being of the form $c_0^{n_1}d_0^{m_1}\cdots c_0^{n_s}d_0^{m_s}$ for some $m_jn_j\in {{\mathbb {N}}}_0$. This implies that

where S is the Bass operator of the oriented graph with constant weight $w=1$.

One sees that $S^{3n}f=2^nf$. Therefore $\langle T^{nl}e,e\rangle \geqslant 2^n$ and so the operator norm satisfies $\Vert T^{nl}\Vert \geqslant 2^n$. The spectral radius therefore satisfies

$$\begin{aligned} r=\lim _n\Vert T^{nl}\Vert ^\frac{1}{nl}\geqslant 2^{\frac{1}{l}}>1. \end{aligned}$$

$\square$

5 Affine buildings

For background on this section, the reader may consult [13]. Let X 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 ${\mathfrak {a}},{\mathfrak {b}}$ containing chambers C, D there is a unique isomorphism ${\mathfrak {a}}\rightarrow {\mathfrak {b}}$ fixing C and D 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 X 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 [6]. 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 [3].

When speaking of “points” in X, we identify the complex X 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 [16, Section 14.4].

Definition 5.1

Generally, there are different families of apartments which make X a building, but there is a unique maximal family [1, Theorem 4.54]. In this paper, we will always choose the maximal family. Let be the automorphism group of the building X, that is, the set of all automorphisms $g:X\rightarrow X$ of the complex X which map apartments to apartments. In the geometric realisation these are cellular maps which are affine on each cell.

Definition 5.2

A choice of types is a labelling that attaches to each vertex v a label ${\text {lab}}(v)$, or type in $\{0,1,\dots ,d\}$ such that for each chamber C the set V(C) of vertices of C is mapped bijectively to $\{0,1,\dots ,d\}$.

Restricting the labelling gives a bijection between the set of all choices of types and the set of all bijections $V(C_0){\mathop {\longrightarrow }\limits ^{\cong }}\{0,1,\dots ,d\}$, where $C_0$ is any given chamber. Therefore the number of different choices of types is $(d+1)!$. 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 [13, Definition 1.2.3].

Definition 5.3

Pick a chamber C and an apartment ${\mathfrak {a}}$ containing C. Let $v_0,v_1,\dots ,v_d$ be the vertices of C with . Pick $v_0$ as origin to give the affine space ${\mathfrak {a}}$ the structure of a vector space. Let denote the open cone in ${\mathfrak {a}}$ spanned by the interior $\mathring{C}$ of the chamber C, i.e.

Further let

For two chambers C, D we finally write

$$\begin{aligned} C\leadsto D \end{aligned}$$

if and only if

Definition 5.4

Let $v_1,\dots ,v_d$ be the other vertices of C and let

$$\begin{aligned} \Lambda =\bigoplus _{j=1}^d{{\mathbb {Z}}}e_j\subset {\mathfrak {a}}, \end{aligned}$$

where $e_j=r_jv_j$ and $r_j>0$ for each $j=1,\dots ,d$ is the largest rational number such that $\Lambda _0\subset \Lambda ,$ where $\Lambda _0$ is the lattice of vertices of type zero.

Let

$$\begin{aligned} {{\mathbb {N}}}^d(\Lambda _0) \end{aligned}$$

denote the set of all $k=(k_1,\dots ,k_d)\in {{\mathbb {N}}}^d$ such that $\sum _{j=1}^d k_je_j$ lies in $\Lambda _0$. Analogously define

$$\begin{aligned} {{\mathbb {N}}}_0^d(\Lambda _0). \end{aligned}$$

For given $k\in {{\mathbb {N}}}_0^d$, the element $\sum _{j=1}^dk_je_j$ is contained in a unique chamber such that as in the picture.

Note that this construction depends on the apartment ${\mathfrak {a}}$ which has been fixed so far. For a given chamber D in the building, we say that D is in relative position k to C, if there exists an apartment ${\mathfrak {a}}$ such that $D=C(k)$ in that apartment. We write this as

$$\begin{aligned} C\leadsto _k D. \end{aligned}$$

We further write

$$\begin{aligned} C\leadsto D, \end{aligned}$$

if there exists some k with $C\leadsto _k D$.

6 Discrete groups

The automorphism group of the building carries a natural topology, the compact-open topology. It is a totally disconnected locally-compact group. A subgroup $\Gamma \subset G$ is discrete if and only if for each chamber C the stabilizer group

$$\begin{aligned} \Gamma _C=\{ \gamma \in \Gamma \,{:}\, \gamma C=C\} \end{aligned}$$

is finite. A discrete group $\Gamma$ is a lattice in G, i.e., there exists a finite G-invariant Radon measure on $G/\Gamma$ if and only if

where the sum runs over the set of all labelled chambers and $\Gamma _C$ is the stabiliser in $\Gamma$ of C.

We fix a discrete group . The subgroup $\Gamma ^\textrm{lab}$ of all $\gamma \in \Gamma$ which preserve a given labelling, is normal and of finite index in $\Gamma$. Replacing $\Gamma$ by $\Gamma ^\textrm{lab}$ we will henceforth assume that $\Gamma$ preserves labellings.

Definition 6.1

Fix a discrete, label preserving subgroup $\Gamma \subset G$. A $\Gamma$-weight is a map

such that

• $w(\gamma C, D)=w(C,\gamma D)=w(C,D)$ for all and all $\gamma \in \Gamma$,

• $\displaystyle w(C,D)\ne 0$ $\Rightarrow$  $C\leadsto \gamma D$ for some $\gamma \in \Gamma$,

• ,

• holds for all which satisfy $C\leadsto D\leadsto E$.

Definition 6.2

Let w be a $\Gamma$-weight. For $k\in {{\mathbb {N}}}_0^d(\Lambda _0)$ we define an operator

by

$$\begin{aligned} T_k(C)=\sum _{C'} w(C,C')C', \end{aligned}$$

where the sum runs over all chambers $C'$ in relative position k to C. Note that for given $k\in {{\mathbb {N}}}_0^d$ in each apartment ${\mathfrak {a}}$ containing C there is at most one $C'$ in position k, but the same $C'$ can lie in infinitely many apartments containing C. As we assume the building to be locally finite, the sum defining $T_k$ is actually finite.

Lemma 6.3

For $k,l\in {{\mathbb {N}}}_0^d(\Lambda _0)$ we have

$$\begin{aligned} T_kT_l=T_{k+l}. \end{aligned}$$

In particular, the operators $T_k$ and $T_l$ commute.

Proof

Fix a chamber D in the double sum $T_k(T_l(C))$. By definition, there exists a uniquely determined chamber $C_l$ in relative position l to C such that D occurs in the sum $T_k(C_l)$. We want to show that D is in relative position $k+l$ to C. For this, consider an apartment ${\mathfrak {a}}$, containing both. All we need to show is, that ${\mathfrak {a}}$ also contains $C_k$. Let b(C) denote the barycentric center of C and b(D) the one of D. Let S be the intersection of all apartments which contain C and D. Then S is a compact convex subset of ${\mathfrak {a}}$, the boundary of which consists of finitely many hypersurfaces along which the building ramifies. Then S contains

since any hypersurface H, which meets the interior of $S_0$, has a translate passing through the special point $v_0$ and this translate also passes through the interior of , but that cannot happen by the minimality of the cone. We claim that the chamber $C_l$ also intersects $S_0$. If not, there must be a hypersurface parting $C_l$ from $S_0$ and then, either a translate of it in ${\mathfrak {a}}$ would meet the interior of , or a translate in an apartment containing $C_l$ would pass through the interior of the corresponding cone. In either case we get a contradiction. Therefore, $C_l$ has a common point with $S_0$, hence $C_l$ lies in the apartment ${\mathfrak {a}}$.

On the other hand, for any chamber D in relative position $k+l$ to C there exist uniquely determined chambers $C_k$ and $C_l$ in relative positions k and l such that each apartment containing C and D also contains $C_k$ and $C_l$. This and the transitivity of w proves the claim.$\square$

Example 6.4

The integral cone ${{\mathbb {N}}}_0(\Lambda _0)$ has unique generators $k^{(1)},\dots ,k^{(d)}$. Let $T_{(j)}=T_{k^{(j)}}$. Fix some $u\in {{\mathbb {C}}}^d$ and set

$$\begin{aligned} w(C,D)=u_1^{n_1}\ldots u_d^{n_d} \end{aligned}$$

if $\gamma D=T_{(1)}^{n_1}\ldots T_{(d)}^{n_d} C$ holds for some $\gamma \in \Gamma$. This defines a $\Gamma$-weight, which is a direct generalisation of the graph case in Definition 3.3.

Definition 6.5

A quasicharacter on ${{\mathbb {N}}}^d(\Lambda _0)$ is a map $\chi :{{\mathbb {N}}}^d(\Lambda _0)\rightarrow {{\mathbb {C}}}^\times$ with . For a given quasicharacter $\chi$ let

be the generalized eigenspace, i.e, the set of all such that for every $k\in {{\mathbb {N}}}^d( \Lambda _0)$ one has

$$\begin{aligned} (T_k-\chi (k))^mv=0 \end{aligned}$$

for some $m\in {{\mathbb {N}}}$. For every non-zero $\chi$ the space is finite-dimensional. Let $m(\chi )\in {{\mathbb {N}}}_0$ denote its dimension. We then get

where the sum runs over the set of all quasi-characters $\chi$.

Proposition 6.6

For $k\in {{\mathbb {N}}}^d(\Lambda _0)$ let $a_k\in {{\mathbb {C}}}$ be bounded, i.e., there exists $M>0$ such that $|a_k|\leqslant M$ holds for all k. Then the operator

is well defined and maps to itself. On this Hilbert space, T is a trace class operator.

Proof

We compute

This implies that T is trace class.$\square$

Definition 6.7

Let denote the unital subring of generated by the translation operators $T_k$ with $k\in {{\mathbb {N}}}^d(\Lambda _0)$. This is a commutative integral domain. Let denote its quotient field.

For indeterminates $u_1,\dots , u_d$ we define the formal power series

where $u^k=u_1^{k_1}\ldots u_d^{k_d}$. Note that the summation only runs over the set ${{\mathbb {N}}}^d(\Lambda _0)$ of all $k\in {{\mathbb {N}}}^d$ such that $\sum _{j=1}^dk_je_j\in \Lambda _0$.

Theorem 6.8

T(u) is a rational function in u. More precisely, there exists a finite set $E\subset \Lambda _0^+$ and $k(e)\in {{\mathbb {N}}}_0^d$ for every $e\in E$ as well as $k(1),\dots ,k(d)\in {{\mathbb {N}}}^d_0{{{\smallsetminus }}}\{0\}$ such that

Proof

The case of trivial weight is [13, Theorem 3.1.4]. The proof given there extends without problems to the case of general weight.$\square$

Definition 6.9

We say that $u\in {{\mathbb {C}}}^d$ is singular if there exists $1\leqslant j\leqslant d$ such that $u^{-k(j)}$ is an eigenvalue of $T_{k(j)}$. Otherwise, u is regular. The singular set is a countable union of complex submanifolds of codimension 1 in ${{\mathbb {C}}}^d$, so the regular set is connected, open and dense.

Proposition 6.10

The family $u\mapsto T(u)$ is a meromorphic family of trace class operators on ${{\mathbb {C}}}^d$. It is holomorphic on the regular set. The map

$$\begin{aligned} Z(u)={\text {tr}}T(u) \end{aligned}$$

is meromorphic on ${{\mathbb {C}}}^d$ and holomorphic on the regular set. We have

Proof

This is clear from the theorem and the fact that all $T_k$ are trace class.$\square$

7 The prime geodesic theorem for a weighted building quotient

Let ${{\mathbb {D}}}^\times =\{z\in {{\mathbb {C}}}\,{:}\, 0<|z|<1\}$.

Theorem 7.1

(Prime geodesic theorem) For $k\in {{\mathbb {N}}}^d(\Lambda _0)$ let

There are $z_1,\dots ,z_N\in {{\mathbb {C}}}^d$ such that

$$\begin{aligned} N(k) \sim \sum _{j=1}^N z_j^k, \end{aligned}$$

as $k_j\rightarrow \infty$ independently. Moreover, there exists a sequence $z_j\in ({{\mathbb {C}}}^\times )^d$ with $\lim _jz_j=0$ such that for every $k\in {{\mathbb {N}}}^d(\Lambda _0)$ we have, with absolute convergence of the sum,

$$\begin{aligned} N(k)=\sum _{j=1}^\infty z_j^k. \end{aligned}$$

Proof

We show the last assertion first. By definition we get and so

As is a quasi-character, there exists $z_\chi \in \left( {{\mathbb {C}}}^\times \right) ^d$ with

$$\begin{aligned} \chi (k)=z_\chi ^k. \end{aligned}$$

Now let $(z_j)$ be the sequence that runs through all $z_\chi$ with $m(\chi )\ne 0$, where each $z_\chi$ is repeated with multiplicity $m(\chi )$. The claim follows. The first claim follows from the fact that $z_j\rightarrow 0$ in ${{\mathbb {C}}}^d$.$\square$

Writing $z_j=(z_{j,1},\dots ,z_{j,d})\in ({{\mathbb {C}}}^\times )^d$ we write the statement of the theorem as

$$\begin{aligned} N(k_1,k_2,\dots ,k_d)=\sum _{j=1}^\infty z_{j,1}^{k_1}\cdots z_{j,d}^{k_d}. \end{aligned}$$