Benjamini–Schramm convergence and zeta functions

Abstract

The equivalence of Benjamini–Schramm convergence and zeta-convergence, known for graphs, is proven for sequences of compact Riemann surfaces. A program is initialized, to extend this connection to arbitrary locally homogeneous spaces.

Introduction

Benjamini–Schramm convergence or BS-convergence of metric probability spaces \(X_n\) to a pointed metric space (Xp) means that for every radius \(R>0\) the probability of a point x having the ball \(B_R(x)\) of radius R isometric with \(B_R(p)\) tends to one, i.e.,

$$\begin{aligned} P_n\left( \big \{x\in X_n: B_R(x)\cong B_R(p)\big \}\right) \longrightarrow 1,\quad n\rightarrow \infty . \end{aligned}$$

For hyperbolic surfaces, Selberg [22] introduced a geometric zeta function which counts closed geodesics. Ihara [14] established a p-adic analog of this, which later was generalized to arbitrary graphs by Hashimoto and Hori [12]. Lenz, Pogorzelski and Schmidt recently proved [17] that a sequence of graphs of bounded valency is BS-convergent to an infinite tree if and only if the corresponding Ihara zeta functions converge to the trivial one.

The present paper may be considered the starting point for a program aiming at generalizing the equivalence

$$\begin{aligned} \hbox {BS-convergence}\ \Leftrightarrow \ \hbox {zeta-convergence} \end{aligned}$$

to arbitrary locally homogeneous spaces, i.e., double quotients \(\Gamma \backslash G/K\), where G is a locally compact group, \(\Gamma , K\) subgroups, where K is compact and \(\Gamma \) discrete. In order to define BS-convergence, one needs a metric, so one assumes that the topology of G/K is induced by a proper metric. It is also necessary for this metric to be left G-invariant in order to induce a derived metric on \(\Gamma \backslash G/K\). The first steps of this program are the introduction of the notions of BS- and zeta-convergence in this context. This is done in Sect. 1, extending a previous notion by the author [9]. Suitable zeta functions are present in the literature, at least for Lie groups [4, 11, 22] or p-adic groups [7, 14]. A general definition for arbitrary locally compact groups is still lacking.

The next step is to show the equivalence of BS- to zeta-convergence for compact hyperbolic surfaces. This is done in Sect. 3. In this case, the bounded valency condition of [17] is replaced by a lower bound on the injectivity radius. Note that BS-convergence means that for every bound R, the number of closed geodesics of length \(\le R\) becomes small. It could still be that closed geodesics cluster at large lengths, but the cluster moves toward \(\infty \). The convergence of the zeta functions, however, takes into account all geodesics at once, with means that such clustering does not happen.

In this paper, the trace formula is used to transfer the question of zeta convergence to a spectral theoretic context. Then a uniform growth estimate on Laplace eigenvalues [13] is used to derive the convergence of the zeta functions. In the first section we introduce notations and collect material from our previous paper [9]. In the second section we extend a statement of [9] in the relative case. In the third section we state and prove the main theorem, and in the fourth section we collect some further questions and projects which might come out of this paper.

Relative Benjamini–Schramm convergence

In this section we extend Proposition 2.4 of [9] to arbitrary locally compact groups and to the relative setting. Thus we get a very useful criterion for BS-convergence, which in [9] has been used to give a very simple and elegant proof of one of the main results of [1]. In the previous section, only convergence to a homogeneous space has been considered, in which case the base point is irrelevant. In [9] the author extended the notion of BS-convergence to a relative situation which was formulated in purely group-theoretical terms. In geometric terms it translates to the following:

Definition 1.1

(BS-convergence without base point) Let \((X_n,d_n,P_n)\) be a sequence of metric probability spaces, which share a common covering \(X_\infty \), i.e., for each \(n\in {{\mathbb {N}}}\) there is a metric covering map \(\pi _n:X_\infty \rightarrow X_n\), which means that \(\pi _n\) is surjective and for each \(x\in X_n\) there exists \(\varepsilon >0\) such that for every \(y\in \pi _n^{-1}(x)\) the map \(\pi _n\) maps the ball \(B_\varepsilon (y)\) isometrically to \(B_\varepsilon (x)\). Then we say that the sequence \((X_n)\) is BS-convergent to \(X_\infty \), if for every \(r>0\),

$$\begin{aligned} P_n\left( \big \{ x\in X_n: \forall _{y\in \pi _n^{-1}(x)}\ \pi _n\big |_{B_r(y)}\,\text { is an isometry } B_r(y)\rightarrow B_r(x)\big \}\right) \ \longrightarrow \ 1 \end{aligned}$$

as \(n\rightarrow \infty \).

Definition 1.2

In the paper [9] the following situation was considered: \((\Gamma _n)\) is a sequence of lattices in a locally compact group G and \(\Gamma _\infty \) is a common normal subgroup. We then say that the sequence \((\Gamma _n)\)Benjamini–Schramm converges to \(\Gamma _\infty \), or BS-converges, written \(\Gamma _n{\mathop {\longrightarrow }\limits ^{BS}}\Gamma _\infty \), if for every compact set \(C\subset G\) the sequence

$$\begin{aligned} P_n\left( \big \{x\in \Gamma _n\backslash G:x^{-1}\left( \Gamma _n\smallsetminus \Gamma _\infty \right) x\cap C=\emptyset \big \}\right) \ \longrightarrow \ 1 \end{aligned}$$

as \(n\rightarrow \infty \). Here \(P_n\) is the normalized Haar measure on \(\Gamma _n\backslash G\).

If \((\Gamma _n)\) is BS-convergent to the trivial group \(\{1\}\), then we say that \((\Gamma _n)\) is a BS-sequence.

The next proposition considerably extends Proposition 2.4 of [9].

Proposition 1.3

Let G be a locally compact group and K a compact subgroup. Assume that the topology on \(X=G/K\) is generated by a G-invariant proper metric. Let \((\Gamma _n)\) be a sequence of lattices in G and let \(\Gamma _\infty \) be a common normal subgroup of the \(\Gamma _n\).

Consider the following statements:

  1. (a)

    The sequence \(\Gamma _n\) is BS-convergent to \(\Gamma _\infty \).

  2. (b)

    The sequence of metric probability spaces \(X_n=\Gamma _n\backslash X\) is BS-convergent to \(X_\infty =\Gamma _\infty \backslash X\).

Then (a) \(\Rightarrow \) (b) unconditionally.

If for every open ball \(\emptyset \ne B\subset X\) and every isometry \(\phi :B\rightarrow B\) there exists a uniquely determined \(g\in G\) such that \(\phi (x)=gx\), \(x\in B\) and every \(\Gamma _n\) is torsion-free, then (b) \(\Rightarrow \) (a) holds as well.

Note that (b) \(\Rightarrow \) (a) holds for X being a symmetric space without compact factors.

Proof

Throughout, we will denote the probability measures on \(\Gamma _n\backslash G\) and on \(\Gamma _n\backslash X\) by the same symbol \(P_n\).

(a) \(\Rightarrow \) (b): Assume that \(\Gamma _n\) is BS-convergent to \(\Gamma _\infty \) and let \(r>0\). As we encounter different metric spaces, we shall write \(B_r(z,Z)\) for the open r-ball in Z around z. The space \(X=G/K\) has a natural base-point \(x_0=eK\). We abbreviate \(B_r=B_r(eK,X)\). Let \(g\in G\) and consider the point \(x=\Gamma _n gK\in X_n\). For a discrete subgroup \(\Gamma \subset G\) and any set \(A\subset X\) we write \(\Gamma \backslash A\) for the image of A in \(\Gamma \backslash X\) or, what amounts to the same, \(A/\sim \), where \(a\sim a'\) if and only if there exists \(\gamma \in \Gamma \) with \(\gamma a=a'\). We use the invariance of the metric to identify

$$\begin{aligned} B_r(\Gamma g K,\Gamma \backslash X)\cong \Gamma \backslash B_r(gK,X) =\Gamma \backslash g B_r\cong \left( g^{-1}\Gamma g\right) \backslash B_r. \end{aligned}$$

This means that \(B_r(\Gamma _n gK,X_n)\) is isometric to some \(B_r(\Gamma _\infty hK,X_\infty )\) if and only if

$$\begin{aligned} \left( g^{-1}\Gamma _ng\right) \backslash B_r\cong \left( h^{-1}\Gamma _\infty h\right) \backslash B_r. \end{aligned}$$

Let \(U_r\) be the pre-image of \(B_r\) under the projection map \(G\rightarrow G/K=X\). Let \(C=\overline{U_r} \overline{U_r}^{-1}\). Then C is a compact subset of G. Let

$$\begin{aligned} A_n(C)=\big \{g\in \Gamma _n\backslash G:g^{-1}\left( \Gamma _n\smallsetminus \Gamma _\infty \right) g\cap C=\emptyset \big \}. \end{aligned}$$

Then \(P_n(A_n(C))\) tends to 1 as \(n\rightarrow \infty \). Let \(C_r\) denote the compact set \(\overline{U_r}\overline{U_r^{-1}}\). Then if \(g\in A_n(C_r)\), for every \(\gamma _n\in \Gamma _n\) one has

$$\begin{aligned} g^{-1}\gamma _n g\in U_r U_r^{-1}\,\Rightarrow \, \gamma _n\in \Gamma _\infty . \end{aligned}$$

Now if \(g^{-1}\Gamma _ng uK=g^{-1}\Gamma _ng vK\) for two \(u,v\in U_r\), then there exists \(\gamma _n\in \Gamma _n\) and \(k\in K\) such that \(g^{-1}\gamma _ng=vku^{-1}\) and hence \(g^{-1}\gamma _ng\in U_rU_r^{-1}\) and so u and v already give the same element in \(g^{-1}\Gamma _\infty g\backslash B_r\). In other words, it follows that

$$\begin{aligned} (g^{-1}\Gamma _n g)\backslash B_r = (g^{-1}\Gamma _\infty g)\backslash B_r. \end{aligned}$$

Let \(T_n(r)\) denote the set of all \(x\in X_n\) such that there exists \(y\in X_\infty \) with \(B_r(x,X_n)\cong B_r(y,X_\infty )\). Then the above entails

$$\begin{aligned} A_n(C_r)K\subset T_n(r). \end{aligned}$$

Hence we get \(P_n(T_n(r))\rightarrow 1\) and so (a) \(\Rightarrow \) (b) is proven.

(b) \(\Rightarrow \) (a): First note that under the given conditions, each \(\Gamma _n\) acts fixed-point-freely on X and there exists a radius \(t_n>0\) such that the projection map \(p_n:X\rightarrow \Gamma _n\backslash X\) induces an isometric isomorphism \(B_{t_n}(x)\rightarrow p_n\big ( B_{t_n}(x)\big )\) for every \(x\in X\).

Let \(C\subset G\) be a compact set. Then there exists some \(r>0\) such that \(C\subset U_r U_r^{-1}\). Let \(x_n\in T_n(r)\) and let \(\phi :B_r(x_n,X_n)\rightarrow B_r(x_\infty ,X_\infty )\) be an isometry. Write \(x_n=\Gamma _n g_nK\) and \(x_\infty =\Gamma _\infty g_\infty K\), then \(\phi \) can be viewed as a map \(g_n^{-1}\Gamma _ng_n\backslash B_r\rightarrow g_\infty ^{-1}\Gamma _\infty g_\infty \backslash B_r\). If \(\phi \) maps the origin \(g_n^{-1}\Gamma _n g_n\) to some \((g_\infty ^{-1}\Gamma _\infty g_\infty )g_0\), then one can replace \(g_\infty \) by \(g_\infty g_0\) and \(\phi \) will preserve origins. Next let \(0<t\le \min (r,t_n)\). Then \(\phi \) induces an isometry of the ball \(B_t\subset X\). Then there exists \(\alpha \in G\) such that, on \(B_t\), the map \(\phi \) is given by \(z\mapsto \alpha z\). Replacing \(g_\infty \) with \(g_\infty \alpha \) we arrive at

$$\begin{aligned} \phi \left( \left( g_n^{-1}\Gamma _ng_n\right) x\right) =\left( g_\infty ^{-1}\Gamma _\infty g_\infty \right) x,\quad x\in B_t, \end{aligned}$$

or, what amounts to the same, the diagram

commutes. Let \(0<t\le r\) be maximal with this property. We claim that \(t=r\). If not, then there exists \(z\in B_r\) with \(d(z,eK)=t<r\). As \(\phi \) is an isometry, the diagram above still commutes with \(B_t\) replaced by the closed ball

$$\begin{aligned} \overline{B}_t=\big \{ x\in X: d(x,eK)\le t\big \}. \end{aligned}$$

Let \(s=\min (t_n,r-t)\). We therefore have \(\phi \left( (g_n^{-1}\Gamma _ng_n)z\right) =(g_n^{-1}\Gamma _ng_n)z\) and as \(\phi \) is an isometry, \(\phi \) maps \(B_s(z)\) to \(B_s(z)\). Again, this isometry is induced by an element \(g\in G\) and we have \(gy=y\) for every y in the non-empty open set \(B_s(z)\cap B_t(x)\). The uniqueness condition implies that \(g=e\) the neutral element in G. This means that the diagram commutes with \(B_t\) replaced by a larger open set which contains z. As this is the case for every z, by compactness we conclude that t was not maximal. Hence the diagram

commutes. The lower triangle implies that the natural map \(\psi \) is an isometry as well, so we end up with the commutative diagram

Now let \(\gamma _n\in \Gamma _n\) and suppose \(g_n^{-1}\gamma _ng_n\in C_r\). By definition, there exist \(u,v\in U_r\) with \(g_n^{-1}\gamma _n g_n=uv^{-1}\), or \((g_n^{-1}\gamma _n g_n)v=u\), which means that the points vK and uK are mapped to the same point in \((g_n^{-1}\Gamma _n g_n)\backslash B_r\), so they map to the same point in \((g_n^{-1}\Gamma _\infty g_n)\backslash B_r\). This means that there exist \(\gamma _\infty \in \Gamma _\infty \) and \(k\in K\) such that \((g_n^{-1}\gamma _\infty g_n)v=uk\). Hence

$$\begin{aligned} \gamma _\infty ^{-1}\gamma _n=g_nvkv^{-1}g_n^{-1} \end{aligned}$$

is contained in a compact group. As \(\Gamma _n\) is torsion-free, it follows \(\gamma _n=\gamma _\infty \) and therefore \(\Gamma _n g_n\in A_n(C_r)\). We have shown

$$\begin{aligned} T_n(r)\subset A_n(C_r)K. \end{aligned}$$

which implies \(P_n(A_n(C_r))\rightarrow 1\). \(\square \)

Definition 1.4

A sequence \((\Gamma _n)\) of lattices is called uniformly discrete, if there exists a unit-neighborhood \(U\subset G\) such that \(x^{-1}\Gamma _nx\cap U=\{1\}\) holds for every \(x\in G\).

Remark 1.5

In [9] it is shown that

$$\begin{aligned}&(\Gamma _n)\text { is BS and uniformly discrete}\\&\Rightarrow (\Gamma _n)\text { Plancherel}\\&\Rightarrow (\Gamma _n)\text { is BS}. \end{aligned}$$

Plancherel and Benjamini–Schramm sequences

Let \(G={\text {SL}}_2({{\mathbb {R}}})\). Then \(K={\text {SO}}(2)\) is a maximal compact subgroup of G. The group G acts on the upper half plane \({{\mathbb {H}}}=\{ z\in {{\mathbb {C}}}:{\text {Im}}(z)>0\}\) via linear fractionals and this action induces an identification of \(G/\{\pm 1\}\) with the group of orientation-preserving isometries of the two-dimensional hyperbolic space. We normalize the Haar measure on K to have volume 1. Next we normalize the Haar measure on G such that it induces the usual \(y^{-2}\,dx\,dy\) on the upper half plane \({{\mathbb {H}}}\cong G/K\).

For a cocompact lattice \(\Gamma \subset G\) the unitary representation of G, given by right translation on \(L^2(\Gamma \backslash G)\) decomposes as a direct sum of irreducibles

$$\begin{aligned} L^2(\Gamma \backslash G)\cong \bigoplus _{\pi \in \widehat{G}}N_\Gamma (\pi )\pi . \end{aligned}$$

The multiplicities \(N_\Gamma (\pi )\) are finite and are zero outside a countable subset of the unitary dual \(\widehat{G}\).

Definition 2.1

The measure on \(\widehat{G}\) given by

$$\begin{aligned} \mu _\Gamma =\sum _{\pi \in \widehat{G}}N_\Gamma (\pi )\,\delta _\pi \end{aligned}$$

is called the spectral measure attached to \(\Gamma \).

Definition 2.2

For \(f\in C_c^\infty (G)\) let \(\pi (f)\) denote the operator defined by integrating the representation \(\pi \) against f. More precisely, for \(v\in V_\pi \) we have

$$\begin{aligned} \pi (f)v=\int _G f(x)\pi (x)v\,\hbox {d}x, \end{aligned}$$

where the integral is a Bochner integral. Another way to say this is that \(\pi (f)\) is the uniquely defined operator such that for any two \(v,w\in V_\pi \) one has

$$\begin{aligned} \left\langle \pi (f)v,w\right\rangle =\int _G f(x)\left\langle \pi (x)v,w\right\rangle \,\hbox {d}x. \end{aligned}$$

For a reductive Lie group like \(G={\text {SL}}_2({{\mathbb {R}}})\), the operator \(\pi (f)\) is known to be a trace class operator for every \(\pi \in \widehat{G}\).

A sequence of cocompact lattices \((\Gamma _n)\) in G is called a Plancherel sequence, if for every \(f\in C_c^\infty (G)\) we have

$$\begin{aligned} \frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\int _{\widehat{G}}{\hat{f}}(\pi )\,d\mu _{\Gamma _n}(\pi )\ \longrightarrow f(e) \end{aligned}$$

as \(n\rightarrow \infty \), where \({\hat{f}}(\pi )={\text {tr}}\pi (f)\).

The unitary dual \(\widehat{G}\) comes equipped with the topology of locally uniform convergence of matrix coefficients, or Fell topology, see [10].

The Plancherel theorem states that there is a unique Borel measure \(\mu _\mathrm {Pl}\) on the unitary dual \(\widehat{G}\) such that for every \(f\in C_c^\infty (G)\) we have \(f(e)=\int _{\widehat{G}}{\hat{f}}(\pi )\,d\mu _\mathrm {Pl}(\pi )\). (For an explicit computation of the Plancherel measure for the group \(G={\text {SL}}_2({{\mathbb {R}}})\), see Section 11.3 of [6].) This means that the sequence \((\Gamma _n)\) is Plancherel if and only if in the dual space of \(C_c^\infty (G)\) one has weak-*-convergence

$$\begin{aligned} \frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\mu _{\Gamma _n}\ \longrightarrow \ \mu _\mathrm {Pl}. \end{aligned}$$

Remark 2.3

  1. (a)

    If a sequence \((\Gamma _n)\) of lattices is a Plancherel sequence, then

    $$\begin{aligned} \frac{1}{{\text {vol}}(G/\Gamma _n)}\mu _{\Gamma _n}(U)\ \longrightarrow \ \mu _\mathrm {Pl}(U) \end{aligned}$$

    for every relatively compact open set \(U\subset \widehat{G}\), whose boundary has Plancherel measure zero. This follows from the density principle of Sauvageot [21].

  2. (b)

    If \((\Gamma _n)\) is a Plancherel sequence, then

    $$\begin{aligned} {\text {vol}}(\Gamma _n\backslash G)\longrightarrow \infty \end{aligned}$$

    as \(n\rightarrow \infty \). This follows from the fact that the spectral measure of each \(\Gamma _n\) is discrete and the Plancherel measure is not.

The Selberg zeta function

Let \((\Gamma _n)\) be a sequence of cocompact lattices in G. For simplicity, we shall assume that each \(\Gamma _n\) is torsion-free, which can easily be arranged as every lattice \(\Gamma \) contains a torsion-free sublattice.

Definition 3.1

For \(\gamma \in \Gamma _n\) the length of \(\gamma \) is defined by

$$\begin{aligned} l(\gamma )=\inf _{x\in X}d(\gamma x,x). \end{aligned}$$

It is known that for torsion-free cocompact lattices \(\Gamma \), every \(\gamma \in \Gamma \) satisfies \(l(\gamma )>0\). The Selberg zeta function for \(\Gamma _n\) is defined for \(s\in {{\mathbb {C}}}\) with \({\text {Re}}(s)>1\) as

$$\begin{aligned} Z_n(s)= \prod _{\gamma }\prod _{k\ge 0} \left( 1-e^{-(s+k)l(\gamma )}\right) , \end{aligned}$$

where the first product runs over all primitive conjugacy classes in \(\Gamma _n\) (see [6], Section 11.6). The product converges for \({\text {Re}}(s)>1\), and the so defined function extends holomorphically to all of \({{\mathbb {C}}}\).

Theorem 3.2

Let \((\Gamma _n)\) be a sequence of torsion-free cocompact lattices in G.

  1. (a)

    If the sequence \((\Gamma _n)\) is uniformly discrete and Plancherel, then

    $$\begin{aligned} \frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\frac{Z_n'}{Z_n}(s) \end{aligned}$$

    converges to zero in the set \(\{{\text {Re}}(s)>1\}\), as n tends to infinity.

  2. (b)

    If

    $$\begin{aligned} \frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\frac{Z_n'}{Z_n}(s) \end{aligned}$$

    converges to zero in the set \(\{{\text {Re}}(s)>1\}\) as \(n\rightarrow \infty \), then the sequence is Plancherel.

In either case, the convergence of \(\frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\frac{Z_n'}{Z_n}(s)\) is uniform on every set of the form \(\{{\text {Re}}(s)\ge \alpha \}\), for \(\alpha >1\).

Proof

For \({\text {Re}}(s)>1\) we have

$$\begin{aligned} \frac{Z_n'}{Z_n}(s)=\sum _{k\ge 0}\sum _{[\gamma ]}\ell (\gamma _0)e^{-(s+k)\ell (\gamma )}=\sum _{[\gamma ]} \ell (\gamma _0)\frac{e^{-s\ell (\gamma )}}{1-e^{-\ell (\gamma )}}, \end{aligned}$$

where the sum runs over all conjugacy classes \([\gamma ]\ne \{1\}\) in \(\Gamma _n\) and \(\gamma _0\) is the underlying primitive of \(\gamma \), i.e., \(\gamma =\gamma _0^m\) for some \(m\in {{\mathbb {N}}}\). Now let \({\text {Re}}(s)\ge \alpha >1\), then \(\left| e^{-(s+k)\ell (\gamma )}\right| =e^{-({\text {Re}}(s)+k)\ell (\gamma )}\le e^{-(\alpha +k)\ell (\gamma )}\) and so the addendum follows.

Let \(K={\text {SO}}(2)\) be the standard maximal compact subgroup of G and let \(\widehat{G}_K\) denote the set of all \(\pi \in \widehat{G}\) such that the representation space \(V_\pi \) contains nonzero K-fixed vectors. Then it is known [15],

$$\begin{aligned} \widehat{G}_K=\left\{ \pi _{ir}: r\in i\left( 0,\frac{1}{2}\right) \cup {{\mathbb {R}}}_{\ge 0}\right\} \cup \{\mathrm {triv}\}. \end{aligned}$$

The map \(\phi :\widehat{G}_K\rightarrow [0,\infty )\), given by

$$\begin{aligned} \phi (\mathrm {triv})&=0,\\ \phi (\pi _{ir})&=\frac{1}{4}+r^2 \end{aligned}$$

is a homeomorphism. We shall from now on identify \(\widehat{G}_K\) with \([0,\infty )\). For brevity, we write \(\mu (f)\) instead of \(\int _Xf\,\hbox {d}\mu \) where \(\mu \) is a measure on X and f a function. We shall also consider \(\mu _n\) and \(\mu _\mathrm {Pl}\) as measures on \(\widehat{G}_K\subset \widehat{G}\).

We now proceed to the proof of part (a) of the theorem. Suppose that the sequence \((\Gamma _n)\) is Plancherel. By [6], Section 11, there are numbers \(r_{n,j}\in {{\mathbb {R}}}\cup i\left( 0,\frac{1}{2}\right) \) such that

$$\begin{aligned} L^2(\Gamma _n\backslash {{\mathbb {H}}})=L^2(\Gamma _n\backslash G)^K={{\mathbb {C}}}\oplus \bigoplus _{j=1}^\infty \pi _{ir_{n,j}}, \end{aligned}$$

where \({{\mathbb {C}}}\) stands for the one-dimensional space of constant functions and \(\pi _{ir}\) is the induced representation (principal or complementary series) with \(r\in {{\mathbb {R}}}\cup i\left( 0,\frac{1}{2}\right) \). Formally we set \(r_{n,0}=\frac{i}{2}\). Fix \(s,b\in {{\mathbb {C}}}\) with \({\text {Re}}(s),{\text {Re}}(b)>1\). By Section 11.6 of [6] we can plug the function

$$\begin{aligned} h_{s,b}(r)=\frac{1}{s^2+r^2}-\frac{1}{b^2+r^2} \end{aligned}$$

into the trace formula and, as in Lemma 11.6.2 of [6], we get

$$\begin{aligned} \frac{1}{s}\frac{Z_n'}{Z_n}\left( s+\frac{1}{2}\right) -\frac{1}{b}\frac{Z_n'}{Z_n} \left( b+\frac{1}{2}\right) =\mu _n(H_{s,b})-\mu _\mathrm {Pl}(H_{s,b}), \end{aligned}$$

where \(H_{s,b}(\lambda )=\frac{1}{s^2+\lambda -\frac{1}{4}}-\frac{1}{b^2+\lambda -\frac{1}{4}}\) and \(\mu _n=\mu _{\Gamma _n}\). This means

$$\begin{aligned} \frac{1}{s}\frac{Z_n'}{Z_n}\left( s+\frac{1}{2}\right)&=\frac{1}{b}\frac{Z_n'}{Z_n} \left( b+\frac{1}{2}\right) -\frac{{\text {vol}}(\Gamma _n\backslash G)}{\pi }\sum _{k=0}^\infty \left( \frac{1}{s+\frac{1}{2}+k} -\frac{1}{b+\frac{1}{2}+k}\right) \\&\quad + \,2\sum _{j=0}^\infty \frac{1}{s^2+r_{n,j}^2}- \frac{1}{b^2+r_{n,j}^2}. \end{aligned}$$

Let \(D_s\) denote the operator \(D_s(\psi )(s)=-\frac{\partial }{\partial s}\left( \frac{1}{s}\psi (s)\right) \). We get

$$\begin{aligned} D_s\left[ \frac{Z_n'}{Z_n}\left( s+\frac{1}{2}\right) \right]&= 4s\sum _{j=0}^\infty \frac{1}{(s^2+r_{n,j}^2)^2} -\frac{{\text {vol}}(\Gamma _n\backslash G)}{\pi }\sum _{k=0}^\infty \frac{1}{(s+\frac{1}{2}+k)^2} \end{aligned}$$

and

$$\begin{aligned} D_s^2\left[ \frac{Z_n'}{Z_n}\left( s+\frac{1}{2}\right) \right] =&-\frac{{\text {vol}}(\Gamma _n\backslash G)}{\pi }\left( \frac{1}{s^2}\sum _{k=0}^\infty \frac{1}{(s+\frac{1}{2}+k)^2}+\frac{2}{s}\sum _{k=0}^\infty \frac{1}{(s+\frac{1}{2}+k)^3} \right) \\&\quad +\,16s\sum _{j=0}^\infty \frac{1}{(s^2+r_{n,j}^2)^3}. \end{aligned}$$

Let

$$\begin{aligned} h_s(\lambda )=\frac{1}{\left( s^2+\lambda -\frac{1}{4}\right) ^3}. \end{aligned}$$

Note that the function \(r\mapsto h_s\left( r^2+\frac{1}{4}\right) \) equals \(\frac{1}{4s}D_s\frac{\partial }{\partial s} h_{s,b}(r)\). Therefore the above formula says

$$\begin{aligned} D_s^2\left[ \frac{Z_n'}{Z_n}\left( s+\frac{1}{2}\right) \right] =\mu _n(h_s)-{\text {vol}}(\Gamma _n\backslash G)\mu _\mathrm {Pl}(h_\lambda ). \end{aligned}$$

For \(s>\frac{1}{2}\) the continuous function \(h_s\) is positive on \([0,\infty )\).

By the trace formula, [6] Section 11.4, we have that \(\sum _{j=0}^\infty |h_s(\lambda _{n,j})|<\infty \) for every \(s>\frac{1}{2}\). Further, in [6] Section 11.3 the Plancherel measure is explicitly computed. As \(h_s\) is decreasing to the power 3, it follows that integral \(\mu _\mathrm {Pl}(h_s)\) is finite.

If \(I\subset [0,\infty )\) is a relatively open, bounded interval, then by [21] we have that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\mu _n(\mathbf{1} _I) =\mu _{\mathrm {Pl}}(\mathbf{1} _I), \end{aligned}$$

where \(\mathbf{1} _I\) is the indicator function of I. By linearity this extends to linear combinations of functions of the form \(\mathbf{1} _I\). There exists a sequence \((L_k)_{k\in {{\mathbb {N}}}}\) of such linear combinations such that \(0\le L_k\nearrow h_s\) outside a countable set S, which is of Plancherel measure zero and can also be chosen to be of \(\mu _{\Gamma _n}\) measure zero for all n and have empty intersection with \({{\mathbb {N}}}\). We can also choose the \(L_k\) so that for each \(T\in {{\mathbb {N}}}_0\) we have that

$$\begin{aligned} \Phi _k(T)=\sup _{x\in [T,T+1)\smallsetminus S}h_s(x)-L_k(x) \end{aligned}$$

tends to zero for \(k\rightarrow \infty \).

For \(T\ge 1\) let \(N_n(T)=\#\big \{ j:\big |\frac{1}{4}+r_{n,j}^2\big |<T\big \}\). Recall that \(\lambda _{n,j}=\frac{1}{4}+r_{n,j}^2\) is the j-th Laplace eigenvalue. The sequence \((\Gamma _n)\) is uniformly discrete, which means that the injectivity radii of the manifolds \(\Gamma _n\backslash {{\mathbb {H}}}\) are bounded below. Therefore, by formula (1.5) of [13], there exists a constant \(C>0\) such that for every \(T\ge 1\) one has

$$\begin{aligned} N_n(T)\le C\,{\text {vol}}(\Gamma _n\backslash G)\, T. \end{aligned}$$

Now, since \(N_n(0)=0\),

$$\begin{aligned} 0\le \mu _n(h_s-L_k)&=\sum _jh_s(\lambda _{n,j})-L_k(\lambda _{n,j})\\&\le \sum _{T=0}^\infty \Phi _k(T)\big (N_n(T+1)-N_n(T)\big )\\&\le C{\text {vol}}(\Gamma _n\backslash G)\sum _{T=0}^\infty \Phi _k(T)(2T+1). \end{aligned}$$

Since \(\Phi _k(T)\le 2h_s(T)\) and \(\sum _{T=0}^\infty h_s(T)(2T+1)<\infty \), we can apply dominated convergence, to get that this sum tends to zero for \(k\rightarrow \infty \). So let \(\varepsilon >0\). Then there is \(k_0\in {{\mathbb {N}}}\) such that for all \(k\ge k_0\) we have that \(0\le \mu _n(h_s)-\mu _n(L_k)<{\text {vol}}(\Gamma _n\backslash G)\varepsilon /3\) holds for all \(n\in {{\mathbb {N}}}\) and that \(|\mu _\mathrm {Pl}(L_k)-\mu _\mathrm {Pl}(h_s)|<\varepsilon /3\).

Fix some \(k\ge k_0\). Then there exists \(n_0\in {{\mathbb {N}}}\) such that for all \(n\ge n_0\) one has

$$\begin{aligned} \left| \frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\mu _n(L_k)-\mu _\mathrm {Pl}(L_k)\right| <\varepsilon /3. \end{aligned}$$

And so

$$\begin{aligned} \left| \frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\mu _n(h_s)-\mu _\mathrm {Pl}(h_s)\right|&\le \frac{1}{{\text {vol}}(\Gamma _n\backslash G)}|\mu _n(h_s)-\mu _n(L_k)|\\&\quad +\left| \frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\mu _n(L_k)-\mu _\mathrm {Pl}(L_k)\right| \\&\quad +|\mu _\mathrm {Pl}(L_k)-\mu _\mathrm {Pl}(h_s)|<\frac{\varepsilon }{3}+\frac{\varepsilon }{3}+\frac{\varepsilon }{3}=\varepsilon . \end{aligned}$$

This means that \(\frac{1}{{\text {vol}}(\Gamma _n\backslash G)}D_s^2\frac{Z_n'}{Z_n}(s)\) converges to zero for \({\text {Re}}(s)>1\). Since \(\frac{Z_n'}{Z_n}(s)=\sum _{k\ge 0}\sum _{[\gamma ]}\ell (\gamma _0)e^{-(s+k)\ell (\gamma )}\), the following lemma proves the ‘only if’ direction of the theorem.

Lemma 3.3

For \(n,k\in {{\mathbb {N}}}\) let \(a_{n,k},b_{n,k}>0\) be real numbers. Suppose that \(L_n(s)=\sum _{k=1}^\infty a_{n,k}e^{-sb_{n,k}}\) converges for \({\text {Re}}(s)>1\) and that \(D_s^2L_n(s)\) tends to zero as \(n\rightarrow \infty \). Then \(L_n(s)\) also tends to zero as \(n\rightarrow \infty \) for every s with \({\text {Re}}(s)>1\).

Proof

As the sum \(L_n(s)\) converges locally uniformly, by the theorem of Weierstrass, we can differentiate under the sum to get for \(s>1\) that

$$\begin{aligned} s^4D_s^2L_n(s)=\sum _{k=1}^\infty a_{n,k}(3b_{n,k}s+b_{n,k}^2s^2+3)e^{-sb_{n,k}}\ge \sum _{k=1}^\infty a_{n,k}e^{-sb_{n,k}}=L_n(s)\ge 0. \end{aligned}$$

Now if the former tends to zero as \(n\rightarrow \infty \), then so will the latter. \(\square \)

(b) For the converse direction assume convergence of \(\frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\frac{Z_n'}{Z_n}(s)\) to zero and let \(f\in C^\infty _c(G)\). As f has compact support, there exists \(c>0\) such that for every \(n\in {{\mathbb {N}}}\) and every \(\gamma \in \Gamma _n\smallsetminus \{1\}\) with \(l(\gamma )>c\) the conjugation orbit \(\{x\gamma x^{-1}:x\in G\}\) has empty intersection with \({\text {supp}}(f)\). Hence for such \(\gamma \) we have that \({\mathcal {O}}_\gamma (f)=0\). Here \({\mathcal {O}}_\gamma (f)=\int _{G/G_\gamma }f(x\gamma x^{-1})\,dx\) is the orbital integral and \(G_\gamma \) is the centralizer of \(\gamma \) in G. As f is bounded and has compact support, there exists \(M>0\) such that \(|{\mathcal {O}}_\gamma (f)|\le M\) for all \(\gamma \in \Gamma \smallsetminus \{1\}\). Note that any cocompact lattice only contains hyperbolic elements besides the trivial one. By Theorem 9.5.1 of [6], the trace formula is valid for every \(f\in C_c^\infty (G)\), i.e.,

$$\begin{aligned} \mu _n(f)=\sum _{[\gamma ]}{\text {vol}}(\Gamma _\gamma \backslash G_\gamma )\,{\mathcal {O}}_\gamma (f), \end{aligned}$$

where \(\Gamma _\gamma =\Gamma \cap G_\gamma \). For \(\gamma =1\) we have \({\text {vol}}(\Gamma _\gamma \backslash G_\gamma )={\text {vol}}(\gamma _n\backslash G)\) and for \(\gamma \ne 1\), the proof of Theorem 11.4.3 implies that \({\text {vol}}(\Gamma _\gamma \backslash G_\gamma )=\ell (\gamma _0)\). So it follows that for given \(s>1\) we have that

$$\begin{aligned} \left| \frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\mu _n(f)-f(e)\right|&\le \frac{M}{{\text {vol}}(\Gamma _n\backslash G)}\sum _{[\gamma ]\ne [e], \ell (\gamma )\le c}\ell (\gamma _0)\\&\le \frac{M}{{\text {vol}}(\Gamma _n\backslash G)}\frac{1-e^{-c}}{e^{-sc}}\sum _{[\gamma ]\ne [e], \ell (\gamma )\le c}\ell (\gamma _0)\frac{e^{-s\ell (\gamma )}}{1-e^{-\ell (\gamma )}}\\&= \frac{M}{{\text {vol}}(\Gamma _n\backslash G)}\frac{1-e^{-c}}{e^{-sc}}\frac{Z_n'}{Z_n}(s)\longrightarrow 0 \end{aligned}$$

as \(n\rightarrow \infty \). The theorem is proven. \(\square \)

At this point one may ask, what happens to the zeta functions at arguments \(s\in {{\mathbb {C}}}\) with \({\text {Re}}(s)\le 1\)? This question is partially answered by the following functional equation, as it implies that \(\frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\frac{Z_n'}{Z_n}(s)\) will converge to a nonzero limit for \(n\rightarrow \infty \), if \({\text {Re}}(s)<0\). This makes the question of convergence in the critical strip \(0<{\text {Re}}(s)<1\) even more mysterious.

Proposition 3.4

For \(s\in {{\mathbb {C}}}\smallsetminus {{\mathbb {Z}}}\) and \(n\in {{\mathbb {N}}}\) we have that

$$\begin{aligned} \frac{Z_n'}{Z_n}(s)+\frac{Z_n'}{Z_n}(1-s)=\left( s-\frac{1}{2}\right) \,{\text {vol}}(\Gamma _n\backslash G)\,\cot (\pi s). \end{aligned}$$

In particular, if \(\frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\frac{Z_n'}{Z_n}(s)\) converges to zero for \(n\rightarrow \infty \) and \({\text {Re}}(s)>1\), then for \({\text {Re}}(s)<0\) and \(s\notin {{\mathbb {Z}}}\) it will converge to \(\left( s-\frac{1}{2}\right) \,\cot (\pi s)\).

Proof

As above, we have for \({\text {Re}}(s)>\frac{1}{2}\) that

$$\begin{aligned} \frac{1}{s}\frac{Z_n'}{Z_n}\left( s+\frac{1}{2}\right)&=\frac{1}{b}\frac{Z_n'}{Z_n} \left( b+\frac{1}{2}\right) -\frac{{\text {vol}}(\Gamma _n\backslash G)}{\pi }\sum _{k=0}^\infty \left( \frac{1}{s+\frac{1}{2}+k} -\frac{1}{b+\frac{1}{2}+k}\right) \\&\quad + 2\sum _{j=0}^\infty \frac{1}{s^2+r_{n,j}^2}- \frac{1}{b^2+r_{n,j}^2}. \end{aligned}$$

Now the right hand side converges for all \(s\in {{\mathbb {C}}}\smallsetminus \left( \frac{1}{2}-{{\mathbb {N}}}\right) \) and thus establishes an analytic continuation of the left hand side. Also, the right hand side is an even function in s with the exception of the first sum. Taking the right hand side for \(s\in {{\mathbb {C}}}\) and subtracting its value at \(-s\) we get

$$\begin{aligned} \frac{1}{s}\frac{Z_n'}{Z_n}\left( s+\frac{1}{2}\right) +\frac{1}{s}\frac{Z_n'}{Z_n}\left( -s+\frac{1}{2}\right)&= -\frac{{\text {vol}}(\Gamma _n\backslash G)}{\pi }\sum _{k=0}^\infty \left( \frac{1}{s-\frac{1}{2}-k} +\frac{1}{s+\frac{1}{2}+k}\right) . \end{aligned}$$

Replacing s with \(s-\frac{1}{2}\) this yields

$$\begin{aligned} \frac{1}{s-\frac{1}{2}}\frac{Z_n'}{Z_n}\left( s\right) +\frac{1}{s-\frac{1}{2}}\frac{Z_n'}{Z_n}\left( 1-s\right)&= -\frac{{\text {vol}}(\Gamma _n\backslash G)}{\pi }\sum _{k=0}^\infty \left( \frac{1}{s-1-k} +\frac{1}{s+k}\right) \\&={\text {vol}}(\Gamma _n\backslash G)\,\cot (\pi s). \end{aligned}$$

\(\square \)

Open questions and further projects

Convergence inside the critical strip

Let \(\Gamma _n\) be a sequence as in Theorem 3.2 such that \(\frac{1}{{\text {vol}}(\Gamma _n\backslash G)}\frac{Z_n'}{Z_n}(s)\) tends to zero for \({\text {Re}}(s)>1\) as \(n\rightarrow \infty \). Then this sequence of functions also converges for \({\text {Re}}(s)<0\), but what happens in between, in the critical strip? It might be possible to answer this question by finding suitable test functions for the trace formula. Note here that the zeros of the zeta function, i.e., poles of \(\frac{Z_n'}{Z_n}\), will accumulate on the critical line, as \(n\rightarrow \infty \).

Uniform discreteness

One assertion of the main theorem was proven under the condition of uniform discreteness, or, equivalently, a lower bound on the injectivity radius. It is not clear whether that condition is necessary. It seems impossible to eliminate the injectivity radius from the eigenvalue estimates, as Theorem 8.1.2 in [3] shows. According to this theorem, for fixed genus g (and therefore fixed volume \({\text {vol}}(\Gamma \backslash G)\)), for every \(\varepsilon >0\) there exist groups \(\Gamma \) of genus g and \(N(1+\varepsilon )\) arbitrarily large. Therefore, the only option seems to lie in an analysis of the Teichmüller space along the lines of [19] and the references therein (although possible critical cases have been excluded in that paper).

General rank one groups

The present proof uses eigenvalue estimates for the Laplacian. For general rank one Lie groups like \({\mathrm {SO}}(n,1)\) the Selberg zeta function is described by the spectrum of generalized Laplacians [2] on certain homogeneous vector bundles. An extension of the present results would therefore require an extension of the eigenvalue estimates to these bundles. One possible path is to use the group Laplacian instead, which would provide a much weaker estimate, as the dimension increases; however, it might be sufficient for the task at hand.

Higher rank

For higher-rank groups the Selberg zeta function has to be replaced by corresponding higher-rank zeta functions as in [5]. As this zeta function only collects closed geodesics which lie in an open Weyl chamber, it will be necessary to consider several zeta functions, one for each conjugacy class of non-compact Cartan subgroups. On the other hand, a simplification may arise by only considering the restriction of these several variable zeta functions to generic lines.

p-adic groups

For p-adic groups the symmetric space is replaced with the Bruhat–Tits building. The rank one case (i.e., the case of graphs) has in great generality been dealt with affirmatively in [17]. The higher-rank case will rely on the several variable zeta functions defined in [8] and otherwise face the same difficulties as in the Lie-group situation except for the fact that small radii of injectivity play no role here.

Locally compact groups

This last and most general case is highly speculative. Is it possible to give a zeta function for any uniform lattice \(\Gamma \) in an arbitrary locally compact group G which reflects the global geometry well enough to detect Benjamini–Schramm convergence as formulated in [9]? There are a possible top-down approach and a bottom-up approach to this problem. The top-down approach uses the data given in the trace formula to define a new type of zeta function, such that the spectral side of the trace formula yields analytic continuation. The bottom-up approach uses the known cases and the structure theory of locally compact groups given, for instance, in [23].

Non-cocompact lattices

For arithmetic congruence groups, the adelic trace formula can be used to show that certain sequences of arithmetic groups are BS, see [18, 20]. An open problem raised in these papers is the question if any sequence \((\Gamma _n)\) of congruence subgroups in a given linear algebraic group G is already BS if the covolumes tend to infinity. A similar statement is known to be wrong without the congruence property.

The connection to Selberg-type zeta functions is more subtle in the non-compact situation, as the trace formula does not provide a direct link between geometric spectral data.

Quantum ergodicity

The interesting paper [16] has opened a new perspective in mixing spectral convergence and quantum ergodicity. In the paper, this is formulated for hyperbolic surfaces and, as in [9], there is room for further developments along the lines described above.

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Deitmar, A. Benjamini–Schramm convergence and zeta functions. Res Math Sci 7, 27 (2020). https://doi.org/10.1007/s40687-020-00225-4

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