Abstract
The equivalence of Benjamini–Schramm convergence and zetaconvergence, known for graphs, is proven for sequences of compact Riemann surfaces. A program is initialized, to extend this connection to arbitrary locally homogeneous spaces.
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1 Introduction
Benjamini–Schramm convergence or BSconvergence of metric probability spaces \(X_n\) to a pointed metric space (X, p) 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.,
For hyperbolic surfaces, Selberg [22] introduced a geometric zeta function which counts closed geodesics. Ihara [14] established a padic 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 BSconvergent 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
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 BSconvergence, 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 Ginvariant 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 zetaconvergence 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 padic 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 zetaconvergence 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 BSconvergence 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.
2 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 BSconvergence, 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 BSconvergence to a relative situation which was formulated in purely grouptheoretical terms. In geometric terms it translates to the following:
Definition 1.1
(BSconvergence 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 BSconvergent to \(X_\infty \), if for every \(r>0\),
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 BSconverges, written \(\Gamma _n{\mathop {\longrightarrow }\limits ^{BS}}\Gamma _\infty \), if for every compact set \(C\subset G\) the sequence
as \(n\rightarrow \infty \). Here \(P_n\) is the normalized Haar measure on \(\Gamma _n\backslash G\).
If \((\Gamma _n)\) is BSconvergent to the trivial group \(\{1\}\), then we say that \((\Gamma _n)\) is a BSsequence.
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 Ginvariant 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:

(a)
The sequence \(\Gamma _n\) is BSconvergent to \(\Gamma _\infty \).

(b)
The sequence of metric probability spaces \(X_n=\Gamma _n\backslash X\) is BSconvergent 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 torsionfree, 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 BSconvergent to \(\Gamma _\infty \) and let \(r>0\). As we encounter different metric spaces, we shall write \(B_r(z,Z)\) for the open rball in Z around z. The space \(X=G/K\) has a natural basepoint \(x_0=eK\). 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
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
Let \(U_r\) be the preimage 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
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
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
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
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 fixedpointfreely 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
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
Let \(s=\min (t_n,rt)\). 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 nonempty 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
is contained in a compact group. As \(\Gamma _n\) is torsionfree, it follows \(\gamma _n=\gamma _\infty \) and therefore \(\Gamma _n g_n\in A_n(C_r)\). We have shown
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 unitneighborhood \(U\subset G\) such that \(x^{1}\Gamma _nx\cap U=\{1\}\) holds for every \(x\in G\).
Remark 1.5
In [9] it is shown that
3 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 orientationpreserving isometries of the twodimensional 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
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
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
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
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
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
Remark 2.3

(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].

(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.
4 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 torsionfree, which can easily be arranged as every lattice \(\Gamma \) contains a torsionfree sublattice.
Definition 3.1
For \(\gamma \in \Gamma _n\) the length of \(\gamma \) is defined by
It is known that for torsionfree 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
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 torsionfree cocompact lattices in G.

(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.

(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
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 Kfixed vectors. Then it is known [15],
The map \(\phi :\widehat{G}_K\rightarrow [0,\infty )\), given by
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
where \({{\mathbb {C}}}\) stands for the onedimensional 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
into the trace formula and, as in Lemma 11.6.2 of [6], we get
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
Let \(D_s\) denote the operator \(D_s(\psi )(s)=\frac{\partial }{\partial s}\left( \frac{1}{s}\psi (s)\right) \). We get
and
Let
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
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
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
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 jth 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
Now, since \(N_n(0)=0\),
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
And so
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
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.,
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
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
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
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
Replacing s with \(s\frac{1}{2}\) this yields
\(\square \)
5 Open questions and further projects
5.1 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 \(n\rightarrow \infty \).
5.2 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).
5.3 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.
5.4 Higher rank
For higherrank groups the Selberg zeta function has to be replaced by corresponding higherrank 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 noncompact Cartan subgroups. On the other hand, a simplification may arise by only considering the restriction of these several variable zeta functions to generic lines.
5.5 padic groups
For padic 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 higherrank case will rely on the several variable zeta functions defined in [8] and otherwise face the same difficulties as in the Liegroup situation except for the fact that small radii of injectivity play no role here.
5.6 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 topdown approach and a bottomup approach to this problem. The topdown 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 bottomup approach uses the known cases and the structure theory of locally compact groups given, for instance, in [23].
5.7 Noncocompact 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 Selbergtype zeta functions is more subtle in the noncompact situation, as the trace formula does not provide a direct link between geometric spectral data.
5.8 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/s40687020002254
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DOI: https://doi.org/10.1007/s40687020002254