Benjamini–Schramm convergence and zeta functions

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 (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., P n x ∈ X n : B R (x) ∼ = B R (p) −→ 1, n → ∞.
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 BS-convergence ⇔ zeta-convergence to arbitrary locally homogeneous spaces, i.e., double quotients \G/K , where G is a locally compact group, , K subgroups, where K is compact and 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 \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 ≤ R becomes small. It could still be that closed geodesics cluster at large lengths, but the cluster moves toward ∞. 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 BSconvergence 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 ∞ , i.e., for each n ∈ N there is a metric covering map π n : X ∞ → X n , which means that π n is surjective and for each x ∈ X n there exists ε > 0 such that for every y ∈ π −1 n (x) the map π n maps the ball B ε (y) isometrically to B ε (x). Then we say that the sequence (X n ) is BS-convergent to X ∞ , if for every r > 0, as n → ∞.

Definition 1.2
In the paper [9] the following situation was considered: ( n ) is a sequence of lattices in a locally compact group G and ∞ is a common normal subgroup. We then say that the sequence ( n ) Benjamini-Schramm converges to ∞ , or BS-converges, written n BS −→ ∞ , if for every compact set C ⊂ G the sequence as n → ∞. Here P n is the normalized Haar measure on n \G. If ( n ) is BS-convergent to the trivial group {1}, then we say that ( 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 ( n ) be a sequence of lattices in G and let ∞ be a common normal subgroup of the n . Consider the following statements: The sequence of metric probability spaces X n = n \X is BS-convergent to X ∞ = ∞ \X.

Then (a) ⇒ (b) unconditionally.
If for every open ball ∅ = B ⊂ X and every isometry φ : B → B there exists a uniquely determined g ∈ G such that φ(x) = gx, x ∈ B and every n is torsion-free, then (b) ⇒ (a) holds as well.
Note that (b) ⇒ (a) holds for X being a symmetric space without compact factors.
Proof Throughout, we will denote the probability measures on n \G and on n \X by the same symbol P n .
(a) ⇒ (b): Assume that n is BS-convergent to ∞ 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 ∈ G and consider the point x = n gK ∈ X n . For a discrete subgroup ⊂ G and any set A ⊂ X we write \A for the image of A in \X or, what amounts to the same, A/ ∼, where a ∼ a if and only if there exists γ ∈ with γ a = a . We use the invariance of the metric to identify This means that B r ( n gK, X n ) is isometric to some B r ( ∞ hK, X ∞ ) if and only if Let U r be the pre-image of B r under the projection map G → G/K = X. Let C = U r U r −1 .
Then C is a compact subset of G. Let Then P n (A n (C)) tends to 1 as n → ∞. Let C r denote the compact set U r U −1 r . Then if g ∈ A n (C r ), for every γ n ∈ n one has Now if g −1 n guK = g −1 n gvK for two u, v ∈ U r , then there exists γ n ∈ n and k ∈ K such that g −1 γ n g = vku −1 and hence g −1 γ n g ∈ U r U −1 r and so u and v already give the same element in g −1 ∞ g\B r . In other words, it follows that Let T n (r) denote the set of all x ∈ X n such that there exists y ∈ X ∞ with B r (x, X n ) ∼ = B r (y, X ∞ ). Then the above entails Hence we get P n (T n (r)) → 1 and so (a) ⇒ (b) is proven.
First note that under the given conditions, each n acts fixed-point-freely on X and there exists a radius t n > 0 such that the projection map p n : X → n \X induces an isometric isomorphism B t n (x) → p n B t n (x) for every x ∈ X.
Let C ⊂ G be a compact set. Then there exists some r > 0 such that C ⊂ U r U −1 r . Let x n ∈ T n (r) and let φ : B r (x n , X n ) → B r (x ∞ , X ∞ ) be an isometry. Write x n = n g n K and x ∞ = ∞ g ∞ K , then φ can be viewed as a map g −1 n n g n \B r → g −1 ∞ ∞ g ∞ \B r . If φ maps the origin g −1 n n g n to some (g −1 ∞ ∞ g ∞ )g 0 , then one can replace g ∞ by g ∞ g 0 and φ will preserve origins. Next let 0 < t ≤ min(r, t n ). Then φ induces an isometry of the ball B t ⊂ X. Then there exists α ∈ G such that, on B t , the map φ is given by z → αz.
or, what amounts to the same, the diagram commutes. Let 0 < t ≤ r be maximal with this property. We claim that t = r. If not, then there exists z ∈ B r with d(z, eK ) = t < r. As φ is an isometry, the diagram above still commutes with B t replaced by the closed ball Let s = min(t n , r − t). We therefore have φ (g −1 n n g n )z = (g −1 n n g n )z and as φ is an isometry, φ maps B s (z) to B s (z). Again, this isometry is induced by an element g ∈ G and we have gy = y for every y in the non-empty open set B s (z) ∩ 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 ψ is an isometry as well, so we end up with the commutative diagram Now let γ n ∈ n and suppose g −1 n γ n g n ∈ C r . By definition, there exist u, v ∈ U r with g −1 n γ n g n = uv −1 , or (g −1 n γ n g n )v = u, which means that the points vK and uK are mapped to the same point in (g −1 n n g n )\B r , so they map to the same point in (g −1 n ∞ g n )\B r . This means that there exist γ ∞ ∈ ∞ and k ∈ K such that (g −1 is contained in a compact group. As n is torsion-free, it follows γ n = γ ∞ and therefore n g n ∈ A n (C r ). We have shown which implies P n (A n (C r )) → 1.
Remark 1.5 In [9] it is shown that ( n ) is BS and uniformly discrete

Plancherel and Benjamini-Schramm sequences
Let G = SL 2 (R). Then K = SO(2) is a maximal compact subgroup of G. The group G acts on the upper half plane H = {z ∈ C : Im(z) > 0} via linear fractionals and this action induces an identification of G/{±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 H ∼ = G/K . For a cocompact lattice ⊂ G the unitary representation of G, given by right translation on L 2 ( \G) decomposes as a direct sum of irreducibles The multiplicities N (π) are finite and are zero outside a countable subset of the unitary dual G.

Definition 2.1
The measure on G given by is called the spectral measure attached to .
let π(f ) denote the operator defined by integrating the representation π against f . More precisely, for v ∈ V π we have where the integral is a Bochner integral. Another way to say this is that π(f ) is the uniquely defined operator such that for any two v, w ∈ V π one has For a reductive Lie group like G = SL 2 (R), the operator π(f ) is known to be a trace class operator for every π ∈ G.
A sequence of cocompact lattices ( n ) in G is called a Plancherel sequence, if for every as n → ∞, wheref (π) = tr π(f ). The unitary dual 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 μ Pl on the unitary dual G such that for every f ∈ C ∞ c (G) we have f (e) = Gf (π) dμ Pl (π). (For an explicit computation of the Plancherel measure for the group G = SL 2 (R), see Section 11.3 of [6].) This means that the sequence ( n ) is Plancherel if and only if in the dual space of C ∞ c (G) one has weak-*-convergence 1 vol( n \G) μ n −→ μ Pl . Remark 2.3 (a) If a sequence ( n ) of lattices is a Plancherel sequence, then for every relatively compact open set U ⊂ G, whose boundary has Plancherel measure zero. This follows from the density principle of Sauvageot [21].
as n → ∞. This follows from the fact that the spectral measure of each n is discrete and the Plancherel measure is not.

The Selberg zeta function
Let ( n ) be a sequence of cocompact lattices in G. For simplicity, we shall assume that each n is torsion-free, which can easily be arranged as every lattice contains a torsion-free sublattice.
Definition 3.1 For γ ∈ n the length of γ is defined by It is known that for torsion-free cocompact lattices , every γ ∈ satisfies l(γ ) > 0. The Selberg zeta function for n is defined for s ∈ C with Re(s) > 1 as where the first product runs over all primitive conjugacy classes in n (see [6], Section 11.6). The product converges for Re(s) > 1, and the so defined function extends holomorphically to all of C.

Theorem 3.2 Let ( n ) be a sequence of torsion-free cocompact lattices in G.
(a) If the sequence ( n ) is uniformly discrete and Plancherel, then Let K = SO(2) be the standard maximal compact subgroup of G and let G K denote the set of all π ∈ G such that the representation space V π contains nonzero K -fixed vectors. Then it is known [15], The map φ : G K → [0, ∞), given by is a homeomorphism. We shall from now on identify G K with [0, ∞). For brevity, we write μ(f ) instead of X f dμ where μ is a measure on X and f a function. We shall also consider μ n and μ Pl as measures on G K ⊂ G.
We now proceed to the proof of part (a) of the theorem. Suppose that the sequence ( n ) is Plancherel. By [6], Section 11, there are numbers r n,j ∈ R ∪ i 0, 1 2 such that where C stands for the one-dimensional space of constant functions and π ir is the induced representation (principal or complementary series) with r ∈ R ∪ i 0, 1 2 . Formally we set r n,0 = i 2 . Fix s, b ∈ C with Re(s), 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 and μ n = μ n . This means 1 s Let D s denote the operator D s (ψ)(s) = − ∂ ∂s 1 s ψ(s) . We get Let h s (λ) = 1 Note that the function r → h s r 2 + 1 4 equals 1 4s D s ∂ ∂s h s,b (r). Therefore the above formula says For s > 1 2 the continuous function h s is positive on [0, ∞). By the trace formula, [6] Section 11.4, we have that ∞ j=0 |h s (λ n,j )| < ∞ for every s > 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 μ Pl (h s ) is finite.
is a relatively open, bounded interval, then by [21] we have that where 1 I is the indicator function of I. By linearity this extends to linear combinations of functions of the form 1 I . There exists a sequence (L k ) k∈N of such linear combinations such that 0 ≤ L k h s outside a countable set S, which is of Plancherel measure zero and can also be chosen to be of μ n measure zero for all n and have empty intersection with N. We can also choose the L k so that for each T ∈ N 0 we have that tends to zero for k → ∞.
For T ≥ 1 let N n (T ) = # j : 1 4 + r 2 n,j < T . Recall that λ n,j = 1 4 + r 2 n,j is the j-th Laplace eigenvalue. The sequence ( n ) is uniformly discrete, which means that the injectivity radii of the manifolds n \H are bounded below. Therefore, by formula (1.5) of [13], there exists a constant C > 0 such that for every T ≥ 1 one has Since k (T ) ≤ 2h s (T ) and ∞ T =0 h s (T )(2T + 1) < ∞, we can apply dominated convergence, to get that this sum tends to zero for k → ∞. So let ε > 0. Then there is k 0 ∈ N such that for all k ≥ k 0 we have that 0 ≤ μ n (h s ) − μ n (L k ) < vol( n \G)ε/3 holds for all n ∈ N and that |μ Pl (L k ) − μ Pl (h s )| < ε/3.
Fix some k ≥ k 0 . Then there exists n 0 ∈ N such that for all n ≥ n 0 one has And so This means that 1 vol( n \G) D 2 s Z n Z n (s) converges to zero for Re(s) > 1. Since [γ ] (γ 0 )e −(s+k) (γ ) , the following lemma proves the 'only if' direction of the theorem.

Lemma 3.3
For n, k ∈ N let a n,k , b n,k > 0 be real numbers. Suppose that L n (s) = ∞ k=1 a n,k e −sb n,k converges for Re(s) > 1 and that D 2 s L n (s) tends to zero as n → ∞. Then L n (s) also tends to zero as n → ∞ for every s with 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 s L n (s) = ∞ k=1 a n,k (3b n,k s + b 2 n,k s 2 + 3)e −sb n,k ≥ ∞ k=1 a n,k e −sb n,k = L n (s) ≥ 0.
Now if the former tends to zero as n → ∞, then so will the latter.
(b) For the converse direction assume convergence of 1 vol( n \G) Z n Z n (s) to zero and let f ∈ C ∞ c (G). As f has compact support, there exists c > 0 such that for every n ∈ N and every γ ∈ n {1} with l(γ ) > c the conjugation orbit {xγ x −1 : x ∈ G} has empty intersection with supp(f ). Hence for such γ we have that is the orbital integral and G γ is the centralizer of γ in G. As f is bounded and has compact support, there exists M > 0 such that |O γ (f )| ≤ M for all γ ∈ {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 ∈ C ∞ c (G), i.e., where γ = ∩ G γ . For γ = 1 we have vol( γ \G γ ) = vol(γ n \G) and for γ = 1, the proof of Theorem 11.4.3 implies that vol( γ \G γ ) = (γ 0 ). So it follows that for given s > 1 we have that as n → ∞. The theorem is proven. At this point one may ask, what happens to the zeta functions at arguments s ∈ C with Re(s) ≤ 1? This question is partially answered by the following functional equation, as it implies that 1 vol( n \G) Z n Z n (s) will converge to a nonzero limit for n → ∞, if Re(s) < 0. This makes the question of convergence in the critical strip 0 < Re(s) < 1 even more mysterious. In particular, if 1 vol( n \G) Z n Z n (s) converges to zero for n → ∞ and Re(s) > 1, then for Re(s) < 0 and s / ∈ Z it will converge to s − 1 2 cot(πs).
Proof. As above, we have for Re(s) > 1 2 that . Now the right hand side converges for all s ∈ C 1 2 − N 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 ∈ C and subtracting its value at −s we get Replacing s with s − 1 2 this yields

Convergence inside the critical strip
Let n be a sequence as in Theorem 3.2 such that 1 vol( n \G) Z n Z n (s) tends to zero for Re(s) > 1 as n → ∞. Then this sequence of functions also converges for 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 Z n Z n , will accumulate on the critical line, as n → ∞.

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 vol( \G)), for every ε > 0 there exist groups of genus g and N (1 + ε) 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 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 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 ( 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.

Funding
Open Access funding provided by Projekt DEAL.