A refinement of strong multiplicity one for spectra of hyperbolic manifolds

Let $\calM_1$ and $\calM_2$ denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on $L^2(\calM_1)$ and $L^2(\calM_2)$ (respectively, multiplicities of lengths of closed geodesics in $\calM_1$ and $\calM_2$) are the same, except for a possibly infinite exceptional set of eigenvalues (respectively lengths). We define a notion of density for the exceptional set and show that if it is below a certain threshold, the two manifolds must be iso-spectral.

The length spectrum (respectively primitive length spectrum) is the set of lengths of closed geodesics (respectively primitive closed geodesics) in M listed with multiplicities. The complex length spectrum is the set of pairs, length and holonomy class, of all closed geodesics listed with multiplicities, where the holonomy class of a closed geodesic is the conjugacy class in SO(d − 1), d = dim(M) obtained by parallel transporting tangent vectors along the geodesic. We say that two spaces are length equivalent if they have the same length spectrum, and that they are complex length equivalent if they have the same complex length spectrum.
When M is a hyperbolic surface, Huber [Hu59] showed that the Laplace spectrum and the length spectrum determine each other (and they also determine the representation and complex length spectra). In higher dimensions the complex length spectrum and the representation spectrum still determine each other (see e.g. [Sa02]), however, the relation between the Laplace spectrum and the length spectrum is more mysterious.
In [Ga77], Gangolli generalized Huber's result to all hyperbolic manifolds, however, this generalization involves a different notion of length spectrum, we will call the 1-length spectrum, assigning to each length a certain real positive number (see remark 1.1). His result implies in particular that the Laplace spectrum of a hyperbolic manifold determines the length set (this is in fact true for all negatively curved manifolds [DG75]). However, the question of whether it determines the multiplicities, and the converse question of whether the length spectrum determines the Laplace spectrum were left open.
In [BR10], Bhagwat and Rajan showed that if two lattices satisfy that all but finitely many Laplace eigenvalues (respectively representations) have the same multiplicities, then they are Laplace equivalent (respectively, representation equivalent 1 ). In [BR11] they studied the length spectrum of real hyperbolic manifolds of even dimension, showing that if two lattices have the same multiplicities for all but finitely many lengths, then they are length equivalent. Similar results were previously shown by Elstrodt, Grunewald, and Mennicke in [EGM,Theorem 3.3] for the Laplace spectrum and 1-length spectrum of hyperbolic 3-manifolds. Elstrodt, Grunewald, and Mennicke then asked if it is possible to prove the same result when the finite set is replaced with an infinite set of sufficiently small density in some suitable sense.
In this paper we give a positive answer to the question of Elstrodt, Grunewald, and Mennicke, for the Laplace spectrum, representation spectrum, 1-length spectrum, and length spectrum. On the way, we also answer a question of Bhagwat and Rajan [BR11] and show that the Laplace spectrum of any compact hyperbolic manifold is completely determined by its length spectrum (cf. [Di89] for a similar result for the spectrum of the Laplacian on forms). The second problem of whether the Laplace spectrum determines the multiplicities in the length spectrum remains open.
Remark 0.1. The results of Bhagwat and Rajan can be thought of as an analogue of the strong multiplicity one theorem for cusp forms. This theorem states that if f and g are two Hecke new-forms for which the eigenvalues of the Hecke operators are equal at all but finitely many primes, then they are equal at all primes, and f = g; see [La76,p. 125]. (The analogy is only with the first part of the theorem, as there are examples of iso-spectral but not isometric hyperbolic manifolds; cf. [Su85,Vi80].) Continuing with this analogy, our result can be compared to Ramakrishnan's refinement [Ra94], stating that the finite set of primes in the strong multiplicity one theorem can be replaced with an infinite set, as long as it has Dirichlet density less than 1/8.
In order to describe our results we need to introduce some more notation. For any uniform lattice Γ ⊂ G and any π ∈Ĝ we denote by m Γ (π) the multiplicity of π in L 2 (Γ\G). For every ℓ ∈ (0, ∞) we denote by m Γ (ℓ) (respectively m o Γ (ℓ)) the number of closed geodesics (respectively primitive closed geodesics) of length ℓ. We also denote by ls Γ (respectively ls o Γ ) the set of lengths of (primitive) closed geodesics in M = Γ\H. Let G = NAK be an Iwasawa decomposition of G and let M denote the centralizer of A in K. For any σ ∈M let π σ,ν , ν ∈ iR and π σ,ν , ν ∈ (0, ρ) denote the unitary principle and complementary series inĜ where ρ = ρ G denotes half the sum of the positive roots for (G, A) (see section 1.1 for more details).
For any two lattices Γ 1 , Γ 2 ⊆ G and every σ ∈M we define the following spectral density function to measure the difference between the (principal part) of the representation spectrum.
We think of the asymptotic growth rate of these functions as T → ∞ as describing the "density" of places for which the multiplicities are different. We note that this captures more information as it also takes into account by how much the multiplicities differ. Using this notion we prove a refinement of [BR10, Theorems 1.1 and 1.2] for real rank one groups.
Theorem 1. Let G denote a real rank one group and Γ 1 , Γ 2 ⊆ G two uniform lattices without torsion.
(2) Let B ⊆M be a finite set. If then Γ 1 and Γ 2 are representation equivalent.
Remark 0.2. The finite set B above can also be replaced by an infinite set satisfying a certain sparsity condition. Moreover, when M = SO(2), SO(3) or SU(2) we can naturally identifyM with N and this condition is then essentially that B is of density zero (see section 2.3).
For any two lattices Γ 1 , Γ 2 ⊆ G we also define a length density function where the sum is over ℓ ∈ ls o Γ 1 ∪ ls o Γ 2 . Our first results on the length spectrum is of a slightly different nature than the results in [BR10,BR11], in the sense that we start with (partial data) on the length spectrum and retrieve the Laplace spectrum.
Corollary. If two compact hyperbolic manifolds are length equivalent then they are Laplace equivalent.
For hyperbolic surfaces the Laplace spectrum determines the length spectrum, hence, for surfaces the threshold in Theorem 2 also implies that the two lattices are length equivalent.
Remark 0.3. It is interesting to compare this to the result of Buser [Bu92,Theorem 10.1.4] showing that there is a constant c(g, ǫ) such that if two hyperbolic surfaces of genus g and injectivity radius ≥ ǫ have the same multiplicities for all lengths ≤ c(g, ǫ), then they are length equivalent.
In higher dimensions, we do not know if the Laplace spectrum determines the length spectrum. Nevertheless, with the exception of the odd dimensional real hyperbolic spaces, if we impose a smaller threshold for the growth rate we are able to recover the length spectrum directly. Specifically, for each of the rank one groups we define the threshold With this threshold we can prove a refinement of [BR11, Theorem 1].
For odd dimensional real hyperbolic space the threshold α 0 = 0 so the statement is empty. In this case, even assuming that m o Γ 1 (ℓ) = m o Γ 2 (ℓ) for all but finitely many values of ℓ we were not able to prove that Γ 1 and Γ 2 are length equivalent. However, from Theorem 2 we know that they must be Laplace equivalent and hence must have the same length set and the same volume. Using this fact we can show Theorem 4. Let Γ 1 , Γ 2 ⊆ SO 0 (2n + 1, 1) denote two uniform lattices without torsion. If m o Γ 1 (ℓ) = m o Γ 2 (ℓ) for all ℓ ∈ {ℓ 1 , . . . ℓ k } then ℓ 1 , . . . , ℓ k are rationally dependent. In particular, if k = 1 then Γ 1 and Γ 2 are length equivalent.
We now discuss the sharpness of our thresholds for the density functions. Regarding Theorems 3 and 4, we note that there are no known examples of hyperbolic manifolds that are Laplace equivalent but not representation equivalent. It is thus possible that the correct threshold is actually the same as in Theorem 2.
For Theorem 2 we recall the Prime Geodesic Theorem [Ga77,Ma69], stating that Our threshold is thus roughly the square root of the trivial bound Remark 0.4. We note that the co-volumes of the lattices we use in the proof go to infinity as ǫ → 0. It is thus still possible that a positive density threshold can hold under the additional assumption that the volumes are uniformly bounded. See section 5.6 for a similar phenomenon in the analogous context of arithmetically equivalent number fields.
For the representation spectrum we suspect that our threshold is not optimal. We recall that the Weyl law for the principal spectrum is with d = dim(G/K) and C an explicit constant depending on G (see [MV83]). Consequently, the trivial bound for D σ (Γ 1 , Γ 2 ; T ) is O(T d ) and the correct threshold could very well be or even a positive density threshold of the form D σ (Γ 1 , Γ 2 ; T ) ≤ cT d with c a sufficiently small constant. We note that the first condition implies the two lattices at least have the same co-volume. We conclude this introduction with a brief outline of the paper. In Section 1 we introduce some notation and recall some basic results on the spectral theory of symmetric spaces. In section 2 we give the proof of Theorem 1 using the Selberg trace formula. Our proof is similar to the original proof of [EGM,Theorem 3.3]; the new ingredient which allows us to improve on their result is the use of more general test functions in the trace formula (instead of just the heat trace). In section 3 we develop a new trace formula in which the geometric side involves the length spectrum directly. The price we have to pay is that on the spectral side, in addition to the Laplace spectrum, we have contribution from other representations occurring in L 2 (Γ\G). In section 4 we show that, by using suitable test functions in the trace formula, we can isolate the contribution of each of those representations. This enables us to prove Theorems 2,3, and 4. Finally, in section 5 we recall the construction in [LMNR] of lattices with the same length sets and use it to prove Theorem 5. plaining their construction of spaces with the same length sets. I also thank Peter Sarnak and Masato Wakayama for clarifying a few points regarding the trace formula. This work was partially supported by NSF grant DMS-1001640.

Notation and preliminaries
1.1. Basic structure on symmetric spaces. Let G denote a connected semisimple Lie group of real rank one, K ⊂ G a maximal compact subgroup and H = G/K the corresponding symmetric space. That is G = SO 0 (n + 1, 1), SU(n + 1, 1), Sp(n, 1) or F II and H is real, complex, quaternionic, or octonionic hyperbolic space respectively.
Fix an Iwasawa decomposition G = NAK and let g = n⊕a⊕k denote the corresponding decomposition of the Lie algebra g. Let M, M * ⊆ K denote the centralizer and normalizer of A in K respectively. Let W = W (G, A) = M * /M denote the baby Weyl group; since we assume that dim(a) = 1 then |W | = 2 and we write W = {1, w}. We denote by Σ = Σ(G, A) the set of restricted roots for the pair (G, A) and by Σ + the set of positive restricted roots, then either Σ + = {α} or Σ + = {α, 2α}. Let ρ = ρ G denote half the sum of the positive roots, that is, ρ = (dim(n 1 ) + 2 dim(n 2 ))α 2 where n = n 1 ⊕ n 2 is the decomposition into the root spaces of α 1 and α 2 respectively. We fix (once and for all) an element H ∈ a with α(H) = 1 and for any t ∈ R we denote by a t = exp(tH) ∈ A. We can identify the dual spaces a * = R and a * C = C via ν = ν(H). With this identification ρ = dim(n 1 )+2 dim(n 2 ) 2 .
The action of M * on M (by conjugation) induces an action of the Weyl group W = {1, w} onM . We note that under this action π σ,iν = π wσ,−iν and that these are the only pairs of equivalent principal series representation. We say that a representation σ ∈M is ramified if σ = wσ and unramified otherwise, and we recall that there are unramified σ ∈M only when G = SO 0 (2m + 1, 1).
We denote byĜ c the set of equivalence classes of the principle and complementary series representations, and by S =Ĝ \Ĝ c (that is, S is the set of equivalence classes of discrete series representation, limits of discrete series, and Langland's quotients).
1.3. Closed geodesics and conjugacy classes. For any γ ∈ Γ we denote by [γ] ∈ Γ # its conjugacy class. Since M is of negative curvature, there is a natural correspondence between conjugacy classes in Γ = π 1 (M), free homotopy classes of closed curves in M, and (oriented) closed geodesics.
We say that an element γ ∈ Γ is primitive if it cannot be written as γ = δ j for some other δ ∈ Γ; note that this only depends on the Γconjugacy class. To any [γ] ∈ Γ # , we define the primitivity index j(γ) as the unique j ∈ N such that γ = δ j with δ ∈ Γ primitive. Under the above correspondence, a closed geodesic is primitive if and only if the corresponding conjugacy class is primitive. Moreover, the primitivity index j(γ) is the number of times the geodesic wraps around itself.
Any hyperbolic γ ∈ Γ is conjugated in G to an element m γ a ℓγ ∈ MA + where A + = {a t |t > 0}. Here ℓ γ is uniquely determined by [γ] and m γ is determined up to conjugacy in M. The pair (ℓ γ , [m γ ]) is then precisely the length and holonomy of the closed geodesic corresponding to [γ].
1.4. The σ-length and representation spectra. To any irreducible representation σ ∈M we attach two function L Γ,σ : R + → R and m Γ,σ : (0, ρ) ∪ iR → N, we call the σ-length spectrum and σ-representation spectrum respectively. These two functions are closely related via the Selberg trace formula.
The σ-length spectrum is defined by where χ σ is the character of σ and is the Weyl discriminant. When σ is unramified we also define L ± Γ,σ (ℓ) = L Γ,σ (ℓ) ± L Γ,wσ (ℓ). Remark 1.1. The definition of the length spectrum of a hyperbolic 3manifold given in [EGM,Definition 3.1] coincides with what we call the 1-length spectrum, that is, the σ-length spectrum for σ = 1 the trivial representation.
In order to define the σ-representation spectrum we fix a virtual representation η = ⊕ τ ∈K a τ τ with (a τ ∈ Z almost all zeros) such that η | M = σ (respectively σ + wσ if σ is unramified). Let Λ σ,ν = χ σ,ν (Ω G ) and let S =Ĝ \Ĝ c . For π ∈ S we denote by α Γ (π) the corrected multiplicity given by with d ω the formal degree of ω. The σ-representation spectrum is given by For σ unramified we also define m ± Γ (ν) by and (1.4) m − Γ,σ (ν) = m Γ (π σ,ν ) − m Γ (π wσ,ν ) We note that by [Mi82, Theorem 1.2] the σ-representation spectrum does not depend on the choice of η. Also, since representations π ∈ S have a minimal K-type (see [Kn86, Chapter XV]), then for any fixed η, there are only finitely many π ∈ S for which [π | K ; η] = 0. In particular, for σ ∈M fixed, m Γ,σ (ν) = m Γ (π σ,ν ) for all but finitely many values of ν. Moreover, for σ = 1 trivial, m Γ,1 (ν) = m Γ (π 1,ν ) for all ν. For any σ ∈M we define the σ-spectral set as 1.5. Trace formula attached to σ. Building on the Selberg trace formula developed in [Wal, War], Sarnak and Wakayama [SW99] derived a trace formula attached to each irreducible representation σ ∈M . They derived this formula in general for a (possibly) nonuniform lattice. We will write it down only for the simpler case of a uniform lattice without torsion. The derivation in this case is much simpler as there are no contribution from continuous spectrum or unipotent elements and the treatment of the multiplicities of discrete series is straight forward.
Remark 1.2. For the case of a uniform lattice this formula, with a special test function coming from the fundamental solution to the heat equational, was already derived in [Mi82,MV83]. We note that, in addition to the treatment of non-uniform lattices, another new features in [SW99] which is crucial for our application is the use of general test functions. See also [Di89] for a similar trace formula.
For Γ ⊆ G a uniform lattice without torsion the trace formula corresponding to σ ∈M takes the following form (see [SW99, Theorem 2 and Theorem 6.5] 2 : • For σ ∈M ramified, for any even g ∈ C ∞ c (R), whereĝ denotes the Fourier transform of g and µ σ (ν)dν is the Plancherel measure.
• For σ ∈M unramified, for any even g ∈ C ∞ c (R) we have the same formula but with m + Γ,σ , L + Γ,σ and µ + σ = 2µ σ . In addition, for any odd g ∈ C ∞ c (R) we have Remark 1.3. For any virtual representation η = a σ σ of M, we can define m Γ,η , L Γ,η and µ η as the corresponding weighted sums. With this convention, the above trace formula holds for any virtual representation and not just the irreducible representations.

Proof of Theorem 1
Let G denote a fixed semisimple group of real rank one and Γ 1 , Γ 2 ⊂ G two uniform lattices without torsion. Throughout this section we will keep the two lattices fixed and suppress them from the notation. In particular, we will denote ∆m 2.1. Density results for a fixed σ. As a first step, for each fixed σ ∈M we will use the trace formula to relate the σ-representation spectrum to the σ-length spectrum. In particular, we show that we can recover the σ-length spectrum from the σ-representation spectrum and vise versa, and moreover, to do that all we need is to know one of them up to an error of density zero.
Proof. Assume first that σ is ramified. The equality of the volumes follows from Weyl's law for the principal series (see [MV83, Corollary 1]). It remains to show the equality for the σ-length spectrum. To do this we will use the trace formula with an appropriate test function.
On the other hand,ĝ T (ν) = 1 Tĝ ( ν T )2 cos(νℓ 0 ) is bounded by 2 T |ĝ( ν T )| for ν ∈ R and by 2 Tĝ ( ν T ) cosh(ρℓ 0 ) for ν ∈ i(0, ρ). We can thus bound, The first (finite) sum goes to zero in the limit and for the second sum, using the fast decay ofĝ(ν) From the assumption that 1 T |ν k |≤T |∆m σ (iν k )| → 0 as T → ∞ it is not hard to see that the above sum also goes to zero in the limit. Comparing this with the right hand side, we get that |∆L σ (ℓ 0 )| = 0.
When σ is unramified we have the same result with ∆m + σ (ν) and ∆L + σ instead. Proof. We will write down the proof for ramified σ, the proof in the unramified case is identical.
To estimate the right hand side, we can bound the integral For the sum, note that g T is given by a convolution So as T → ∞ the right hand side goes to zero implying that ∆m σ (ν 0 ) = 0 as well.
Next we estimate the left hand side. Let δ > 0 be such that The left hand side of the trace formula can be written as The first term is ∆m σ (iν 0 )(1+o(1)) as T → ∞. For the rest of the sum we can bound |∆m σ | ≤ m Γ 1 ,σ + m Γ 2 ,σ and for each of the two lattices we have Since for N sufficiently large (depending only on the dimension) the sum converges we get that the left hand side of the formula converges to ∆m σ (iν 0 ), and since the right hand side goes to zero we have ∆m σ (iν 0 ) = 0.

2.2.
Proof of Theorem 1. The first part is a special case of Theorem 6 with σ = 1 (recall that m Γ,1 (ν) = m Γ (π 1,ν ) is the multiplicity of the eigenvalue ρ 2 −ν 2 ). We also note that the second condition in Theorem 6 with σ = 1 gives an analogous result for the 1-length spectrum. For the second part, let B ⊂M be a finite set and assume that Dσ(Γ 1 ,Γ 2 ;T ) T → 0 for all σ ∈M \ B. From Theorem 6 we get that vol(Γ 1 \G) = vol(Γ 2 \G) and L Γ 1 ,σ = L Γ 2 ,σ for all σ ∈M \ B. We will show that Γ 1 and Γ 2 have the same complex length spectrum and are hence representation equivalent.
Fix an arbitrary length ℓ 0 ∈ (0, ∞) and holonomy m 0 ∈ M. Since there are only finitely many geodesics of length ℓ 0 we can find for all σ ∈ B. Since these are finitely many conditions we can clearly find such a function. We then have an expansion absolutely converges andF (σ) = 0 for all σ ∈ B.
Next let g ∈ C ∞ c (R) be even and supported on a small enough neighborhood of ℓ 0 such that ls Γ 1 ∪ ls Γ 2 intersects its support at {ℓ 0 } and satisfy g(ℓ 0 ) = ℓ −1 0 D(m 0 a ℓ 0 ). We then have .
Expand F (m γ ) and change the order of summation (note that all series converges absolutely) to get Since the right hand side is the same for Γ = Γ 1 and Γ = Γ 2 then so is the left hand side, implying that Since this is true for any pair (ℓ 0 , m 0 ) we get that the complex length spectrum is the same.
2.3. Further refinement. We note that the above proof will still work if we replace the finite set B with an infinite set, as long as it is sufficiently sparse so that for any small neighborhood B ⊂ M # we can find a smooth class function F supported on B withF (σ) = 0 for all σ ∈ B.
sin(θ) . For these cases the following lemma shows that we can take the set B to be any set satisfying that Then for any δ > 0 and Proof. Since the Fourier transform of f (θ − θ 0 ) vanishes together with the Fourier transform of f (θ) we may assume that θ 0 = 0. Also, since we may always renormalize, it is enough to show that there is f ∈ C ∞ c (R) not identically zero that is supported on (−δ, δ) withf (n) = 0 for n ∈ B.
Next consider the function defined by the infinite product The condition (2.3) implies that the product converges (uniformly on compacta) to an entire function with exponential growth |f 2 (z)| e C|z| α . This function clearly vanishes on B (but it is not the Fourier transform of a compactly supported function). We now definef (z) =f 1 (z)f 2 (z), thenf is entire of exponential growth |f (z)| e δ|z|/2+C|z| α e δ|z| , and it still vanishes on B. Moreover, on the real line it decays like Consequently, the Paley Wiener Theorem (see [Ru66,Theorem 19.3]) implies thatf is the Fourier transform of a smooth function f that is supported on (−δ, δ).

Alternating trace formula
In the proof of Theorem 1 we used the fact that the σ-representation spectrum appearing in the trace formula is essentially given by the multiplicities of the principal series representations. The relation between the σ-length spectrum and the length spectrum is not that straight forward (even for trivial σ). In this section we develop a new formula (which is an alternating sum of trace formulas corresponding to certain virtual representations of M) where the geometric side involves the length spectrum directly. Precisely we show Theorem 7. Let G denote a real rank one group and Γ ⊂ G a uniform lattice without torsion. Let m = ρ G − α 0 (G) ∈ N, then there are m + 1 virtual representations η 0 , η 1 , . . . , η m of M, with η 0 the trivial representation such that for any even g ∈ C ∞ c (R) we have whereĝ q (ν) =ĝ(ν + i(m−q)) +ĝ(ν + i(q −m)), and the weight function ψ is given by Proof. In order to derive this alternating formula we expand the Weyl discriminant (1.1) as a sum of characters of representations of M. Specifically, we will show in Proposition 3.5 below that with η 0 , . . . , η m certain virtual representations of M with η 0 trivial. Now, since any γ ∈ Γ can be written in a unique way as γ = δ j with δ ∈ Γ primitive, we have Plugging in the expansion for D(γ) = D(γ) from (3.2) in the numerator we get where g q (ℓ) = g(ℓ)(e (m−q)ℓ + e (q−m)ℓ ). We conclude the proof by applying the trace formula attached to each of the virtual representation η q separately.
Remark 3.1. For the odd dimensional orthogonal groups the representations η 1 , . . . , η m are actual irreducible representation. For the other groups, it is also possible to obtain a similar formula using irreducible representations instead of virtual representations by using appropriate linear combinations of theĝ q 's.
The rest of this section will be devoted to the proof of the expansion (3.2) for the Weyl discriminant.
3.1. The Adjoint representation. In order to expand the Weyl discriminant as a sum of characters we first need to understand Ad(MA) and in particular its restriction to n 1 and n 2 . Denote by n 1 = dim n 1 and n 2 = dim n 2 . From the definition of n 1 and n 2 , for a ℓ = exp(ℓH) ∈ A we have that Ad(a ℓ ) |n 1 = e ℓ I and Ad(a ℓ ) |n 2 = e 2ℓ I. The following proposition describes the action of Ad(M). Proof. For the orthogonal group this is clear. For the unitary group G = SU(n + 1, 1), we write its Lie algebra as and the Lie algebras in the Cartan decomposition g = p⊕k are given by with v ∈ C n and c ∈ C. Then a ⊂ p is the real subspace , c ∈ R and the root spaces n 1 and n 2 are given by of dimension n 1 = 2n and which is one dimensional. We get that M, the centralizer of A in K, is given by A simple computation then shows that the adjoint action of M is trivial on n 2 and that Ad(m)X v = mX v m −1 = X x * uv . Fixing the standard basis X e j , X ie j , j = 1, . . . , n for n 1 (recalling the natural inclusion U(n) ⊆ SO(2n)) we get a homomorphism from M to SO(2n). For G = Sp(n, 1) repeating the same arguments replacing C with the quaternions H we get of dimension n 1 = 4(n − 1), of dimension n 2 = 3, and On n 1 we have Ad(m)X v = X uvx * and fixing a suitable basis for H n ∼ = C 2n ∼ = R 4n gives a homomorphism from M to SO(4(n − 1)). On n 2 the action is given by Ad(m)Y d = Y xdx * and choosing a suitable basis this gives a homomorphism into SO(3). Finally, for G = F II we have that n 1 is 8-dimensional and n 2 is 7-dimensional. The smallest irreducible representation of M = Spin (7) is the orthogonal representation which is 7-dimensional and the only 8-dimensional irreducible representation is the spin representation. Let t denote a maximal commutative subspace of m so that h = t ⊕ a is a Cartan algebra for g. Since the root spaces for h are one dimensional, any subspace of n j on which ad(t) acts trivially is at most one dimensional. Consequently, the action of ad(m) is not trivial on n 1 and n 2 , and hence the restriction of Ad(M) to n 2 gives the orthogonal representation and the restriction to n 1 is either the spin representation or a direct sum of the orthogonal and trivial representation. Finally, we note that the latter cannot hold as it would imply that the center of M acts trivially on n.
Remark 3.2. For the unitary and symplectic groups the homomorphism ι 1 : M → SO(n 1 ) is not the standard inclusion U(n) ⊂ SO(2n) (respectively, Sp(n − 1) ⊆ SO(4(n − 1)). In particular, when n = 2 this homomorphism has a nontrivial kernel. The homomorphism ι 2 is trivial for the unitary group and factors through the natural isomorphism Sp(1) ∼ = SO(3) for the symplectic group.

3.2.
Representations of the orthogonal groups. Next, we want to expand each one of these determinants as a linear combination of irreducible characters. To do this we recall some facts about the representation theory of the orthogonal groups that we will need. We refer to [Kn02, Chapter V] for background and more details on representation theory of compact groups.
The irreducible representations of the orthogonal group are parameterized by their highest weights: When n = 2m the possible highest weighs are so(2m) = { j a j e j : a 1 ≥ . . . ≥ a n−1 ≥ |a n |, a i − a j ∈ Z, 2a j ∈ Z}, and when n = 2m + 1 they are so(2m + 1) = { j a j e j : a 1 ≥ . . . ≥ a n ≥ 0, a i − a j ∈ Z, 2a j ∈ Z}.
There is a one to one correspondence between these highest weights and the irreducible representations of the simply connected Lie group Spin(n); a representation of Spin(n) factors through a representation of SO(n) if and only if all the coefficient are integral.
For any θ ∈ (Z/2πZ) m let − sin(θ j ) cos(θ j ) . When n = 2m is even, the map θ → u θ is an isomorphism of (Z/2πZ) m with the maximal torus of SO(2m). When n = 2m+ 1 is odd this isomorphism is given by θ →ũ θ = u θ 0 0 1 . For any weight λ, let χ λ denote the character of the irreducible representation of highest weight λ (that we may think of as a function on (Z/2πZ) m by restriction to the maximal torus). We recall the Weyl character formula: where ξ λ (θ) = exp(i j a j θ j ), W is the Weyl group of SO(n), D is the Weyl discriminant of SO(n) given by and δ = 1 2 α∈∆ + α is half the sum of the positive roots. For any fixed ℓ ∈ R consider the function F n (ℓ, ·) : SO(n) → R, F (ℓ, u) = | det(e −ℓ/2 I n − e ℓ/2 u)|. This is clearly a class function on SO(n) and we can write it as a linear combination of irreducible characters.
Proof. We first treat the even case. Evaluating F 2m (ℓ, ·) at u θ we get We can think of the functions S m,k as class functions on SO(2m) and write them as a linear combination of irreducible characters. To do this, for any n, k ∈ N with 2k ≤ n we define (3.7) N 0 (n, k) = #{(j 1 , . . . , j k ) ∈ {1, . . . , n − 1} k |j i+1 ≥ j i + 2} (3.8) N(n, k) = 1 k = 0 N 0 (n, k) + N 0 (n − 2, k − 1) k ≥ 1 We show in Lemma 3.4 below that Plugging this in the above expression we get The result now follows from the identity which can be easily proved by induction from the recursion relation together with the simple observation that N(2k, k) = 2 for all k ≥ 1. The second line of (3.5) is a consequence of the relation which follows from the decomposition of the restriction of τ k to SO(2m) into irreducible representations of SO(2m).
We still need to prove the identity (3.9) and the recursion (3.10).
Proof. To simplify notation we denote by χ k = χ σ k and χ ± m = χ σ ± m . Let W denote the Weyl group of SO(2m), which acts on the weights e 1 , . . . , e m by permutations and even sign changes. We recall the Weyl character formula where ξ e j (u θ ) = e iθ j , δ = m j=1 (m − j)e j is half the sum of the positive roots, and D(u) = w∈W sgn(w)ξ wδ (u) denotes the Weyl discriminant of SO(2m).
We first prove the formula for k < m. Let denote the subgroup of positive elements and observe that we can write where N k = #{w ∈ W + |w(e 1 + . . . + e k ) = e 1 + . . . + e k }.
In this case, we get a contribution from all λ ∈ E m satisfying (3.12) and (3.13), but also from the weight λ = e 1 + . . . + e m−1 − e m , giving the formula 3.3. The Weyl discriminant. Combining these results we get the following expression for the Weyl discriminant. Proof. We prove it separately for the orthogonal, unitary, symplectic and exceptional groups.
Orthogonal groups: For G = SO 0 (n + 1, 1) we have that ρ = n 2 and m = [ n 2 ]. The formula follows immediately from Corollary 3.1 and Proposition 3.2 where η q = σ q when n = 2m, and it is the virtual representation of SO(2m + 1) whose restriction to SO(2m) is σ q when n = 2m + 1.

Proof of Theorems 2, 3 and 4
We can now use the alternating trace formula to relate the length spectrum with some combination of the representation spectrum. The remarkable point is that we can actually retrieve each one of the multiplicities m Γ,ηq appearing in the formula (rather than just their combination). The key to retrieving the individual multiplicities is the fact that when dilating the functionsĝ q (T ν) they grow exponentially in T and different values of q correspond to different rates of exponential growth. Theorems 2, 3 and 4 will all follow from the following result.
Assume by contradiction that m Γ 1 ,ηq = m Γ 2 ,ηq for some q < ρ − α and let q 0 denote smallest such q. We first show that the bound (4.1) implies that an appropriate grouping together of the complementary series must cancel. To do this we consider the difference between the alternating trace formulas for Γ 1 and Γ 2 , We can rewrite the third sum as m q=0 (−1) q ν k ∈(0,ρ) Note that C(x) = 0 only on a finite (possibly empty) set which is contained in (−m, ρ + m). We show that in fact C(x) = 0 for all |x| > m − q 0 . Otherwise, let x 0 ∈ (−m, ρ + m) with |x 0 | > m − q 0 be such that C(x 0 ) = 0 and that C(x) = 0 for all |x| > |x 0 | (we can find such a point as C(x) = 0 on a finite set).
Since we assume that q 0 < m + α 0 − α and that |x 0 | > m − q 0 , we get that α − |x 0 | < α 0 and we can find some c ∈ [0, 1) such that α − |x 0 | < α 0 c < α 0 (where c = 0 iff α 0 = 0). We now estimate the two sides of the alternating trace formula with the test function where 1 1 cT,T denotes the indicator function of Using the fact that |g T (ℓ)| e −T |x 0 | for ℓ ∈ [cT − 1, T + 1] and g T (ℓ) = 0 for ℓ ∈ [cT − 1, T + 1] we can bound the left hand side of the trace formula by some constant times which goes to zero as T → ∞. For the right side of the trace formula, for any ν ∈ R we can bound implying that for all q ≥ q 0 as T → ∞ ν k ∈R |∆m ηq (iν k )||ĝ T,q (ν k )| → 0.
If q 0 > 0 then the minimality of q 0 implies that ∆m η 0 = 0, and since η 0 = 1 is the trivial representation we get that ∆V = 0 (e.g., from Weyl's law). In the case when q 0 = 0, then m − |x 0 | < 0 and the bound |ĝ T,q (ν)| T e T (m−|x 0 |) |ĝ q (ν)| implies that Rĝ T,q (ν)µ q (ν)dν → 0. Consequently, the only part of the right hand side of the formula that does not vanish in the limit is the finite sum |x k |≤|x 0 |ĝ T (ix k )C(x k ) which converges to C(x 0 ). Since the left hand side of the formula goes to zero we get that C(x 0 ) = 0 in contradiction.
For the left hand side, using (4.1) together with the bound for any c ′ ∈ (c, 1) and the fact that g(ℓ) = 0 for ℓ ∈ (−c, c), the same argument as above gives a bound of O(e −T ((m−q 0 )c ′ +α 0 c−α) ). Since q 0 < m − α + α 0 c we can choose c ′ sufficiently close to one to insure that the left hand side goes to zero as T → ∞.
Next we estimate the right hand side. For Since we already showed that C(x) = 0 for |x| > m − q 0 we get that the contribution of the complementary series ĝ T (ix k )C(x k ) → 0 as T → ∞.

4.2.
Proof of Theorem 3. When G/K is not an odd dimensional real hyperbolic space the threshold α 0 > 0. Assume that ℓ≤T |∆L o (ℓ)| e αT , with 0 < α < α 0 . Then Theorem 8 implies that m Γ 1 ,ηq = m Γ 2 ,ηq for q = 0, . . . , m. Since these are all the representation appearing in the alternating trace formula we get that for any even test function But this can only happen if ∆m o (ℓ) = 0 for all ℓ ∈ R (otherwise, taking g supported around the smallest ℓ where ∆m o (ℓ) = 0 will give a contradiction).

Spaces with similar length spectrum
In order to prove Theorem 5 (which excludes the possibility of a positive density threshold in Theorem 2) we need to find pairs of lattices for which the length spectrum is as similar as possible. To do this, we borrow the examples constructed by Leininger, McReynolds, Neumann, and Reid [LMNR] of non iso-spectral manifolds having the same length sets. Their construction is based on a modification of the Sunada method [Su85], for constructing non isometric manifolds having the same length spectrum. We start by recalling the Sunada method and in particular his formula for multiplicities of the length spectrum of a finite cover.

Splitting of primitive geodesics in covers.
Let M = Γ\H be as above, let Γ 0 ⊆ Γ denote a finite index subgroup and let M 0 → M the corresponding finite cover. We recall the analogy between the splitting of prime ideals in number field extensions and splitting of primitive geodesics in finite covers.
Let p denote a primitive conjugacy class in Γ (corresponding to a primitive geodesic in M). We say that a primitive conjugacy class P in Γ 0 lies above p (denoted by P|p) if there is some γ ∈ p and a natural number f such that q = [γ f ] Γ 0 . We call f the degree of P and denote f = deg(P) (this does not depend on the choice of representative γ ∈ p). We note that if P 1 , . . . , P r are all the primitive classes above p, then r j=1 f j = [Γ : Γ 0 ]. Assume further that Γ 0 is normal in Γ. Let P 1 , . . . , P r denote all the primitive conjugacy classes in Γ 0 lying above p. Then the group A = Γ/Γ 0 acts naturally on {P 1 , . . . , P r } by permutation. To any class P|p we attache an element σ P ∈ A which generates the stabilizer of P in A (specifically, if P = [γ f ] Γ 0 for some γ ∈ p then σ P is the class of γ in A = Γ/Γ 0 ). We note that the conjugacy class of σ P in A depends only on p and we denote it by σ p .
The analogy with splitting of prime ideals in extensions of number fields is evident from our choice of notation: primitive classes correspond to prime ideals, (normal) covers to (normal) field extensions, and the element σ P and class σ p to the Frobenius element and Frobenius class in the Galois group.
5.2. The Sunada construction. We recall the general setup for the Sunada construction. Let M = Γ\H as above, let Γ 0 ⊂ Γ denote a finite index normal subgroup and let A = Γ/Γ 0 . Let B ⊂ A denote a subgroup and let Γ 0 ⊂ Γ B ⊂ Γ such that Γ B /Γ 0 = B. Fix a primitive class p in Γ and a class P|p in Γ 0 . Let q 1 , . . . , q r denote all the classes in Γ B above p and let f j = deg(q j ).
The length spectrum for Γ B can be recovered from information on Γ and the data on the splitting degrees as follows: and using Möbius inversion we get where µ(m) is the Möbius function.
Since the value of χ B (a) is determined by |[a] ∩ B|, we see that if B 1 , B 2 ⊆ A satisfy that |[a] ∩ B 1 | = |[a] ∩ B 2 | for all a ∈ A then Γ B 1 and Γ B 2 are length equivalent. In fact, this condition implies that the two spaces are also representation equivalent (see [Su85]).

Lattices with the same length sets. Leininger McReynolds
Neumann and Reid showed in [LMNR] that, in the same setup, if we assume the weaker condition that |[a] ∩ B 1 | = 0 if and only if |[a]∩B 2 | = 0 for any a ∈ A, then Γ B 1 and Γ B 2 have the same length sets. Moreover, under additional assumptions on the groups A, B 1 , B 2 (see [LMNR,Definition 2.2]) one can guarantee that the primitive length sets are the same. In the same paper they constructed explicit examples of triples B 1 , B 2 ⊆ A satisfying these conditions and showed that these groups can be realized as quotients of lattices in any of the rank one groups.
We briefly recall their examples. Let p ∈ Z denote a prime number and assume that it is inert in the imaginary quadratic extension K = Q(i). In this case we have that the ideal (p) = pO is a prime ideal in O = Z[i] and the quotient O/pO = F q is the finite field with q = p 2 elements. We then have the following inclusion of exact sequences and and similarly V q ∼ = sl 2 (F q ). Inside sl 2 (F p ) ∼ = F 3 p we have the plane x y −y −x ) ∈ sl 2 (F p )|x, y ∈ F p }. As shown in [LMNR], in any rank one group G there is a uniform lattice Γ that surjects onto A = PSL 2 (O/p 2 O) for infinitely many values of p. In this case, the lattices Γ B 1 and Γ B 2 , corresponding to B 1 = V p and B 2 = {I + pX|X ∈ R ⊥ (F p )}, have the same primitive length sets. We now take a closer look at this example and analyze the difference between the multiplicities. 5.4. Computing the splitting types. Let Γ denote a uniform lattice in a rank one group G and assume that there is a surjection Consider the following finite index subgroup: Γ 0 = ker(ϕ), Γ Vq = ϕ −1 (V q ) and Γ j = ϕ −1 (B j ), j = 1, 2 where B 1 = V p = {I + pX|X ∈ sl 2 (F p )} and B 2 = {I + pX|X ∈ R ⊥ (F p )} as above. We note that Γ Vq and Γ 0 are both normal subgroups of Γ.
To get hold of the splitting types of primitive classes p in Γ we need to count how many elements lie in the intersection of each conjugacy class with each of the above subgroups. Since V q is normal in PSL 2 (O/p 2 O) a conjugacy class intersects V q if and only if it is contained there. Furthermore, since V q is commutative, the action of PSL 2 (O/p 2 O) on V q by conjugation factors through the quotient PSL 2 (F q ). The following lemma describes the PSL 2 (F q ) conjugacy classes in V q = sl 2 (F q ).
is a conjugacy class. In addition to these classes and the trivial class there are two nilpotent classes with representatives ( 0 0 1 0 ) and ( 0 0 ε 0 ) with ε ∈ F * q not a square. Moreover, for any X ∈ sl 2 (F q ) the size of its conjugacy class is given by We now compute the size of the intersection of each of these classes with V p ∼ = sl 2 (F p ) and R ⊥ (F p ).
Lemma 5.2. For any X ∈ sl 2 (F q ) we have Proof. The first part follows from the previous lemma together with the observation that C Q (F q ) ∩ sl 2 (F p ) = C Q (F p ) and that the intersection of a nilpotent conjugacy class with sl 2 (F p ) is the corresponding SL 2 (F p ) conjugacy class.
For the second part, note that x y −y −x ) |x, y ∈ F p , x 2 − y 2 = Q}, which is of size p − 1 for any Q ∈ F * p . The only nontrivial elements in R ⊥ not yet accounted for are the ones with zero determinant. There are 2(p − 1) such elements and since X is conjugated to ( 0 0 1 0 ) if and only if εX is conjugated to ( 0 0 ε 0 ) we have exactly p − 1 elements in each of the remaining two conjugacy classes.
Fix a primitive conjugacy class p in Γ, a primitive class P in Γ 0 above p, and let σ P ∈ A denote the "Frobenius" element. (That is, σ P = ϕ(γ) for γ ∈ p satisfying that [γ deg(P) ] Γ 0 = P). Since Γ Vq is normal in Γ, there is a natural number d 0 = d 0 (p) such that any γ ∈ p satisfies that γ d 0 is primitive in Γ Vq . Consequently, we have that σ d P ∈ V q if and only if d 0 |d and that A Vq (p, d) = |A| |Vq| δ d,d 0 . We now compute the splitting types for B 1 and B 2 .
Lemma 5.3. For j = 1, 2. If σ d 0 p = 1 then A B j (p, d) = |A| d 0 |B j | δ d,d 0 . Otherwise Proof. Since σ d p ∈ V q if and only if d 0 |d we get that χ B j (σ d p ) = 0 unless d = kd 0 for some k ∈ N, and hence A B j (p, d) = 0 unless d = md 0 for some m ∈ N.
If σ d 0 = 1 then σ kd 0 p = 1 for all k ∈ N, and hence χ B j (σ kd 0 p ) = |A| |B j | for all k ∈ N. Using (5.4) we get that in this case When σ d 0 P = 1 we write σ d 0 p = I + pX 0 with X 0 ∈ sl 2 (F q ) nontrivial. In this case, σ kd 0 p = (I + pX 0 ) k = I + pkX 0 . In particular, if p|k then σ kd 0 p = 1 and again χ B j (σ kd 0 p ) = |A| |B j | . For k prime to p, if X 0 ∈ C Q then kX 0 ∈ C k 2 Q and if X ∼ ( 0 0 c 0 ) then kX ∼ ( 0 0 kc 0 ). Thus, the sizes of the conjugacy classes of [kX 0 ] and [kX 0 ] ∩ B j don't depend on k so χ B j (σ kd 0 p ) = χ B j (σ d 0 p ). Using (5.4) we get When (m, p) = 1, the second sum is empty and the first sum is k|m µ( m k ) = δ m,1 . When m = pm with (m, p) = 1, the second sum is δm ,1 and the first sum is Finally, when m = p km with k > 1, both sums vanish. We thus get that in any case where X 0 = 0 we have 5.5. Proof of Theorem 5. Theorem 5 follows from the following proposition with p a sufficiently large inert prime.
Proof. We separate the primitive classes in Γ into 5 types as follows. Let d 0 = d 0 (p) as above and write σ d 0 P = I + pX 0 with X 0 ∈ sl 2 (F q ). We then say that p is of trivial type if X 0 = 0, and of nilpotent type if X 0 is nilpotent, and we say that p is of irregular type, quadratic type, or non-quadratic type if X 0 ∈ C Q (F q ) with Q ∈ F * q \ F * p , Q ∈ (F * p ) 2 or Q ∈ F * p \ (F * p ) 2 respectively (recall that F * p ⊂ (F * q ) 2 ). Note that even though X 0 depends on the choice of P above p, this characterization depends only on p.
Consider the partial counting functions m tr , m ir , m qr , m nq and m ni each given by m I (ℓ, d) = #{p of type I|d 0 (p) = d and ℓ p = ℓ/d}. Let K/Q denote a finite field extension and O K the corresponding ring of integers. For any prime number p ∈ Z we have a decomposition pO K = p e 1 1 · · · p er r with p j ⊆ O K the prime ideals dividing p and e j the ramification degrees (which are all one except for the ramified primes dividing the discriminant). Each quotient O K /p j is a finite field of order p f j where f j denotes the inertia degree of p j . The splitting type of p in K is then given by the numbers A K (p, d) = #{j|f j = d} for d = 1, . . . , n = [K : Q].
Let K 1 , K 2 denote two number fields. We say that a prime number p ∈ Z have the same splitting types in K 1 and K 2 if A K 1 (p, d) = A K 2 (p, d) for all d = 1, . . . , n. In [Pe77, Theorem 1], Perlis showed that if all but finitely many primes have the same splitting types in K 1 and K 2 , then all primes have the same splitting types. In this case the fields have the same Dedekind Zeta functions, but they are not (necessarily) isomorphic; such fields are called arithmetically equivalent fields.
The finite set of exceptions in Perlis's theorem can be replaced with an infinite set of density zero. A positive density threshold that is independent of the number fields probably does not hold. Nevertheless, it is possible to give a positive density threshold which depends on the degree of the smallest Galois field L/Q containing K 1 and K 2 . Specifically, one can show Proposition 5.5. In the above setting, for any 1 ≤ d ≤ n if the set P bad (d) = {p unramified|A K 1 (p, d) = A K 2 (p, d)} has Dirichlet density smaller then 1/|Gal(L/Q)| then P bad (d) = ∅. In particular, if P bad = ∪ n d=1 P bad (d) has density below 1/|Gal(L/Q)| then K 1 and K 2 are arithmetically equivalent.
Proof. The proof is a direct result of Chebotarev's Density Theorem combined with the following observation. For any prime p denote by σ p the Frobenius conjugacy class in Gal(L/Q). Denote by A = Gal(L/Q) and by B j = Gal(L/K j ). Then (using the same argument as in section 5.2) the condition A K 1 (p, d) = A K 2 (p, d) is equivalent to the condition