On Small Gaps in the Length Spectrum

We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrary small gaps is topologically generic: this is established both for surfaces of constant negative curvature (Theorem 3.1), and for the space of negatively curved metrics (Theorem 4.1). While arbitrary small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric. One result in this direction is presented in Section 5.


Introduction: geodesic length separation in negative curvature
On negatively curved manifolds, the number of closed geodesics of length ≤ T grows exponentially in T. (We refer the reader to [Mar04, for a comprehensive discussion about the growth and distribution of closed geodesics).
The abundance of closed geodesics leads to the natural question about the sizes of gaps in the length spectrum. In the current note we present a number of results related to this question. In some situations we are able to control the gaps from below, while in other we show that such control is not possible in general.
We note that a presence of exponentially large multiplicities in the length spectrum of a Riemannian manifold (which can be considered as a limiting case of small gaps) changes the level spacings distribution of Laplace eigenvalues on that manifold, see e.g. [L-S].
For generic Riemannian metrics, the length spectrum is simple [Abr,A2], so for any closed geodesic γ, only γ −1 will have the same length. So, by the Dirichlet box principle, there exist exponentially small gaps between the lengths of different geodesics.
Accordingly, it seems interesting to investigate manifolds where the gaps between the lengths of different geodesics have exponential lower bound: there exist constants C, β > 0, such that for any l 1 = l 2 ∈ Lsp(M) (length spectrum of the negatively curved manifold M), we have (1.1) |l 2 − l 1 | > Ce −β·max(l 1 ,l 2 ) .
This assumption is satisfied for arithmetic hyperbolic groups by the trace separation criterion (cf. [Tak] and [Hej,§18]). In Section 2 we explain (see Theorem 2.6) why the assumption (1.1) holds for hyperbolic manifolds whose fundamental group has algebraic elements.
In particular, the surfaces satisfying (1.1) form a dense set in the corresponding Teichmuller space. On the other hand the existence of arbitrary small gaps is topologically generic as is shown in Theorem 3.1 for surfaces of constant negative curvature and in Theorem 4.1 for the space of negatively curved metrics endowed with C r -topology, for any r > 0.
While arbitrary small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric. One result in this direction is presented in Section 5 there we obtain an explicit lower bound for the gaps valid for almost every hyperbolic surface.
Length separation between closed geodesics is relevant for the study of wave trace formulas on negatively-curved manifolds: to accurately study contributions from exponentially many closed geodesics to the wave trace formula, it is necessary to separate contributions from geodesics which differ either on the length axis, or in phase space. We remark that a suitable version of (1.1) always holds in phase space: small tubular neighbourhoods of closed geodesics in phase space are disjoint, as shown in [JPT]. Since there exist metrics for which the size of the length gaps cannot be controlled (Theorem 4.1), the authors in [JPT] established microlocal wave trace formula, and used the separation of closed trajectories in phase space in the proof.
2.1. Distances between algebraic numbers. In this section we consider gaps in the length spectrum for manifolds whose fundamental group admits algebraic generators. But first we provide a few general results about the algebraic numbers.
Proof. Let α j be the roots of P counted with multiplicities. We claim that |α j | ≤ R := 1 + j |a j |. Indeed if |x| > R then since R > 1 we get The result follows since j |α j | = |a 0 |.
Given a field K which is an extension of Q of degree d let H(L, N, p) be the set of all elements of K of the form β N p where β ∈ O K and for each automorphism σ j of K we have |σ j (β)| ≤ L.
Let I (L, N, p, D) be the set of numbers which satisfy α E + a E−1 α E−1 + · · · + a 0 = 0 where E ≤ D and a j ∈ H (L, N, p).

2.2.
Manifolds with algebraic generators of π 1 . We now formulate the main result of this section.
Theorem 2.6. Let X be a hyperbolic manifold such that the generators of π 1 (X) belong to P SO n,1 (Q). Then (1.1) holds.
We remark that in dimension 2 groups satisfying the assumptions of Theorem 2.6 form a dense set in the corresponding Teichmuller space T g . This can be established, for example, by the arguments of Section 5.
If n ≥ 3 then [G-R, Theorem 0.11] building on earlier results of of Selberg [Sel] and Mostow ([Most]) shows the conditions of Theorem 2.6 are satisfied for all finite volume hyperbolic manifolds. Hence we obtain Corollary 2.7. (1.1) holds for finite volume hyperbolic manifolds of dimension n ≥ 3.
The proof of Theorem 2.6 is similar to the proof of Proposition 3 in [GJS], where it is shown that the rotation matrices in SU(2) ∩ M 2 (Q) satisfy the Diophantine condition defined in [GJS]. Related results for other Lie groups were established in [ABRS,Br11,Var]. Related questions were also discussed in [Glu].
Proof. Let γ 1 and γ 2 be two closed geodesics. Let l j be the length of γ j , W j be the word fixing γ j , B j be the matrix corresponding to W j , m j be the word length of W j and r j = l j /2. To establish (1.1) it suffices to show that (2.8) |e r 1 − e r 2 | ≥Ce −c max(r 1 ,r 2 ) .
Without a loss of generality we assume that m j ≫ 1. By ([Miln, Lemma 2]) we know that if trivial unless m j and m 2 are comparable. Let us assume to fix our ideas that m 2 ≥ m 1 . By assumption there is a finite extension K of Q and numbers L and N such that all entries of the generators belong to H(L, N, 1). Accordingly the entries of B j belong to Closed geodesics on X correspond to loxodromic elements of π 1 (X) ⊂ P SO n,1 (also called boosts) that that fix no points in H n and fix two points in ∂H n . It is shown in the proof of Thm. I.5.1] that B j has precisely two positive real eigenvalues α 1,j = e r j and α 2,j = e −r j ; all other eigenvalues of B j have modulus one. Since the coefficients of the characteristic polynomial of B j are the sums of minors we have e r j ∈ I((L(n + 1)) (n+1)m j (n + 1)!, N, m j (n + 1)).
Remark 2.10. In dimension two the proof can be simplified slightly by remarking that 2 cosh(l j /2) = trB j ∈ K. An alternative proof of Theorem 2.6 could proceed by using explicit formulas for the lengths of closed geodesics on hyperbolic manifolds (see e.g. [P-R, (3), p. 246]) and the estimates for linear forms in logarithms (see e.g. Chapter 2]). The proof we give is more elementary, using only basic facts about algebraic numbers and matrix eigenvalues; and fairly concrete.
3. Small gaps for surface of constant negative curvature. Let Theorem 3.1. The set of tuples (A 1 , A 2 . . . A 2g ) ∈ G g where (1.1) fails is topologically generic.
Proof. Let γ A denote the closed geodesic whose lift to the fundamental cover joins q and Aq. Let L denote the length spectrum of the geodesics γ A where A belongs to a subgroup generated by A 1 and A 2 . Note that for a dense set of tuples it holds that for each δ there exists L such that for l > L the set [l, l + δ] intersects L. One way to see this is to consider the geodesics γ A k where e λ j is the leading eigenvalue of A j . Note that for typical A 1 , A 2 we have κ(A 1 , A 2 ) = 0 and λ 1 and λ 2 are non commensurable. Consider a geodesicγ = γ A 3 A n 1 where n is very large. By the foregoing discussion there exists l ∈ L such that |l − Lγ| < δ. Now consider the After applying a small perturbation if necessary we can assume that all entries of this matrix have the same order as its trace. Then tr(A 3 (η)A n 1 ) = tr(A 3 A n 1 ) + ηc, so by a small perturbation we can make L γ A 3 (η)A n 1 as close to l as we wish. Now the result follows by a standard Baire category argument (cf. Section 4).

Constructing metrics with small gaps in the length spectrum
This section is devoted to the proof of the following fact.
Theorem 4.1. For any r > 3 for any negatively-curved C r metric g, for any function F (t) (which we assume is monotone and fast decreasing), and a number δ > 0, there exists a metricḡ, such that ||ḡ − g|| C r < δ and there exists an infinite sequence of pairs of closed g-geodesics γ 1,j , γ 2,j with Lḡ(γ i,j ) → ∞ as j → ∞, and This shows that, in general, one cannot obtain good lower bounds for gaps in the length spectrum for a C r open set of negatively curved metrics.
Theorem 4.1 follows from the lemma below by a standard Baire category argument.
Lemma 4.3. Given a metric g and numbers L and δ there is a metric g such that ||g −g|| C r ≤ δ and there are twog-geodesics γ 1 and γ 2 such that Lg(γ 1 ) = Lg(γ 2 ) > L.
We also need the following fact Lemma 4.4. Let g andg be two negatively curved metrics such that ||g −g|| ∞ ≤ δ and γ andγ be two closed geodesics for g andg respectively of lengths L and L. If γ andγ are homotopic then Proof. Recall that for negatively curved there exists a unique geodesic in each homotopy class and this geodesic is length minimizing. The second inequality follows since the length of γ with respect tog is at most L(1 + δ) andγ is shorter. The first inequality follows from the second by interchanging the roles of g andg.
For j ≥ 1 we apply Lemma 4.3 to find g j such that and there are two geodesics γ 1,j , γ 2,j such that Then g k satisfies the required properties since, by Lemma 4.4, the lengths of g i,l have changed by less than F (L g l (γ 1,l )+1)/2 in the process of making consecutive inductive steps.
Remark 4.6. In particular if we continue the above procedure for the infinite number of steps then the limiting metric will satisfy the conditions of Theorem 4.1.
The proof of Lemma 4.3 relies on two facts. If γ is a closed geodesic let ν γ denote the invariant measure for the geodesic flow supported on γ. Let h denote the topological entropy of the geodesic flow. Let µ denote the Bowen-Margulis measure. Recall [P-P] that µ the measure of maximal entropy for the geodesic flow. It has a full support in the unit tangent bundle SM.
Lemma 4.8. For each q 0 ∈ M there exists ε such that for each L there is periodic geodesic γ such that L(γ) > L and γ does not visit an ε neighborhood of q 0 .
In the proof of Lemma 4.8 we need several facts about the dynamics of the geodesic flow which we call φ t . Recall [A1] that φ t is uniformly hyperbolic. In particular, there is a cone field K(x) and λ > 0 such that for u ∈ K, ||dφ t (u)|| ≥ e λt ||u||. Moreover the cone field K can be chosen in such a way that if x = (q, v) and u = (δq, δv) ∈ K(x) then (4.9) ||δq|| ≥ c||δv|| and ∠(δq, v) ≥ π 4 We call a curve σ unstable ifσ ∈ K. By the foregoing discussion if σ is un unstable curve then the length of the projection of φ t (σ) on M is longer than ce λt .
Proof of Lemma 4.8. We first show how to construct a not necessary closed geodesic avoiding B(q 0 , ε) and then upgrade the result to get the existence of a closed geodesics.
The first part of the argument is similar to [B-S, D]. Pick a small κ > 0. Take an unstable curve σ of small length κ. We show that if κ and ε are sufficiently small then σ contains a point such that the corresponding geodesic avoids B(q 0 , ε). Let T 1 be a number such that |φ T 1 (σ)| = 1 where φ denotes the geodesic flow. Note that T 1 = O(| ln κ|). Also observe that due to (4.9) there exists a number r 0 such that ifσ is an unstable curve and x ∈σ is such that d(q(x), q 0 ) < ε then for all y ∈σ such that Cε ≤ d(y, x) ≤ r 0 we have d(q(φ t y), q 0 ) > ε for |t| < r 0 where d denotes the distance in the phase space (just take r 0 much smaller than the injectivity radius of q 0 ).
Suppose now that dim(M) = 2. Let H r (M) denote the space of C r metrics with positive topological entropy. This set is C r open ( [K]) and dense. (If genus(M) ≥ 2 then every metric has positive topological entropy [K]. For torus the density of H r (M) follows from [Ban] and for sphere it follows from [K-W]).  Proof of Theorem 4.11. By [K] if g ∈ H r (M) then there is a hyperbolic basic set Λ for the geodesic flow. Since Lemmas 4.3, 4.7, 4.8 and 4.10 remain valid in the setting of hyperbolic sets the proof is similar to the proof of Theorem 4.2. (In the proof of Lemma 4.8 we need to take σ 1 so that it crosses completely an element of some Markov partition Π such that all elements of Π have unstable length between κ and Cκ. The number of eligible segments now is not O(1/κ) but O(1/κ a ) for some a > 0 but this is still much larger than | ln κ|.)

Small gaps for hyperbolic surfaces, continued
Here we show that for Lebesgue-typical hyperbolic surface the gaps in the length spectrum cannot be too small. Our argument in similar to [K-R]. Related results are obtained in [Var].
Proof. By induction. For n = 0 or 1 the result follows from Proposition 5.1.
Proposition 5.3. (Remez inequality) (see [B-G] or [Yom, Theorem 1.1]) Let B be a convex set in R n , Ω ⊂ B, and P be a polynomial of degree D. Then Corollary 5.5. If P N ∈ Z[x 1 , x 2 , . . . , x n ] are polynomials of degree D N and ε N is a sequnces such that N ε 1/D N < ∞ then |P (x 1 , . . . x n )| < ε N has only finitely many solutions for almost every (x 1 . . . x n ) ∈ R n .
Proof. It suffices to show this for a fixed cube B with side 2. Then Corollaries 5.2 and 5.4 give so the statement follows from Borel-Cantelli Lemma.
Corollary 5.6. Let m be a fixed number.
Proof. (a) It suffices to prove the statement under the assumption that |a j | > δ for some fixed δ > 0. Then d j = 1+b j c j a j and so the result follows from Corollary 5.5. (b) Rewriting the equations defining G g in the form we can express the entries of A 2g as rational functions of the entries of the other matrices. Arguing as in part (a) we can reduce the inequality Corollary 5.7. For each η > 0 for almost every A 1 , . . . A m ∈ SL 2 (R) the inequality holds for all except for finitely many words W.
Proof. If ||W (A 1 , . . . A m ) − I|| ≤ ε then all entries of W − I are ε close to I. Conisdering for example, the condition W 11 (A 1 , . . . A m ) − 1 we get a polynomial of degree |W |. Therefore, by Corollary 5.6 it suffices to check that Corollary 5.8. For A = (A 1 . . . A 2g ) ∈ G g let S A be the surface defined by A. Given a word W let l(W, A) be the length of the closed geodesic in the homotopy class defined by W. Then for each η > 0 the following holds for almost all A ∈ G g There exists a constant K = K(A) such that for each pair W 1 , W 2 either l(W 1 , A) = l(W 2 , A) or (5.9) |l(W 1 , A) − l(W 2 , A)| ≥ K(4g − 1) −[(2g+4)(4g−2)+η] max 2 (|W 1 |,|W 2 |) .
Remark 5.10. Recall that [Ran] shows that for any hyperbolic surface the length spectrum has unbounded multiplicity so there are many pairs of non conjugated words there the first alternative of the corollary holds.

< ∞
There are at most (4g − 1) 2k pairs (W 1 , W 2 ) with k = max(W 1 , W 2 ) so the last sum is estimated by proving the result.
(1) A suitable version of Theorem 2.6 should hold for other symmetric spaces. In particular, recall that arithmetic manifolds appear as fundamental domains G/Γ where G is a connected semi-simple algebraic R-group without compact factors of R-rank ≥ 2, and Γ is a lattice in G (cf. [Mar74,Mar75,Mar77]). Thus we expect that a version of Theorem 2.6 should hold in higher rank setting. Note however, that for higher rank symmetric spaces closed orbits are not isolated but appear in families.
(2) The proof of Theorem 4.1 relies on localized perturbations. Therefore it does not work in the analytic category. We expect that Theorem 4.1 is still valid for analytic metrics but the proof would require new ideas.
(3) It is likely that an explicit lower bound for the gaps in the length spectrum could also be obtained for prevalent set of negatively curved metrics (see [Kal] for related results) but we do not pursue this question here.