Length Spectrum

Abstract

In this chapter, we prove a version of the main result using the length spectrum instead of the Laplace spectrum, in case the manifold is negatively curved. We introduce the L-series corresponding to a unitary representation of the fundamental group, recall its convergence properties in relation to volume entropy, and its behaviour in finite covers.

In this chapter, we prove a version of the main result using the length spectrum instead of the Laplace spectrum, in case the manifold is negatively curved. We introduce the L-series corresponding to a unitary representation of the fundamental group, recall its convergence properties in relation to volume entropy, and its behaviour in finite covers.

1 L-Series

We assume that M is a connected negatively curved oriented closed Riemannian manifold. Then the following properties hold:

• 1. each free homotopy class [γ] contains a unique closed geodesic [56, Thm. 3.8.14]. We write ℓ(γ) for the length of that geodesic.

• 2. the geodesic flow is of Anosov type, and the topological entropy equals the volume entropy, defined as $h_M:=\lim \limits _{R \rightarrow + \infty } (\log \mathrm {vol}(B(x,R)))/R,$ where B(x, R) is a geodesic ball of radius R in the universal cover $\widetilde M$ of M centered at some point $x \in \widetilde M$ [65].

Recall some elements of the theory of prime geodesics (for general M but an abelian cover, we treated this in Sect. 10.1). We call [γ] prime if the associated geodesic is not a multiple of another geodesic (in the sense of oriented cycles). Let ϖ: M′→ M denote a G-cover for a finite group G. Notice that $h_{M'}=h_M$ by definition. Above each fixed prime geodesic of M lie finitely many prime geodesics of M′, the set of which carries a transitive G-action. For a fixed γ′ mapping to γ, we let (ϖ|γ) denote any element of G that generates the stabiliser of γ′. All such elements are conjugate in G (in the abelian case, the element does not depend on the choice of γ′, cf. Sect. 10.1).

Let $\rho \colon G \rightarrow \operatorname {U}(N, \mathbf {C})$ denote a representation. Since the determinant takes the same value on conjugate matrices, we have a well-defined associated L-series

$$\displaystyle \begin{aligned} L_M(\rho,s):= \prod_{[\gamma]} \det(1_N - \rho((\varpi|{\gamma})) e^{-s \ell(\gamma)} )^{-1}, \end{aligned}$$

where 1_N is the identity matrix of size N × N. If ρ ₁ and ρ ₂ are two representations of G, then [76, p, 135]

$$\displaystyle \begin{aligned} L_M(\rho_1 \oplus \rho_2) = L_M(\rho_1)L_M(\rho_2).\end{aligned} $$

(12.1)

Choosing ρ = 1, L _M(1, s) is related to analogues of the Selberg zeta function. Standard theory of Dirichlet series (applying Möbius inversion to the coefficients of the logarithmic derivative) implies that knowledge of L _M(1, s) is equivalent to knowledge of the multiset of lengths {ℓ(γ)}. Parry and Pollicott and Adachi and Sunada have proven:

Lemma 12.1.1 ([76, Thm. 1 and 2] and [1, Thm. A])

The function L _M(ρ, s) converges absolutely for Re(s) > h _M and can be analytically continued to an open set D that contains the half-plane Re(s) ≥ h _M . For an irreducible representation ρ, L(ρ, s) is holomorphic in D unless ρ = 1 . Furthermore, L _M(1, s) has a simple pole at s = h _M.

2 Main Result for the Length Spectrum

In this setup, the analogue of Lemma 3.9.1 is the following:

Lemma 12.2.1

Let G be a finite group acting by fixed-point free isometries on a negatively curved Riemannian manifold M′ with quotient M = G∖M′. Set $h:=h_M=h_{M'}$ . If ρ is any unitary representation of G, then the multiplicity 〈ρ, 1〉 of the trivial representation in the decomposition of ρ into irreducibles equals −ord_s=h L _M(ρ, s), the order of the pole of L _M(ρ, s) at s = h.

Proof

Let D denote the extended region of convergence as in Lemma 12.1.1. Decompose $\rho = \bigoplus \limits _{i=1}^N \langle \rho _i,\rho \rangle \rho _i$ as a sum over irreducible representations ρ _i. Then by formula (12.1), we have

$$\displaystyle \begin{aligned} L_M(\rho,s) = \prod_{i=1}^N L(\rho_i,s)^{\langle \rho_i,\rho\rangle}, \end{aligned}$$

and set ρ ₁ = 1 for convenience. Applying Lemma 12.1.1, we find from this product decomposition that ord_s=h L _M(ρ, s) = 〈1, ρ〉. □

We have the following further two analogues of results previously shown for the Laplace spectra. The analogue of Lemma 3.8.1 for L-series is the following.

Lemma 12.2.2 ([91, Remark 2 After Lemma 1])

If M → M ₁ → M ₀ is a tower of finite Riemannian coverings and M → M ₀ is Galois with group G, M → M ₁ with group H, and $\rho \colon H \rightarrow \operatorname {U}(N,\mathbf {C})$ a representation, then

$$\displaystyle \begin{aligned} L_{M_0}(\operatorname{Ind}_{H}^G \rho,s) = L_{M_1}(\rho,s). \end{aligned}$$

The analogue of Proposition 4.1.1 for L-series is the following.

Lemma 12.2.3

Suppose that we have diagram (1.2) and set $h:=h_{M_1}=h_{M_2}=h_M$ ; then, for two linear characters and , a representation isomorphism $\operatorname {Ind}_{H_1}^G \chi _1 \cong \operatorname {Ind}_{H_2}^G \chi _2$ is equivalent to

$$\displaystyle \begin{aligned} \mathrm{ord}_{s=h} L_{M_i}(\overline \chi_i \otimes \operatorname{Res}_{H_i}^G \operatorname{Ind}_{H_j}^G \chi_j,s) \end{aligned} $$

(12.2)

being the same for the pairs (i, j) given by (1, 1), (2, 1) and (1, 2), (2, 2).

Proof

The proof is essentially the same as that of Proposition 4.1.1, but now using Lemma 12.2.1 instead of Lemma 3.9.1. □

We can adopt the reasoning in the proof of the main theorem to find:

Theorem 12.2.4

Suppose we have a diagram ( 1.1 ) of negatively curved Riemannian manifolds with (common) volume entropy h, and suppose that the action of G on M in the extended diagram ( 1.2 ) is homologically wide. Then M ₁ and M ₂ are equivalent Riemannian covers of M ₀ if and only if the pole orders at s = h of a finite number of specific L-series of representations on M ₁ and M ₂ is equal.

More specifically, using the notation of Theorem 6.4.1 , M ₁ and M ₂ are equivalent Riemannian covers of M ₀ if and only if there exists a linear character $\chi \colon \widetilde H_2 \rightarrow {\mathbf {C}}^*$ such that

$$\displaystyle \begin{aligned} \mathrm{ord}_{s=h} L_{M_1}(\overline \Xi \otimes \operatorname{Res}_{\widetilde H_1}^{\widetilde G} \operatorname{Ind}_{\widetilde H_1}^{\widetilde G} \Xi) = \mathrm{ord}_{s=h} L_{M_2}(\overline \chi \otimes \operatorname{Res}_{\widetilde H_2}^{\widetilde G} \operatorname{Ind}_{\widetilde H_1}^{\widetilde G} \Xi) \end{aligned}$$

and

$$\displaystyle \begin{aligned} \mathrm{ord}_{s=h} L_{M_1}(\overline \Xi \otimes \operatorname{Res}_{\widetilde H_1}^{\widetilde G} \operatorname{Ind}_{\widetilde H_2}^{\widetilde G} \chi) = \mathrm{ord}_{s=h} L_{M_2}(\overline \chi \otimes \operatorname{Res}_{\widetilde H_2}^{\widetilde G} \operatorname{Ind}_{\widetilde H_2}^{\widetilde G} \chi). \end{aligned}$$

There are $\ell |H_2^{\operatorname {ab}}|$ linear characters χ on $\widetilde H_2$ , and the dimension of the representations involved is the index [G : H ₂].

Remark 12.2.5

In case of a surface of constant curvature − 1, Theorems 1.2.1 and 12.2.4 are equivalent using a twisted version of the Selberg trace formula as in [49, III.4.10].

Open Problem

In [28, Theorem 3.1], it was proven that two global function fields (corresponding to smooth projective algebraic curves over finite fields) are isomorphic if and only if there is a group isomorphism between their abelianised absolute Galois groups such that the corresponding L-series are equal. In [13], it is shown that one may restrict to unramified characters by admitting extensions of the ground field.

For negatively curved (e.g., hyperbolic) manifolds M ₁ and M ₂, the corresponding question is whether they are isometric if and only if there is an isomorphism $\psi \colon \operatorname {H}_1(M_1,\mathbf {Z}) \cong \operatorname {H}_1(M_2,\mathbf {Z})$ such that

$$\displaystyle \begin{aligned} L_{M_2}(\chi,s)\!=\!L_{M_1}(\chi \circ \psi,s) \end{aligned}$$

$$\displaystyle \begin{aligned} \hspace{-97pt}\mbox{ for all }\chi {\in} \operatorname{Hom}(\pi_1(M_2),\!{\mathbf{C}}^*) \!=\! \operatorname{Hom}(\operatorname{H}_1(M_2,\mathbf{Z}),\!{\mathbf{C}}^*). \end{aligned}$$

Open Problem

The geodesic length function defines the marked length spectrum [γ]↦ℓ(γ) as a function from conjugacy classes in π ₁(M) to R _>0. Croke and Otal [30] [74] showed that this characterises the isometry class of M in dimension two (in arbitrary dimension, this is an open conjecture of Burns and Katok, see [22, Problem 3.1]; a local version was proven in [44]). This motivates the question: what is the relation between the marked length spectrum and the information encoded in the L-series L _M(ρ, s) where ρ runs over all unitary representations of π ₁(M) (this information might be called the “twisted length spectrum”)?