THE MARKED LENGTH SPECTRUM VERSUS THE LAPLACE

SPECTRUM ON FORMS ON RIEMANNIAN NILMANIFOLDS

Ruth Gornet

Texas Tech University

Abstract. The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In contrast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spectrum on one-forms. Outside of the standard spheres vs. the Zoll spheres, which are not even isospectral, this is the only example of a pair of Riemannian manifolds with the same marked length spectrum, but not the same spectrum on forms. This partially extends and partially contrasts the work of Eberlein, who showed that on two-step nilmanifolds, the same marked length spectrum implies the same Laplace spectrum both on functions and on forms.

Section 1: Introduction.

The spectrum of a closed Riemannian manifold (M, g), denoted spec(M, g), is the collection of eigenvalues with multiplicities of the associated Laplace–Beltrami operator acting on smooth functions. Two Riemannian manifolds (M, g) and (M′, g′) are said to be isospectral if spec(M, g) = spec(M′, g′).

The Laplace–Beltrami operator may be extended to act on smooth p-forms by Δ = dδ + δd, where δ is the adjoint of d and p is a positive integer. We call its eigenvalue spectrum the p-form spectrum.

The length spectrum of a Riemannian manifold is the set of lengths of smoothly closed geodesics, counted with multiplicity. The multiplicity of a length is defined as the number of distinct free homotopy classes that contain a closed geodesic of that length. We denote the length spectrum of (M, g) by [L]-spec(M, g). This is a natural notion, since the geodesic of shortest length in a free homotopy class is just the shortest loop representing that class. (Note that other definitions of multiplicity appear in the literature.)

1991 Mathematics Subject Classification. Primary 53C22, 58F17, 58G25; Secondary 53C30. Key words and phrases. Marked length spectrum, Length spectrum, Laplace spectrum, Laplace

spectrum on forms, Closed geodesics, Nilpotent Lie groups

Research at MSRI supported in part by NSF grant DMS-9022140. Research at MSRI and Texas Tech supported in part by NSF grant DMS-9409209.

Typeset by AMS-TEX

Two Riemannian manifolds (M_{1}, g_{1}) and (M_{2}, g_{2}) have the same marked length spectrum if there exists an isomorphism between the fundamental groups of M_{1} and M_{2} such that corresponding free homotopy classes contain smoothly closed geodesics of the same length. Clearly, manifolds with the same marked length spectrum necessarily have the same length spectrum.

The purpose of this paper is to study the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions and the Laplace spectrum on forms on Riemannian nilmanifolds.

The relationship between the Laplace spectrum and lengths of closed geodesics arises from the study of the wave equation (see [DGu], [GuU]), and in the case of compact, hyperbolic manifolds, from the Selberg Trace Formula (see [C], Chapter XI). Colin de Verdiere [CdV] has shown that generically, the Laplace spectrum determines the length spectrum. On Riemann surfaces, Huber showed that the length spectrum and the Laplace spectrum are equivalent notions (see [Bu] for an exposition).

The Poisson formula gives the relationship between the Laplace spectrum and length spectrum of flat tori, with the result that pairs of flat tori are isospectral if and only if they share the same length spectrum (see [CS], [G3]). Pesce [P2] has computed a Poisson-type formula relating the Laplace spectrum and length spectrum of Heisenberg manifolds, and has also shown that pairs of Heisenberg manifolds that are isospectral must have the same lengths of closed geodesics. Previously, Gordon [G1] exhibited the first examples of isospectral manifolds that do not have the same length spectrum. These Heisenberg manifolds have the same lengths of closed geodesics. However, the length spectra often differ in the multiplicities that occur. All known examples of manifolds that are isospectral have the same lengths of closed geodesics.

The marked length spectrum contains significantly more geometric information than the length spectrum. Croke [Cr] and Otal [Ot1], [Ot2] independently showed that if a pair of compact surfaces with negative curvature have the same marked length spectrum, they are necessarily isometric. The same is true for flat tori (see [G3]). Recently Eberlein [E1] showed that for two-step nilmanifolds, the same marked length spectrum implies the same Laplace spectrum both on functions and on p-forms for all p. (See Section 3 for more details.)

However, the standard sphere and the Zoll sphere (see [Bes]) have the same marked length spectrum (trivially so, as they are both simply connected and by definition have the same lengths of closed geodesics), yet they are not even isospectral on functions. Indeed, any manifold isospectral to a standard sphere of dimension less than or equal to six must be isometric to it (see [B2]). Examples of pairs of Riemannian manifolds that are isospectral on functions but not on forms are sparse. Most constructions for producing pairs of isospectral manifolds can be explained by Sunada’s method [S] or its generalizations [DG], [GW1], [B3]. Pairs of manifolds constructed by the Sunada techniques necessarily have the same p-form spectrum for all p.

For any choice of P ∈ Z^{+}, Ikeda [I2] has constructed examples of isospectral lens spaces that are isospectral on p-forms for p = 0, 1, ⋯, P but not isospectral on (P + 1)-forms. A straightforward argument shows that for the family of lens spaces considered by Ikeda, if a pair of lens spaces in this family has the same marked length spectrum, they are necessarily isometric. Gordon [G2] has constructed pairs of Heisenberg manifolds that are isospectral on functions, but not isospectral on one-forms. A consequence of Eberlein’s theorem is that Heisenberg manifolds with the same marked length spectrum are necessarily isometric. (See Section 3 for more details.) The only other known examples of manifolds that are isospectral on functions but not isospectral on forms are pairs of isospectral three-step nilmanifolds presented and studied in [Gt3]. These examples are studied further here.

This paper focuses almost exclusively on three-step nilmanifolds. The main results are a partial extension and a partial converse to Eberlein’s theorem for higher-step nilmanifolds.

Main Theorem 3.2.2. For a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions.

Main Example. Example V in the table below exhibits the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum, but not the same spectrum on one-forms.

These results have led to the following.

Conjecture. Pairs of Riemannian nilmanifolds with the same marked length spectrum are necessarily isospectral on functions.

Background ideas and notation are established and explained in Section 2. In [Gt3], we presented a new construction for producing pairs of isospectral nilmanifolds of arbitrary-step. In Section 3, this construction together with Eberlein’s theorem and techniques from Riemannian geometry are used to prove the Main Theorem.

Also in [Gt3], we presented new examples of isospectral three-step nilmanifolds with combinations of properties described in the table below. For consistency, the numbering of the examples in this paper coincides with the numbering of the examples in [Gt3]. Note that Example V is also the Main Example.

The spectrum on functions, spectrum on forms, quasi-regular representations, and fundamental groups of these examples were examined in [Gt3]. In Section 4 and 5 we compare the length spectrum and marked length spectrum of these examples. The pairs of isospectral manifolds described below have the same lengths of closed geodesics. However, the length spectra often differ in the multiplicities that occur.

Table I: New Examples of Isospectral Manifolds

Pair of 3-Step  ∀p Same Rep. Equiv. Isomorphic  Same  Same
Isospectral  p-form Fundamental Fundamental Length Marked Length
Nilmanifolds  Spectrum  Groups  Groups  Spectrum  Spectrum
I(7 dim)  Yes  Yes  No  No  No
II(5 dim)  Yes  Yes  Yes  Yes  No
III\IV(7\5 dim)  No  No  No  No  No
V(7 dim)  No  No  Yes  Yes  Yes

All of the examples described in the above table are of the form (Γ\G, g), where G is a three-step nilpotent Lie group, Γ is a cocompact, discrete subgroup of G (i.e. Γ\G compact) and g arises from a left invariant metric on G. Two cocompact, discrete subgroups Γ_{1} and Γ_{2} of a Lie group G are called representation equivalent if the associated quasi-regular representations are unitarily equivalent. If Γ_{1} and Γ_{2} are representation equivalent, then (Γ_{1}\G, g) and (Γ_{2}\G, g) are necessarily isospectral on functions and on smooth p-forms for any choice of left invariant metric g on G. Remark. Example I provided the first example of a pair of representation equivalent subgroups of a solvable Lie group producing nilmanifolds with unequal length spectra. This cannot happen in the two-step nilpotent case. The relationship between the quasi-regular representation and the length spectrum and marked length spectrum of nilmanifolds is studied in [Gt2], where we also present the first examples of pairs of representation equivalent subgroups of two-step nilpotent Lie groups that do not produce nilmanifolds with the same marked length spectrum. Example I is also the first example of a pair of nonisomorphic, representation equivalent subgroups of a solvable Lie group. See [Gt1] for more details. Note that nilpotent Lie groups are necessarily solvable.

Some of the contents of this paper are contained in the author’s thesis at Washington University in St. Louis in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The author wishes to express deep gratitude to her advisor, Carolyn S. Gordon, for all of her suggestions, encouragement, and support. The author also wishes to thank Patrick Eberlein for helpful conversations.

Section 2: Background and Notation.

Section 2.1: Definitions.

Let G be a simply connected Lie group with Lie algebra 𝔤. A metric on G is left invariant if left translations are isometries. Note that a left invariant metric is determined by a choice of orthonormal basis of the Lie algebra 𝔤 of G.

Let Γ be a a cocompact, discrete subgroup of G. A left invariant metric g on G descends to a Riemannian metric on Γ\G, which we also denote by g. This paper focuses exclusively on manifolds of the form (Γ\G, g), where g arises from a left invariant metric on G.

On manifolds of the form (Γ\G, g), the Laplace–Beltrami operator is

Δ = − ∑_{i=1}^{n} E_{i}^{2},

where {E_{1}, ⋯, E_{n}} is an orthonormal basis of the Lie algebra 𝔤 of G.

Recall that the free homotopy classes of a manifold Γ\G correspond to the conjugacy classes in Γ. We will denote by [γ]_{Γ} the free homotopy class of Γ\G represented by γ ∈ Γ. That is, [γ]_{Γ} = {γ̂γγ̂^{−1} : γ̂ ∈ Γ}.

We write λ ∈ [γ]_{Γ} if there exists a closed geodesic of length λ > 0 in the free homotopy class [γ]_{Γ} of (Γ\G, g).

Let γ be an element of Γ. We say a geodesic σ of (G, g) is translated by the element γ with period λ > 0 if

γσ(s) = σ(s + λ) ∀s ∈ R.

If σ is a unit speed geodesic, then σ projects to a closed geodesic on (Γ\G, g) of length λ, and σ is contained in the free homotopy class [γ]_{Γ}.

As the projection (G, g) → (Γ\G, g) is a Riemannian covering, all closed geodesics of (Γ\G, g) must arise in this fashion. So to study the closed geodesics of (Γ\G, g), it is enough to study the γ-translated geodesics of (G, g).

Let σ(s) be a geodesic of G through p = σ(0). Let σ̂(s) = p^{−1}σ(s). As left translations are isometries, σ̂ is a geodesic of G through e. If σ is translated by γ with period λ, then σ̂ is translated by (p^{−1}γp), also with period λ. To see this, note that if γσ(s) = σ(s + λ), then

(p^{−1}γp)σ̂(s) = (p^{−1}γp)p^{−1}σ(s) = p^{−1}γσ(s) = p^{−1}σ(s + λ) = σ̂(s + λ).

2.1.1 Notation. In summary, λ ∈ [γ]_{Γ} if and only if there exists x = p^{−1}γp ∈ [γ]_{G} and a unit speed geodesic σ(s) on (G, g) through e = σ(0) such that xσ(s) = σ(s + λ), ∀s ∈ R. That is, x translates σ with period λ. Here [γ]_{G} denotes the conjugacy class of γ in G.

With this notation, a pair of manifolds (Γ_{1}\G_{1}, g_{1}) and (Γ_{2}\G_{2}, g_{2}) share the same marked length spectrum if and only if there exists an isomorphism Φ : Γ_{1} → Γ_{2} such that for all γ ∈ Γ_{1},

λ ∈ [γ]_{Γ_{1}} if and only if λ ∈ [Φ(γ)]_{Γ_{2}}.

We say that the isomorphism Φ marks the length spectrum between (Γ_{1}\G_{1}, g_{1}) and (Γ_{2}\G_{2}, g_{2}).

Section 2.2 Nilmanifolds.

Let 𝔤 be a Lie algebra. We denote by 𝔤^{(1)} the derived algebra [𝔤, 𝔤] of 𝔤. That is, 𝔤^{(1)} is the Lie subalgebra of 𝔤 generated by all elements of the form [X, Y] for X, Y in 𝔤. Inductively, define 𝔤^{(k+1)} = [𝔤, 𝔤^{(k)}]. The Lie algebra 𝔤 is said to be k-step nilpotent if 𝔤^{(k)} ≡ 0 but 𝔤^{(k−1)} ≢ 0. A Lie group G is called k-step nilpotent if its Lie algebra is.

If G is a nilpotent Lie group with cocompact, discrete subgroup Γ, the locally homogeneous space Γ\G is called a nilmanifold. If G is an abelian Lie group, then Γ is merely a lattice of rank n in G, where n is the dimension of G. In this case, log Γ is also a lattice in 𝔤.

Let exp denote the Lie algebra exponential from 𝔤 to G. The Campbell-Baker-Hausdorff formula gives us the group operation of G in terms of 𝔤. Namely, for X, Y ∈ 𝔤 :

exp(X) exp(Y) = exp(X + Y + \frac{1}{2}[X, Y] + \frac{1}{12}[X, [X, Y]] + \frac{1}{12}[Y, [Y, X]] + ⋯),

where the remaining terms are higher-order brackets. Note that for two-step nilpotent Lie groups, only the first three terms in the right-hand side are nonzero. For three-step groups, only the first five terms are nonzero. If 𝔤 is nilpotent and G is simply connected, then exp is a diffeomorphism from 𝔤 onto G. Denote its inverse by log .

If G_{1} and G_{2} are nilpotent Lie groups with cocompact, discrete subgroups Γ_{1} and Γ_{2}, respectively, any abstract group isomorphism Φ : Γ_{1} → Γ_{2} lifts uniquely to a Lie group automorphism Φ : G_{1} → G_{2}.

For details of cocompact, discrete sugbroups of nilpotent Lie groups, see [Ra].

2.2.1 Definition. Let Φ be a Lie group automorphism of G. Let Γ be a cocompact, discrete subgroup of G.

(i) We call Φ an almost inner automorphism if for all elements x of G there exists a_{x} in G such that Φ(x) = a_{x}xa_{x}^{−1}.

(ii) We say Φ is a Γ-almost inner automorphism if for all elements γ of Γ there exists a_{γ} in G such that Φ(γ) = a_{γ}γa_{γ}^{−1}.

Denote by IA(G) (respectively, AIA(G), Γ-AIA(G) ) the group of inner automorphisms (respectively, almost inner automorphisms, Γ-almost inner automorphisms) of G. Note that IA(G) ⊂ AIA(G) ⊂ Γ-AIA(G).

2.2.2 Theorem (Gordon and Wilson, Gordon [GW1],[G1]). Let G be an exponential solvable Lie group, and let Γ_{1} and Γ_{2} be cocompact, discrete subgroups of G. Let Φ be a Γ_{1}-almost inner automorphism of G such that Φ(Γ_{1}) = Γ_{2}. Then (Γ_{1}\G, g) and (Γ_{2}\G, g) are isospectral on functions and on forms for any choice of left invariant metric g on G. Moreover, the automorphism Φ marks the length spectrum between (Γ_{1}\G, g) and (Γ_{2}\G, g).

Note that a nilpotent Lie group is necessarily exponential solvable.

Section 3: The Marked Length Spectrum vs. the Laplace Spectrum on Functions of Three-Step Nilmanifolds.

Throughout this section, G is a simply connected, k-step nilpotent Lie group, with Lie algebra 𝔤, Γ is a cocompact, discrete subgroup of G, and g is a left invariant metric on G which descends to a metric on Γ\G, also denoted by g. We denote the center of 𝔤 by 𝔷 and the center of G by Z(G). Let L_{x} denote left multiplication by x ∈ G. As g is left invariant, L_{x} is always an isometry of (G, g). Let G^{(k)} = exp(𝔤^{(k)}) denote the kth derived subgroup of G. Note that if G is k-step nilpotent, then G^{(k−1)} ⊂ Z(G).

§3.1 Preliminaries.

3.1.1 Theorem. Let G be a three-step nilpotent Lie group with left invariant metric g. Let σ be a geodesic on (G, g) that is translated by the element γ ∈ G with period λ > 0. Let p = σ(0). Then

⟨L_{p∗} ([log (p^{−1}γp) , 𝔤]) , σ̇(0)⟩_{p} ≡ 0.

Remark. This is the three-step generalization of a result due to Eberlein [E1]. Recently Dorothee Schueth [Sch] has given an elegant proof, which generalizes the result to nilpotent Lie groups of arbitrary step.

Outline of Proof of 3.1.1.

We briefly describe the basic steps in the original three-step proof. For details, see [Gt4], Chapter 4.

Let G be a simply connected, three-step nilpotent Lie group with Lie algebra 𝔤 and left invariant metric g. Let 𝔤 = ν ⊕ 𝔤^{(1)}, where ν is the orthogonal complement of 𝔤^{(1)} in 𝔤. Let 𝔤^{(1)} = ζ ⊕ 𝔤^{(2)}, where ζ is the orthogonal complement of 𝔤^{(2)} in 𝔤^{(1)}. Thus 𝔤 = ν ⊕ ζ ⊕ 𝔤^{(2)}.

Let {X_{1}, X_{2}, ⋯, X_{J}} be an orthonormal basis of ν. Let {Z_{1}, Z_{2}, ⋯, Z_{K}} be an orthonormal basis of ζ, and let {W_{1}, W_{2}, ⋯, W_{T} } be an orthonormal basis of 𝔤^{(2)}. Throughout this proof the indices i, j, and l run from 1 to J, the indices h and k run from 1 to K, and the indices t and r run from 1 to T.

Define A_{ij}^{k}, B_{ij}^{t}, C_{ik}^{t} by

[X_{i}, X_{j}] = ∑_{k} A_{ij}^{k}Z_{k} + ∑_{t} B_{ij}^{t}W_{t}
[X_{i}, Z_{k}] = −[Z_{k}, X_{i}] = ∑_{t} C_{ik}^{t}W_{t}.

As [X_{i}, X_{j}] = −[X_{j}, X_{i}], we have A_{ij}^{k} = −A_{ji}^{k} and B_{ij}^{t} = −B_{ji}^{t}. By the Jacobi equation [𝔤^{(1)}, 𝔤^{(1)}] ⊂ [𝔤, 𝔤^{(2)}] ≡ 0. Thus [Z_{k}, Z_{h}] = 0. Finally, by applying the Jacobi equation to X_{i}, X_{j}, X_{k} and examining the W_{t} coefficient, we obtain:

0 = ∑_{k} (A_{jl}^{k}C_{ik}^{t} + A_{ij}^{k}C_{lk}^{t} + A_{li}^{k}C_{jk}^{t}) .

For Lie algebras with a left invariant metric, the covariant derivatives can be calculated via

< ∇_{V} Y, U >= \frac{1}{2} < [U, V], Y > +\frac{1}{2} < [U, Y], V > +\frac{1}{2} < [V, Y], U > for U, V, Y in 𝔤. We obtain the covariant derivatives:

\begin{aligned}∇_{X_{i}}X_{j} & = \frac{1}{2} ∑_{k} A_{ij}^{k}Z_{k} + \frac{1}{2} ∑_{t} B_{ij}^{t}W_{t}, \\ ∇_{X_{i}}Z_{k} & = \frac{1}{2} ∑_{j} A_{ji}^{k}X_{j} + \frac{1}{2} ∑_{t} C_{ik}^{t}W_{t}, \\ ∇_{Z_{k}}X_{i} & = \frac{1}{2} ∑_{j} A_{ji}^{k}X_{j} − \frac{1}{2} ∑_{t} C_{ik}^{t}W_{t}, \\ ∇_{X_{i}}W_{t} & = ∇_{W_{t}}X_{i} = \frac{1}{2} ∑_{j} B_{ji}^{t}X_{j} − \frac{1}{2} ∑_{k} C_{ik}^{t}Z_{k}, \\ ∇_{Z_{k}}Z_{h} & = ∇_{W_{t}}W_{r} = 0, \\ ∇_{Z_{k}}W_{t} & = ∇_{W_{t}}Z_{k} = \frac{1}{2} ∑_{j} C_{jk}^{t}X_{j}. \end{aligned}

For x ∈ G, x = exp (∑_{j} x_{j}X_{j} + ∑_{k} z_{k}Z_{k} + ∑_{t} w_{t}W_{t}) gives us a global coordinate system on G. With this coordinate system, a straightforward computation shows us that

X_{j} = \frac{∂}{∂x_{j}} + ∑_{k} (\frac{1}{2} ∑_{i} x_{i}A_{ij}^{k}) \frac{∂}{∂z_{k}}

+ ∑_{t} (\frac{1}{2} ∑_{i} x_{i}B_{ij}^{t} − \frac{1}{2} ∑_{k} C_{jk}^{t}z_{k} + \frac{1}{12} ∑_{i,l,k} x_{i}C_{ik}^{t}x_{l}A_{lj}^{k}) \frac{∂}{∂w_{t}} ,

Z_{k} = \frac{∂}{∂z_{k}} + ∑_{t} (\frac{1}{2} ∑_{i} x_{i}C_{ik}^{t}) \frac{∂}{∂w_{t}} ,

W_{t} = \frac{∂}{∂w_{t}} .

Let σ(s) = exp (∑_{j} x_{j}(s)X_{j} + ∑_{k} z_{k}(s)Z_{k} + ∑_{t} w_{t}(s)W_{t}) be a geodesic of (G, g) with initial velocity σ̇(0) = ∑_{j} x̄_{j}X_{j} +∑_{k} z̄_{k}Z_{k} +∑_{t} w̄_{t}W_{t}. A straightforward computation of ∇_{σ̇(s)}σ̇(s) ≡ 0 produces the following geodesic equations for a three-step nilpotent Lie group, reduced to a system of n-ordinary differential equations.

ẋ_{j}(s) = − ∑_{l,k} x_{l}(s)A_{jl}^{k}z̄_{k} − ∑_{l,t} x_{l}(s)B_{jl}^{t}w̄_{t} − ∑_{k,t} z_{k}(s)C_{jk}^{t}w̄_{t}

− \frac{1}{2} ∑_{i,l,k,t} x_{i}(s)x_{l}(s)w̄_{t}C_{ik}^{t}A_{jl}^{k} + x̄_{j}

ż_{k}(s) = \frac{1}{2} ∑_{i,j} x_{i}(s)ẋ_{j}(s)A_{ij}^{k} + ∑_{j,t} x_{j}(s)w̄_{t}C_{jk}^{t} + z̄_{k}

ẇ_{t}(s) = \frac{1}{2} ∑_{i,j} x_{i}(s)ẋ_{j}(s)B_{ij}^{t} − \frac{1}{2} ∑_{j,k} ẋ_{j}(s)z_{k}(s)C_{jk}^{t} + \frac{1}{2} ∑_{j,k} x_{j}(s)ż_{k}(s)C_{jk}^{t}

− \frac{1}{6} ∑_{i,j,k,l} x_{i}(s)ẋ_{j}(s)x_{l}(s)C_{ik}^{t}A_{lj}^{k} + w̄_{t}

If we assume that a geodesic σ(s) starts at the identity and is translated by the element γ, then a lengthy but straightforward (brute-force) calculation yields

⟨[log(γ), 𝔤] , σ̇(0)⟩_{e} ≡ 0.

Here one uses the extensively the fact that if γσ(s) = σ(s + λ), then L_{γ∗} (σ̇(s)) = σ̇(s + λ).

In the general case, let σ(s) be a geodesic of G through p = σ(0). Let α(s) = p^{−1}σ(s). Then α is a geodesic of G through e. If σ is translated by γ with period λ, then α is translated by p^{−1}γp, also with period λ. Thus

⟨[log (p^{−1}γp) , 𝔤] , α̇(0)⟩_{e} ≡ 0.

But α̇(0) = (L_{p^{−1}})_{∗} (σ̇(0)) . As our metric is left invariant, we obtain

⟨L_{p∗} ([log (p^{−1}γp) , 𝔤]) , σ̇(0)⟩_{p} = 0,

as desired.

Remark. Ron Karidi [K] has recently given a formulation of the geodesic equations for an arbitrary nilpotent Lie group with a left invariant metric. As above, this formulation is in terms of an orthonormal basis and structure constants of the Lie algebra.

3.1.2 Notation. Let π denote the projection from G onto Ḡ = G/G^{(k−1)}. For Γ a cocompact, discrete subgroup of G, denote by Γ̄ the image of Γ under the canonical projection from G onto Ḡ. The group Γ̄ is then a cocompact, discrete subgroup of Ḡ. Let ḡ denote the metric on Ḡ defined by restricting the left invariant metric g to an orthogonal complement of 𝔤^{(k−1)} ⊂ 𝔷, where 𝔤 is the Lie algebra of G. With this choice of metric ḡ on Ḡ, the mapping

π : (G, g) → (Ḡ, ḡ)

is a Riemannian submersion with totally geodesic fibers.

If Φ : G_{1} → G_{2} is a Lie group mapping, then necessarily Φ : G_{1}^{(k−1)} → G_{2}^{(k−1)}. Let Φ̄ denote the canonical projection of Φ onto Φ̄ = π ∘ Φ : Ḡ_{1} → Ḡ_{2}.

The Lie algebra of Ḡ is 𝔤̄ = 𝔤/𝔤^{(k−1)}. We denote elements of 𝔤̄ by Ū where Ū is the image of U ∈ 𝔤 under the canonical projection from 𝔤 onto Ḡ. Similarly, we will denote elements of Ḡ by x̄ where x̄ is the image of x ∈ G under the canonical projection from G onto Ḡ.

All of the nilpotent Lie groups studied here have the following property.

3.1.3 Definition. Let G be a simply connected, k-step nilpotent Lie group. We say G is strictly nonsingular if the following property holds: for all z in Z(G) and for all noncentral x in G there exists a in G such that

[a, x] = z.

Here [a, x] = axa^{−1}x^{−1}. Equivalently, the Lie algebra 𝔤 is strictly nonsingular if for all noncentral X in 𝔤,

𝔷 ⊂ ad(X)(𝔤).

That is, for all X in 𝔤 − 𝔷 and all Z in 𝔷 there exists Y in 𝔤 such that [X, Y] = Z.

Note that for strictly nonsingular nilpotent Lie algebras, 𝔷 = 𝔤^{(k−1)}.

3.1.4 Corollary. Let G be a simply connected, strictly nonsingular three-step nilpotent Lie group with left invariant metric g. Consider the Riemannian submersion (G, g) → (Ḡ, ḡ). If σ is a geodesic on G such that γσ(s) = σ(s + λ) for some noncentral γ in G and some λ > 0, then σ is a horizontal geodesic. That is,

⟨L_{σ(s)∗}(𝔷), σ̇(s)⟩ ≡ 0 ∀s ∈ R.

Before proving Corollary 3.1.4, recall the following properties of Riemannian submersions.

3.1.5 Proposition (see [GHL]). Let (M, g) → (M̄, ḡ) be a Riemannian submersion.

(i) Let α be a geodesic of (M, g). If the vector α̇(0) is horizontal, then α̇(s) is horizontal for all s, and the curve π ∘ α is a geodesic of (M̄, ḡ) of the same length as σ.

(ii) Conversely, let p ∈ M and let σ be a geodesic of (M̄, ḡ) with σ(0) = π(p). Then there exists a unique local horizontal lift σ̂ of σ through p = σ̂(0), and σ̂ is also a geodesic of (M, g).

Proof of Corollary 3.1.4.

By Theorem 3.1.1

⟨L_{p∗} ([log(p^{−1}γp), 𝔤]) , σ̇(0)⟩_{p} ≡ 0,

where p = σ(0). By strict nonsingularity

𝔷 = 𝔤^{(2)} ⊂ [log(p^{−1}γp), 𝔤] .

Thus

⟨L_{p∗}(𝔷), σ̇(0)⟩_{p} ≡ 0.

Thus σ̇(0) is horizontal. By Proposition 3.1.5, we know that σ̇(s) is horizontal for all s ∈ R.

§3.2 Main Theorem.

On two-step nilmanifolds, we have the following relationship betweeen the marked length spectrum and the p-form spectrum.

3.2.1 Theorem (Eberlein [E]). Let Γ_{1}, Γ_{2} be cocompact, discrete subgroups of simply connected, two-step nilpotent Lie groups G_{1}, G_{2} with left invariant metrics g_{1}, g_{2} respectively. Assume that (Γ_{1}\G_{1}, g_{1}) and (Γ_{2}\G_{2}, g_{2}) have the same marked length spectrum, and let Φ : Γ_{1} → Γ_{2} be an isomorphism inducing this marking. Then Φ = (Φ_{1} ∘ Φ_{2})|_{Γ_{1}}, where Φ_{2} is a Γ_{1}-almost-inner automorphism of G_{1}, and Φ_{1} is an isomorphism of (G_{1}, g_{1}) onto (G_{2}, g_{2}) that is also an isometry. Moreover, this factorization is unique. In particular, (Γ_{1}\G_{1}, g_{1}) and (Γ_{2}\G_{2}, g_{2}) have the same spectrum of the Laplacian on functions and on p-forms for all p.

Remark. Note that if Γ-AIA(G) = IA(G), then the elements of Γ-AIA(G) are isometries of (G, g), where g is any choice of left invariant metric g of G. So by Theorem 3.2.1, any two-step nilmanifold with the same marked length spectrum as (Γ\G, g) is necessarily isometric to it. Note that this property applies to Heisenberg groups. Thus pairs of Heisenberg manifolds with the same marked length spectrum are necessarily isometric.

We may now state the main result of this paper.

3.2.2 Main Theorem. Let G be a simply connected, strictly nonsingular, threestep nilpotent Lie group. Let Γ_{1} and Γ_{2} be cocompact, discrete subgroups of G such that Γ_{1} ∩Z(G) = Γ_{2} ∩Z(G). If (Γ_{1}\G, g) and (Γ_{2}\G, g) have the same marked length spectrum, then (Γ_{1}\G, g) and (Γ_{2}\G, g) are isospectral on functions.

To prove Theorem 3.2.2, we need the following.

3.2.3 Theorem [Gt3, Theorem 3.2]. Let G be a simply connected, strictly nonsingular nilpotent Lie group with left invariant metric g. If Γ_{1} and Γ_{2} are cocompact, discrete subgroups of G such that

Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) and spec (Γ̄_{1}\Ḡ, ḡ) = spec (Γ̄_{2}\Ḡ, ḡ) ,

then

spec (Γ_{1}\G, g) = spec (Γ_{2}\G, g) .

3.2.4 Theorem. Let G be a simply connected, strictly nonsingular three-step nilpotent Lie group with cocompact, discrete subgroup Γ and left invariant metric g. Let γ be a noncentral element of Γ. Then we have the following condition:

λ ∈ [γ]_{Γ} if and only if λ ∈ [π(γ)]Γ̄.

Assume for the moment that Theorem 3.2.4 is true.

3.2.5 Corollary. Let G_{1} and G_{2} be simply connected, strictly nonsingular, threestep nilpotent Lie groups with cocompact, discrete subgroups Γ_{1} and Γ_{2} and left invariant metrics g_{1} and g_{2}, respectively. Let Φ mark the length spectrum between (Γ_{1}\G_{1}, g_{1}) and (Γ_{2}\G_{2}, g_{2}). Then Φ̄ must mark the length spectrum between (Γ̄_{1}\Ḡ_{1}, ḡ_{1}) and (Γ̄_{2}\Ḡ_{2}, ḡ_{2}).

Proof of Corollary 3.2.5.

Let λ ∈ [π(γ)]Γ̄_{1}, π(γ) ≠ 0. By (3.2.4) λ ∈ [γ]_{Γ_{1}}. By hypothesis λ ∈ [Φ(γ)]_{Γ_{2}}. By (3.2.4) again λ ∈ [π(Φ(γ))]Γ̄_{2} = [Φ̄(π(γ))]Γ̄_{2}.

Reversing the roles of Γ̄_{1} and Γ̄_{2}, we obtain the desired result.

Proof of Main Theorem 3.2.2.

Let Φ mark the length spectrum between (Γ_{1}\G, g) and (Γ_{2}\G, g). By (3.2.5) we know that Φ̄ must mark the length spectrum between (Γ̄_{1}\Ḡ, ḡ) and (Γ̄_{2}\Ḡ, ḡ) . By Theorem 3.2.1 spec(Γ̄_{1}\Ḡ, ḡ) = spec(Γ̄_{2}\Ḡ, ḡ).

The result now follows directly from Theorem 3.2.3.

It remains only to prove Theorem 3.2.4, which follows directly from the following two lemmas.

3.2.6 Lemma. Let G be a simply connected, strictly nonsingular three-step nilpotent Lie group with cocompact, discrete subgroup Γ and left invariant metric g. Let γ be a noncentral element of Γ. With the above notation, if λ ∈ [γ]_{Γ} then λ ∈ [π(γ)]Γ̄.

Proof of Lemma 3.2.6.

If λ ∈ [γ]_{Γ}, then there exists a unit speed geodesic σ(s) of G through e such that

p^{−1}γpσ(s) = σ(s + λ)

for some p ∈ G.

By (3.1.4), σ(s) is a horizontal geodesic, and by (3.1.5), π ∘ σ(s) is a unit speed geodesic of (Ḡ, ḡ).

But π(p^{−1}γpσ(s)) = π(p^{−1})π(γ)π(p)π(σ(s)) = π(σ(s + λ)). Thus π(σ) is a unit speed geodesic translated by π(p^{−1})π(γ)π(p) with period λ. That is, λ ∈ [π(γ)]Γ̄, as desired.

3.2.7 Lemma. For G a simply connected, strictly nonsingular k-step nilpotent Lie group. Using the above notation, let λ ∈ [γ̄]Γ̄, where γ̄ ≠ e. Then λ ∈ [γ]_{Γ} for all γ ∈ π^{−1}(γ̄).

Proof of Lemma 3.2.7.

Let σ be a unit speed geodesic of (Ḡ, ḡ) through ē = σ(0) and translated by p^{−1}γ̄p for some p ∈ Ḡ.

By (3.1.5), the unique horizontal lift σ̂ of σ with σ̂(0) = e is a geodesic of (G, g). As both G and Ḡ are complete, we see that σ̂ is defined for all s ∈ R. We

also have π ∘ σ̂(s) = σ(s) for all s ∈ R. To see this, note that the set S of all such s is nonempty as 0 ∈ S, open by completeness, and closed by uniqueness and smoothness. Thus, S = R.

Now π(σ̂(λ)) = p^{−1}γ̄p. Let p̂ be such that π(p̂) = p.

Let γ ∈ π^{−1}(γ̄). Then π(p̂^{−1}γp̂) = p^{−1}γ̄p = π(σ̂(λ)). Thus (σ̂(λ))(p̂^{−1}γp̂)^{−1} is a central element of G.

By strict nonsingularity, there exists x ∈ G such that

x^{−1}(p̂^{−1}γp̂)x(p̂^{−1}γp̂)^{−1} = σ̂(λ)(p̂^{−1}γp̂)^{−1},

that is x^{−1}(p̂^{−1}γp̂)x = σ̂(λ).

If we let p′ = p̂x, then σ̂(λ) = p′^{−1}γp′. Note that π(p′^{−1}γp′) = π(σ̂(s)) = p^{−1}γ̄p. We now show that p′^{−1}γp′σ̂(s) = σ̂(s + λ) for all s ∈ R. Let

α(s) = (p′^{−1}γp′)^{−1}σ̂(s + λ).

Now α(0) = (p′^{−1}γp′)^{−1}σ̂(λ) = e. Also, α(s) is horizontal since g is left invariant and α is just a left translate of the horizontal curve σ̂. Moreover,

π(α(s)) = π((p′^{−1}γp′)^{−1}σ̂(s + λ)) = p^{−1}γ̄^{−1}pσ(s + λ)
= p^{−1}γ̄^{−1}pp^{−1}γ̄pσ(s) = σ(s).

Thus α is a horizontal geodesic through e ∈ G whose projection agrees with σ. By uniqueness in Proposition 3.1.5, α(s) = σ̂(s) ∀s ∈ R.

Consequently,

p′^{−1}γp′σ̂(s) = σ̂(s + λ)

for all s ∈ R. Thus

λ ∈ [γ]_{Γ},

as desired.

§3.3 Three-step Nilmanifolds with a One-Dimensional Center.

3.3.1 Theorem. Let G be a simply connected, strictly nonsingular, three-step nilpotent Lie group with a one-dimensional center. Let Γ_{1} and Γ_{2} be cocompact, discrete subgroups of G such that Γ_{1}∩Z(G) = Γ_{2}∩Z(G). Let g be any left invariant metric on G. Then (Γ_{1}\G, g) and (Γ_{2}\G, g) have the same marked length spectrum if and only if there exists an isomorphism Φ : Γ_{1} → Γ_{2} such that Φ̄ : Γ̄_{1} → Γ̄_{2} marks the length spectrum between (Γ̄_{1}\Ḡ, ḡ) and (Γ̄_{2}\Ḡ, ḡ).

Proof of Theorem 3.3.1.

The forward direction follows immediately from Corollary 3.2.5.

For the converse direction, assume that there exists an isomorphism Φ : Γ_{1} → Γ_{2} such that Φ̄ marks the length spectrum between (Γ̄_{1}\Ḡ, ḡ) and (Γ̄_{2}\Ḡ, ḡ).

We need to show that for all γ ∈ Γ_{1}, λ ∈ [γ]_{Γ_{1}} if and only if λ ∈ [Φ(γ)]_{Γ_{2}}.

We consider two cases:

Case 1: γ ∈ Γ_{1} ∩ Z(G).

If λ ∈ [γ]_{Γ_{1}}, then there exists a geodesic σ(s) of G such that γσ(s) = σ(s + λ). As Φ is an isomorphism, we know that Φ(Γ_{1} ∩ Z(G)) = Γ_{2} ∩ Z(G) = Γ_{1} ∩ Z(G),

and hence, Φ must map a generator of Γ_{1} ∩ Z(G) into a generator of Γ_{1} ∩ Z(G).

There are only two such generators. Thus for all γ ∈ Γ_{1} ∩ Z(G), either Φ(γ) = γ or Φ(γ) = γ^{−1}.

Hence [Φ(γ)]_{Γ_{2}} = [γ]_{Γ_{2}} or [Φ(γ)]_{Γ_{2}} = [γ^{−1}]_{Γ_{2}}.

If [Φ(γ)]_{Γ_{2}} = [γ]_{Γ_{2}}, then the geodesic σ(s) of G projects to a closed geodesic of (Γ_{2}\G, g) of length λ in the free homotopy class [γ]_{Γ_{2}}.

If [Φ(γ)]_{Γ_{2}} = [γ^{−1}]_{Γ_{2}}, then the geodesic α(s) = σ(−s) of G projects to a closed geodesic of (Γ_{2}\G, g) of length λ in [γ^{−1}]_{Γ_{2}}.

This argument also works for Φ^{−1} : Γ_{2} → Γ_{1}, which must necessarily mark the length spectrum. Consequently, for all γ ∈ Γ_{1} ∩ Z(G),

λ ∈ [γ]_{Γ_{1}} if and only if λ ∈ [Φ(γ)]_{Γ_{2}}.

Case 2: γ ∉ Z(G)

Let λ ∈ [γ]_{Γ_{1}}. By strict nonsingularity and Theorem 3.2.4, we know that λ ∈ [π(γ)]Γ̄_{1}. By assumption (ii), we know that λ ∈ [Φ̄(π(γ))]Γ̄_{2}. Now π(Φ(γ)) = Φ̄(π(γ)). Thus by Theorem 3.2.4 again we know λ ∈ [Φ(γ)]_{Γ_{2}}. Reversing the roles of Γ_{1} and Γ_{2} in the above, we see that for all γ ∈ Γ_{1} γ ∉ Γ_{1},

λ ∈ [γ]_{Γ_{1}} if and only if λ ∈ [Φ(γ)]_{Γ_{2}},

as desired.

§4 The Marked Length Spectrum vs. the One-Form Spectrum.

The example below is the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum, but not the same spectrum on one-forms. Outside of the standard vs. Zoll spheres, which are not even isospectral for dimension less than or equal to six, this is the only example of a pair of Riemannian manifolds that have the same marked length spectrum but not the same spectrum on one-forms.

Example V.

We use the notation of Section 3.

Consider the simply connected, strictly nonsingular, three-step nilpotent Lie group G with Lie algebra

𝔤 = span_{R}{X_{1}, X_{2}, Y_{1}, Y_{2}, Z_{1}, Z_{2}, W }

and Lie brackets

[X_{1}, Y_{1}] = [X_{2}, Y_{2}] = Z_{1}
[X_{1}, Y_{2}] = Z_{2}
[X_{1}, Z_{1}] = [X_{2}, Z_{2}] = [Y_{1}, Y_{2}] = W

and all other basis brackets zero.

We fix a left invariant metric on G by letting {E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6}, E_{7}} be an orthonormal basis of 𝔤 where

E_{1} = X_{1} − \frac{1}{2}X_{2} − \frac{1}{4}Y_{2},
E_{2} = X_{2} − \frac{1}{4}Y_{1},
E_{3} = Y_{1},
E_{4} = Y_{1} + Y_{2},
E_{5} = Z_{1},
E_{6} = \frac{1}{2}Z_{1} + Z_{2},
E_{7} = W.

Let Φ be the automorphism of G defined on the Lie algebra level by

X_{1} → −X_{1} + X_{2} + \frac{1}{4}Y_{1} + \frac{1}{2}Y_{2},
X_{2} → X_{2} − \frac{1}{2}Y_{1} + \frac{1}{4}Z_{1},
Y_{1} → −Y_{1},
Y_{2} → 2Y_{1} + Y_{2} + Z_{2},
Z_{1} → Z_{1} + \frac{1}{2}W,
Z_{2} → −Z_{1} − Z_{2} + \frac{1}{4}W,
W → −W.

A straightforward calculation shows that Φ_{∗}([U, V]) = [Φ_{∗}(U), Φ_{∗}(V)] for all U, V in 𝔤. Thus Φ is indeed a Lie group automorphism.

Let Γ_{1} be the cocompact, discrete subgroup of G generated by

{exp(2X_{1}), exp(2X_{2}), exp(Y_{1}), exp(Y_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)},

and let Γ_{2} = Φ(Γ_{1}). Note that Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G) = {exp(jW) : j ∈ Z}.

Let Φ̄ be the projection of Φ onto Ḡ. Then Φ̄ factors as Φ̄ = Ψ_{1} ∘ Ψ_{2} where Ψ_{1} is the automorphism of Ḡ given on the Lie algebra level by

X̄_{1} → −X̄_{1} + X̄_{2} + \frac{1}{4}Ȳ_{1} + \frac{1}{2}Ȳ_{2},
X̄_{2} → X̄_{2} − \frac{1}{2}Ȳ_{1},
Ȳ_{1} → −Ȳ_{1},
Ȳ_{2} → 2Ȳ_{1} + Ȳ_{2},
Z̄_{1} → Z̄_{1},
Z̄_{2} → −Z̄_{1} − Z̄_{2},

and Ψ_{2} is the automorphism of Ḡ given on the Lie algebra level by

X̄_{1} → X̄_{1},
X̄_{2} → X̄_{2} + \frac{1}{4}Z̄_{1},
Ȳ_{1} → Ȳ_{1},
Ȳ_{2} → Ȳ_{2} − Z̄_{1} − Z̄_{2},
Z̄_{1} → Z̄_{1},
Z̄_{2} → Z̄_{2}.

By rewriting Ψ_{1} in terms of the orthonormal basis {Ē_{1}, Ē_{2}, Ē_{3}, Ē_{4}, Ē_{5}, Ē_{6}} of ḡ, one easily sees that Ψ_{1}(Ē_{i}) = ±Ē_{i} for i = 1, …, 6. Thus the automorphism Ψ_{1} is also an isometry of Γ̄. A simple calculation shows that Ψ_{2} is an almost inner automorphism of Ḡ. Thus by (3.2.1), Φ̄ marks the length spectrum between (Γ̄_{1}\Ḡ, ḡ) and (Γ̄_{2}\Ḡ, ḡ). By (3.3.1), Φ marks the length spectrum between (Γ_{1}\G, g) and (Γ_{2}\G, g).

By (3.2.2), (Γ_{1}\G, g) and (Γ_{2}\G, g) must be isospectral on functions.

In contrast, we have the following.

4.1 Theorem [Gt3, Proposition 4.11]. The manifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) are not isospectral on one-forms.

§5 The (Marked) Length Spectrum and Previous Examples.

We now compare the length spectra and marked length spectra of Examples I-IV described in Table I. The spectrum on functions, spectrum on one-forms, quasiregular representations and fundamental groups of these examples were studied in [Gt3].

We use the notation of Section 3.

All of these examples are described by Theorem 3.2.3. In particular, Examples I-IV have the property Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G).

Let λ ∈ [L]-spec(Γ_{i}\G, g). Let m_{i}(λ) denote the multiplicity of λ in [L]-spec(Γ_{i}\G, g). We decompose m_{i}(λ) as

(5.1)  m_{i}(λ) = m′_{i}(λ) + m′′_{i}(λ)

where m′′_{i}(λ) is the number of central free homotopy classes in which λ occurs, and m′_{i}(λ) is the number of noncentral free homotopy classes in which λ occurs.

5.2 Proposition. For pairs of isospectral manifolds constructed using Theorem 3.2.3, the central multiplicities are equal; that is, m′′_{1}(λ) = m′′_{2}(λ).

Proof of Proposition 5.2.

If γ ∈ Γ_{1} ∩ Z(G) = Γ_{2} ∩ Z(G), then by (2.1.1), λ ∈ [γ]_{Γ_{1}} if and only if λ ∈ [γ]_{Γ_{2}}. As the conjugacy classes of γ in Γ_{1} and Γ_{2} respectively contain only the element γ, we have a natural correspondence between the central conjugacy classes in Γ_{1} containing a closed geodesic of length λ and the central conjugacy classes in Γ_{2} containing a closed geodesic of length λ.

Thus, for the examples below, we need only compare m′_{1}(λ) and m′_{2}(λ).

Example I: Remarks.

Let

𝔤 = span_{R}{X_{1}, X_{2}, Y_{1}, Y_{2}, Z_{1}, Z_{2}, W }

with Lie brackets

[X_{1}, Y_{1}] = [X_{2}, Y_{2}] = Z_{1}
[X_{1}, Y_{2}] = Z_{2}
[X_{1}, Z_{1}] = [X_{2}, Z_{2}] = [Y_{1}, Y_{2}] = W

and all other basis brackets zero.

Clearly 𝔤 is a strictly nonsingular, three-step nilpotent Lie algebra.

Let Γ_{1} be the cocompact, discrete subgroup of G generated by

{exp(2X_{1}), exp(2X_{2}), exp(Y_{1}), exp(Y_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)},

and let Γ_{2} be the cocompact, discrete subgroup of G generated by

\{exp(2X_{1}), exp(2X_{2}), exp(Y_{1}), exp(Y_{2} + \frac{1}{2}Z_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)\}.

The fundamental groups and the quasi-regular representations of Example I are studied extensively in [Gt1]. There we showed that Γ_{1} and Γ_{2} are not abstractly isomorphic, hence (Γ_{1}\G, g) and (Γ_{2}\G, g) cannot possibly have the same marked length spectrum for any choice of left invariant metric.

Let g be the left invariant metric on G defined by letting

{X_{1}, X_{2}, Y_{1}, Y_{2}, Z_{1}, Z_{2}, W }

be an orthonormal basis of 𝔤.

In [Gt2], we showed that (Γ_{1}\G, g) and (Γ_{2}\G, g) do not even have the same length spectrum. Although the same lengths of closed geodesics occur, the multiplicities of certain lengths differ.

Example I provided the first example of a pair of representation equivalent subgroups of a solvable Lie group producing manifolds with unequal length spectra. Note that nilpotent Lie groups are necessarily solvable.

Example II: The (Marked) Length Spectrum.

Let

𝔤 = span_{R}{X_{1}, Y_{1}, Y_{2}, Z, W }

with Lie brackets given by

[X_{1}, Y_{1}] = Z
[X_{1}, Z] = [Y_{1}, Y_{2}] = W

and all other basis brackets zero.

Clearly 𝔤 is a strictly nonsingular, three-step nilpotent Lie algebra.

Let Γ_{1} be the cocompact, discrete subgroup of G generated by

{exp(2X_{1}), exp(Y_{1}), exp(Y_{2}), exp(Z), exp(W)}

and let Γ_{2} be the cocompact, discrete subgroup of G generated by

\{exp(2X_{1}), exp(Y_{1} + \frac{1}{2}Z), exp(Y_{2}), exp(Z), exp(W)\}.

Note that these generating sets are canonical in the sense that every element of Γ_{1} can be written in the form exp(2n_{1}X_{1}) exp(m_{1}Y_{1}) exp(m_{2}Y_{2}) exp(kZ) exp(jW) for some integers n_{1}, m_{1}, m_{2}, k, j. Likewise for Γ_{2}.

5.3 Proposition. The above nilmanifolds have the same length spectrum, that is

[L]-spec(Γ_{1}\G, g) = [L]-spec(Γ_{2}\G, g)

for any choice of left invariant metric g of G.

We showed in [Gt3] that Γ_{1} and Γ_{2} are isomorphic as groups. Thus a natural question to ask is, if a pair of nilmanifolds have the same length spectrum and have isomorphic fundamental groups, must they necessarily have the same marked length spectrum? We know already from [Gt2] that this need not be true even in the two-step case. This example is a higher-step example with the same property.

5.4 Proposition. The manifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) do not have the same marked length spectrum for any choice of left invariant metric g on G.

Proof of Proposition 5.4.

Let g be any left invariant metric on G, and assume Ψ : Γ_{1} → Γ_{2} marks the length spectrum between (Γ_{1}\G, g) and (Γ_{2}\G, g). Extend Ψ to the Lie group isomorphism Ψ : G → G such that Ψ(Γ_{1}) = Γ_{2}.

We showed in [Gt3] Proposition 4.6 that any isomorphism Ψ : Γ_{1} → Γ_{2} must be given at the Lie algebra level by:

Ψ_{∗}(W) = ±W,

Ψ_{∗}(Z) = ±Z + h_{0}W

Ψ_{∗}(Y_{2}) = ±Y_{2} mod 𝔤^{(1)}

Ψ_{∗}(Y_{1}) = ±(Y_{1} + \frac{1}{2}Z) + h_{1}Y_{2} + h_{2}Z mod 𝔤^{(2)}

Ψ_{∗}(X_{1}) = ±X_{1} + \frac{1}{2}h_{3}Y_{1} + \frac{1}{2}h_{4}Y_{2} mod 𝔤^{(1)} where h_{0}, h_{1}, h_{2}, h_{3} and h_{4} are integers and h_{3}^{2} + h_{4}^{2} ≠ 0.

By Corollary 3.2.5 and Theorem 3.2.1, Ψ̄ = Φ_{1} ∘ Φ_{2}, where Φ_{1} : Ḡ → Ḡ is an isomorphism that is also an isometry of (Ḡ, ḡ), and Φ_{2} ∈ Γ̄_{1}-AIA(Ḡ). As Ȳ_{1} and Ȳ_{2} are not in [X̄_{1}, 𝔤̄], we must have

Φ_{1}_{∗}(X̄_{1}) = ±X̄_{1} + \frac{1}{2}h_{3}Ȳ_{1} + \frac{1}{2}h_{4}Ȳ_{2} + z_{1}Z̄,

Φ_{1}_{∗}(Ȳ_{1}) = ±Ȳ_{1} + h_{1}Ȳ_{2} + z_{2}Z̄,

Φ_{1}_{∗}(Ȳ_{2}) = ±Ȳ_{2} + z_{3}Z̄,

Φ_{1}_{∗}(Z̄) = ±Z̄, for some z_{1}, z_{2}, z_{3} ∈ R.

Now Φ_{1} an isometry implies that for all Ū, V̄ in 𝔤̄,

(∗)  ⟨Ū, V̄ ⟩ = ⟨Φ_{1}_{∗}(Ū), Φ_{1}_{∗}(V̄)⟩ .

Letting Ū = Z̄ and V̄ = Ȳ_{2} in (∗), we see that z_{3} = 0. Letting Ū = Ȳ_{2} and V̄ = h_{1}Ȳ_{2}+z_{2}Z̄ in (∗), we obtain h_{1} = z_{2} = 0. Finally by letting Ū = \frac{1}{2}h_{3}Ȳ_{1}+\frac{1}{2}h_{4}Ȳ_{2}+z_{1}Z̄ and V̄ = X_{1} in (∗) we see that z_{1} = h_{2} = h_{4} = 0, which contradicts h_{3}^{2} + h_{4}^{2} ≠ 0.

Before proving Proposition 5.3, we need the following.

5.5 Proposition (see [Gt3,Proposition 2.1]). Let Γ_{1} and Γ_{2} be cocompact, discrete subgroups of the Lie group G with left invariant metric g. If for each x in G we have

# { [γ]_{Γ_{1}} ⊂ [x]_{G}} = # { [γ]_{Γ_{2}} ⊂ [x]_{G}} ,

then

[L]-spec(Γ_{1}\G, g) = [L]-spec(Γ_{2}\G, g).

Here # { [γ]_{Γ_{i}} ⊂ [x]_{G}} denotes the number of distinct conjugacy classes in Γ_{i} contained in the conjugacy class of x in G.

Proof of Proposition 5.3.

Let x ∈ G. We count the number of distinct conjugacy classes in Γ_{1} and Γ_{2} contained in [x]_{G}.

Let γ_{1} = exp(2n_{1}X_{1}) exp(m_{1}Y_{1}) exp(m_{2}Y_{2}) exp(kZ) exp(jW) ∈ Γ_{1} for n_{1}, m_{1}, m_{2}, k ∈ Z. Define the mapping F : Γ_{1} → Γ_{2} by

F (γ_{1}) = exp(2n_{1}X_{1}) exp(m_{1}(Y_{1} + \frac{1}{2}Z)) exp(m_{2}Y_{2}) exp(kZ) exp(jW).

The mapping F gives us a correspondence between the elements of Γ_{1} and the elements of Γ_{2}. Note that F is not a Lie group isomorphism.

Now γ_{1} and F (γ_{1}) = γ_{2} are conjugate in G. In particular, F (γ_{1}) = aγ_{1}a^{−1} where a = e if m_{1} = 0, and a = exp(\frac{1}{2}X_{1}) exp((\frac{1}{8} + \frac{k}{2m_{1}} )Y_{2}) if m_{1} ≠ 0. Thus [γ_{1}]_{Γ_{1}} ⊂ [x]_{G} if and only if [F (γ_{1})]_{Γ_{2}} ⊂ [x]_{G}.

To use Proposition 5.5, we must now compare the number of distinct conjugacy classes in Γ_{1} and Γ_{2} respectively that are contained in a fixed [x]_{G}.

Using the Campbell-Baker-Hausdorff formula, two elements

γ_{1} = exp(2n_{1}X_{1}) exp(m_{1}Y_{1}) exp(m_{2}Y_{2}) exp(kZ) exp(jW)

and

γ′_{1} = exp(2n′_{1}X_{1}) exp(m′_{1}Y_{1}) exp(m′_{2}Y_{2}) exp(k′Z) exp(j′W),

of Γ_{1} are conjugate in Γ_{1} if and only if there exist integers n̄_{1}, m̄_{1}, m̄_{2}, k̄ such that

n′_{1} = n_{1}, m′_{1} = m_{1}, m′_{2} = m_{2},
k′ = k + 2m_{1}n̄_{1} − 2n_{1}m̄_{1},
j′ = j + m_{2}m̄_{1} − m_{1}m̄_{2} + 2kn̄_{1} − 2n_{1}k̄
+ 2m_{1}n̄_{1}^{2} − 4n_{1}n̄_{1}m̄_{1} + 2n_{1}^{2}m̄_{1}.

Let K = gcd(2n_{1}, 2m_{1}). From the above, we see that every conjugacy class in Γ_{1} contains at least one representative such that k ∈ {1, 2, ⋯, K}. We call such a representative nice. Two nice representatives are in the same conjugacy class in Γ_{1} if and only if k = k′ and there exist integers n̄_{1}, m̄_{1}, m̄_{2}, k̄ such that m_{1}n̄_{1} −n_{1}m̄_{1} = 0 and

\begin{aligned}j′ & = j + m_{2}m̄_{1} − m_{1}m̄_{2} + 2kn̄_{1} − 2n_{1}k̄ \\ & + 2m_{1}n̄_{1}^{2} − 4n_{1}n̄_{1}m̄_{1} + 2n_{1}^{2}m̄_{1} \end{aligned}

Similarly, two elements of Γ_{2}

γ_{2} = exp(2n_{1}X_{1}) exp(m_{1}(Y_{1} + \frac{1}{2}Z)) exp(m_{2}Y_{2}) exp(kZ) exp(jW),

and

γ′_{2} = exp(2n′_{1}X_{1}) exp(m′_{1}(Y_{1} + \frac{1}{2}Z)) exp(m′_{2}Y_{2}) exp(k′Z) exp(j′W),

are conjugate in Γ_{2} if and only if there exist integers n̄_{1}, m̄_{1}, m̄_{2}, k̄ so that

n′_{1} = n_{1}, m′_{1} = m_{1}, m′_{2} = m_{2},
k′ = k + 2m_{1}n̄_{1} − 2n_{1}m̄_{1},
j′ = j + (m_{1}n̄_{1} − n_{1}m̄_{1}) + m_{2}m̄_{1} − m_{1}m̄_{2} + 2kn̄_{1} − 2n_{1}k̄
+ 2m_{1}n̄_{1}^{2} − 4n_{1}n̄_{1}m̄_{1} + 2n_{1}^{2}m̄_{1}.

Again we see that every conjugacy class in Γ_{2} contains at least one nice representative, that is, a representative such that k ∈ {1, 2, ⋯, K}, where K = gcd(2n_{1}, 2m_{1}) as above. Again, two nice representatives are in the same conjugacy class in Γ_{2} if and only if k = k′ and there exist integers n̄_{1}, m̄_{1}.m̄_{2}, k̄ such that m_{1}n̄_{1} − n_{1}m̄_{1} = 0 and

\begin{aligned}j′ & = j + m_{2}m̄_{1} − m_{1}m̄_{2} + 2kn̄_{1} − 2n_{1}k̄ \\ & + 2m_{1}n̄_{1}^{2} − 4n_{1}n̄_{1}m̄_{1} + 2n_{1}^{2}m̄_{1} \end{aligned}

Note that the correspondence F : Γ_{1} → Γ_{2} sends nice representatives to nice representatives. Thus if we restrict ourselves to nice representatives, the conjugacy conditions are equivalent. That is, two nice representatives γ_{1} and γ′_{1} are in the same conjugacy class in Γ_{1} if and only if the corresponding elements F (γ_{1}) and F (γ_{2}) are in the same conjugacy class in Γ_{2}.

Let γ_{1}, γ_{2}, …, γ_{L} be nice representatives of the L distinct conjugacy classes in Γ_{1} contained in [x]_{G}. Then F (γ_{1}), F (γ_{2}), …, F (γ_{L}) are nice representatives of L distinct conjugacy classes in Γ_{2}. The same applies to F^{−1} : Γ_{2} → Γ_{1}.

Thus

# { [γ]_{Γ_{1}} ⊂ [x]_{G}} = # { [γ]_{Γ_{2}} ⊂ [x]_{G}} ,

as desired.

Example III: The Length Spectrum.

Let

𝔤 = span_{R}{X_{1}, X_{2}, Y_{1}, Y_{2}, Z_{1}, Z_{2}, W }

with Lie brackets

[X_{1}, Y_{1}] = [X_{2}, Y_{2}] = Z_{1}
[X_{1}, Y_{2}] = Z_{2}
[X_{1}, Z_{1}] = [X_{2}, Z_{2}] = [Y_{1}, Y_{2}] = W

and all other basis brackets zero.

Clearly 𝔤 is a strictly nonsingular, three-step nilpotent Lie algebra.

Let Γ_{1} be the cocompact, discrete subgroup of G generated canonically by

{exp(2X_{1}), exp(2X_{2}), exp(Y_{1}), exp(Y_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)},

and let Γ_{2} be the cocompact, discrete subgroup of G generated canonically by

{exp(X_{1}), exp(X_{2}), exp(2Y_{1}), exp(2Y_{2}), exp(Z_{1}), exp(Z_{2}), exp(W)}.

Let g be the left invariant metric on G defined by letting

{X_{1}, X_{2}, Y_{1}, Y_{2}, Z_{1}, Z_{2}, W }

be an orthonormal basis of 𝔤.

5.6 Proposition. The nilmanifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) do not have the same length spectrum. In particular, the multiplicity of the length 1 in [L]-spec(Γ_{1}\G, g) is greater than its multiplicity in [L]-spec(Γ_{2}\G, g).

Proof of Proposition 5.6.

By Proposition 5.2 we need only consider the noncentral free homotopy classes. That is, we need only show m′_{1}(1) > m′_{2}(1).

Let

γ = exp(A_{1}n_{1}X_{1}) exp(A_{2}n_{2}X_{2}) exp(B_{1}m_{1}Y_{1}) exp(B_{2}m_{2}Y_{2}) exp(k_{1}Z_{1}) exp(k_{2}Z_{2}) exp(jW)

for integers n_{1}, n_{2}, m_{1}, m_{2}, k_{1}, k_{2}, j and A_{1}, A_{2}, B_{1}, B_{2} ∈ {1, 2}. Note that γ ∈ Γ_{1} if and only if

(∗)  A_{1} = A_{2} = 2, B_{1} = B_{2} = 1

and if γ ∈ Γ_{2} if and only if

(∗∗)  A_{1} = A_{2} = 1, B_{1} = B_{2} = 2.

By Theorem 3.2.4, to determine if 1 ∈ [γ]_{Γ_{i}} for noncentral γ ∈ Γ_{i}, we need only determine if 1 ∈ [γ̄]Γ̄_{i}. That is, rather than looking at the lengths of closed geodesics on the three-step nilmanifolds (Γ_{i}\G, g), we instead look at the lengths of closed geodesics on the quotient two-step nilmanifolds (Γ̄_{i}\Ḡ, ḡ) for i = 1, 2.

The Lie algebra of Ḡ is 𝔤̄ = 𝔤/𝔤^{(2)} = span_{R}{X̄_{1}, X̄_{2}, Ȳ_{1}, Ȳ_{2}, Z̄_{1}, Z̄_{2}} with Lie brackets

[X̄_{1}, Ȳ_{1}] = [X̄_{2}, Ȳ_{2}] = Z̄_{1}
[X̄_{1}, Ȳ_{2}] = Z̄_{2},

and all other basis brackets zero.

We may now use the following result due to Eberlein.

5.7 Theorem [E1]. Let N be a simply connected, two-step nilpotent Lie group with Lie algebra 𝔫 and left invariant metric g. Let Γ be a cocompact, discrete subgroup of N. Let 𝔷 be the center of 𝔫 and 𝔳 the orthogonal complement of 𝔷 in 𝔫. Any element γ ∈ Γ may be expressed uniquely as exp(V^{∗} + Z^{∗}) where V^{∗} ∈ 𝔳 and Z^{∗} ∈ 𝔷. Let Z^{∗∗} be the component of Z^{∗} orthogonal to [V^{∗}, 𝔫]. Let λ > 0.

(1) If λ ∈ [γ]_{Γ}, then |V^{∗}| ≤ λ ≤ \sqrt{|V^{∗}|^{2} + |Z^{∗∗}|^{2}}.

(2) The period λ = |V^{∗}| ∈ [γ]_{Γ} if and only if |Z^{∗∗}| = 0.

(3) The period λ = \sqrt{|V^{∗}|^{2} + |Z^{∗∗}|^{2}} ∈ [γ]_{Γ}.

Here 𝔫 = 𝔤̄ and the metric ḡ is determined by the orthonormal basis of 𝔤̄

{X̄_{1}, X̄_{2}, Ȳ_{1}, Ȳ_{2}, Z̄_{1}, Z̄_{2}}.

By Theorem 5.7, to find γ̄ such that 1 ∈ [γ̄]Γ̄_{i}, we need γ̄ = exp(V̄^{∗} + Z̄^{∗}) such that |V̄^{∗}|^{2} ≤ 1 ≤ |V̄^{∗}|^{2} + |Z̄^{∗∗}|^{2}, where V̄^{∗} ∈ span_{R}{X̄_{1}, X̄_{2}, Ȳ_{1}, Ȳ_{2}} and Z̄^{∗} ∈ span_{R}{Z̄_{1}, Z̄_{2}}.

For both Γ_{1} and Γ_{2}, V̄^{∗} = A_{1}n_{1}X̄_{1} + A_{2}n_{2}X̄_{2} + B_{1}m_{1}Ȳ_{1} + B_{2}m_{2}Ȳ_{2}, where n_{1}, n_{2}, m_{1}, m_{2} ∈ Z Note that if |V^{∗}| ≠ 0, |V̄^{∗}|^{2} = A_{1}^{2}n_{1}^{2} +A_{2}^{2}n_{2}^{2} +B_{1}^{2}m_{1}^{2} +B_{2}^{2}m_{2}^{2} ≥ 1. So |V̄^{∗}|^{2} ≤ 1 if and only if |V̄^{∗}|^{2} = 1. By Theorem 5.4, λ = 1 = |V̄^{∗}| ∈ [γ̄]Γ̄_{i} if and only if |Z̄^{∗∗}| = 0.

So if γ̄ = exp(V̄^{∗} +Z̄^{∗}) with |V̄^{∗}| ≠ 0, then 1 ∈ [γ̄]Γ̄_{i} if and only if where |V̄^{∗}| = 1 and |Z̄^{∗∗}| = 0.

We consider two cases.

Case 1: (n_{1})^{2} + (m_{2})^{2} ≠ 0.

In this case, 𝔷̄ = [log γ̄, 𝔤̄], so Z̄^{∗∗} is automatically zero. Applying the condition |V̄^{∗}| = 1 and lifting to the three-step level, we have 1 ∈ [γ_{1}]_{Γ_{1}} if and only if (see (∗))

γ_{1} = exp(±Y_{2}) exp(k_{1}Z_{1}) exp(k_{2}Z_{2}) exp(jW),

and 1 ∈ [γ_{2}]_{Γ_{2}} if and only if (see (∗∗))

γ_{2} = exp(±X_{1}) exp(k_{1}Z_{1}) exp(k_{2}Z_{2}) exp(jW).

We must now compare the number of distinct free homotopy classes of Γ_{1} and Γ_{2} that take on one of these forms.

Another element γ′_{1} = exp(±Y_{2}) exp(k′_{1}Z_{1}) exp(k′_{2}Z_{2})exp(j′W) of Γ_{1} is conjugate to γ_{1} in Γ_{1} if and only if there exist integers n̄_{1}, n̄_{2}, m̄_{1} and k̄_{1} such that

k′_{1} = k_{1} ± 2n̄_{2}; k′_{2} = k_{2} ± 2n̄_{1}; j′ = j ± m̄_{1} + 2k_{1}n̄_{1} + 2k_{2}n̄_{2} ± 4n̄_{1}n̄_{2}.

Another element γ′_{2} = exp(±X_{1}) exp(k′_{1}Z_{1}) exp(k′_{2}Z_{2}) exp(j′W) of Γ_{2} is conjugate to γ_{2} in Γ_{2} if and only if there exist integers n̄_{1}, n̄_{2}, m̄_{1} and m̄_{2} such that

k′_{1} = k_{1} ∓2m̄_{1}; k′_{2} = k_{2} ∓2m̄_{2}; j′ = j ∓k̄_{1} +m̄_{1} +k_{1}n̄_{1} +k_{2}n̄_{2} ∓2m̄_{1}n̄_{1} ∓2m̄_{2}n̄_{2}.

For Γ_{1} we have two choices {−1, +1} for the coefficient of Y_{2}, two choices for k_{1}, two choices for k_{2} and one choice for j for a total of 8 distinct free homotopy classes. For Γ_{2} we have two choices {−1, +1} for the coefficient of X_{1}, two choices for k_{1}, two choices for k_{2} and one choice for j for a total of 8 distinct free homotopy classes. Thus, the multiplicities of 1 coming from this case are equal.

Case 2: n_{1}^{2} + m_{2}^{2} = 0 but n_{2}^{2} + m_{1}^{2} ≠ 0.

In this case, [log γ̄, 𝔤̄] = span_{R}{Z̄_{1}}, so Z̄^{∗∗} = 0 if and only if k_{2} = 0. Applying the condition |V̄^{∗}| = 1 and lifting to the three-step level, we have 1 ∈ [γ_{1}]_{Γ_{1}} if and only if (see (∗))

γ_{1} = exp(±Y_{1}) exp(k_{1}Z_{1}) exp(jW),

and 1 ∈ [γ_{2}]_{Γ_{2}} if and only if (see (∗∗))

γ_{2} = exp(±X_{2}) exp(k_{1}Z_{1}) exp(jW).

We must now count the number of distinct free homotopy classes of Γ_{1} and Γ_{2} that take on one of these forms.

Another element γ′_{1} = exp(±Y_{1}) exp(k′_{1}Z_{1}) exp(j′W) of Γ_{1} is conjugate to γ_{1} in Γ_{1} if and only if there exist integers n̄_{1}, m̄_{2} such that

k′_{1} = k_{1} ± 2n̄_{1}; j′ = j ∓ m̄_{2} + 2k_{1}n̄_{1} ± 2n̄_{1}^{2}.

Another element γ′_{2} = exp(±X_{2}) exp(k′_{1}Z_{1}) exp(j′W) in Γ_{2} is conjugate to γ_{2} in Γ_{2} if and only if there exist integers n̄_{1}, m̄_{2} and k̄_{2} such that

k′_{1} = k_{1} ∓ 2m̄_{2}; j′ = j ∓ k̄_{2} + k_{1}n̄_{1} ∓ 2n̄_{1}m̄_{2}

For Γ_{1} we have two choices {−1, +1} for the coefficient of Y_{1}, two choices for k_{1}, and one choice for j for a total of 4 distinct free homotopy classes. For Γ_{2} we have two choices {−1, +1} for the coefficient of X_{2}, two choices for k_{1}, and one choice for j for a total of 4 distinct free homotopy classes. Again, the multiplicities of 1 coming from this case are equal.

Case 3: |V̄^{∗}| = 0, |Z̄^{∗}| ≠ 0.

Let γ = exp(k_{1}Z_{1}) exp(k_{2}Z_{2}) exp(jW), for k_{1}, k_{2}, j ∈ Z. Note that γ ∈ Γ_{1} ∩ Γ_{2}. Thus by (2.1.1), any period occurring in [γ]_{Γ_{1}} will also occur in [γ]_{Γ_{2}}. Let γ′ = exp(k′_{1}Z_{1}) exp(k′_{2}) exp(j′W) be another element of Γ_{1} ∩ Γ_{2}, where k′_{1}, k′_{2}, j′ ∈ Z.

Now γ′ is conjugate to γ in Γ_{1} if and only if there exists integers n̄_{1}, n̄_{2} such that

k′_{1} = k_{1};  k′_{2} = k_{2};  j′ = j + 2(k_{1}n̄_{1} + k_{2}n̄_{2}).

However γ′ is conjugate to γ in Γ_{2} if and only if there exists integers n̄_{1}, n̄_{2} such that

k′_{1} = k_{1};  k′_{2} = k_{2};  j′ = j + (k_{1}n̄_{1} + k_{2}n̄_{2}).

Note that there are twice as many distinct conjugacy classes represented by elements of the form γ = exp(k_{1}Z_{1}) exp(k_{2}Z_{2}) exp(jW) for Γ_{1} as for Γ_{2}. Thus to show the multiplicities are not equal here, we need to exhibit a closed geodesic of length 1 in just one free homotopy class of this form.

Note that |Z̄^{∗∗}|^{2} = |Z̄^{∗}|^{2} = k_{1}^{2} + k_{2}^{2}. By Theorem 5.7(iii) and lifting to the threestep level, we see \sqrt{k_{1}^{2} + k_{2}^{2}} ∈ [γ]_{Γ_{1}} and \sqrt{k_{1}^{2} + k_{2}^{2}} ∈ [γ]_{Γ_{2}}. Thus 1 ∈ [exp(±Z_{i})]_{Γ_{j}}, i, j = 1, 2.

Therefore, for Case 3, 1 occurs with twice the multiplicity in [L]-spec(Γ_{1}\G, g) as it does in [L]-spec(Γ_{2}\G, g).

As the multiplicities of 1 are equal in all of the other cases, the multiplicities of 1 is not equal, as claimed.

Example IV: The Length Spectrum.

Here the Lie algebra is the same Lie algebra as Example II, that is

𝔤 = span_{R}{X_{1}, Y_{1}, Y_{2}, Z, W }

with Lie brackets

[X_{1}, Y_{1}] = Z
[X_{1}, Z] = [Y_{1}, Y_{2}] = W

and all other basis brackets zero.

Let Γ_{1} be the cocompact, discrete subgroup of G generated canonically by

{exp(2X_{1}), exp(Y_{1}), exp(Y_{2}), exp(Z), exp(W)},

and let Γ_{2} be the cocompact, discrete subgroup of G generated canonically by

{exp(X_{1}), exp(2Y_{1}), exp(2Y_{2}), exp(Z), exp(W)}.

Let g be the left invariant metric on G defined by letting

{X_{1}, Y_{1}, Y_{2}, Z, W }

be an orthonormal basis of 𝔤.

5.8 Proposition. The nilmanifolds (Γ_{1}\G, g) and (Γ_{2}\G, g) do not have the same length spectrum. In particular, the multiplicity of the length λ = \sqrt{4π(7 − π)} in [L]-spec(Γ_{1}\G, g) is greater than its multiplicity in [L]-spec(Γ_{2}\G, g).

Proof of Proposition 5.8.

By Proposition 5.2, we only consider the noncentral free homotopy classes. In particular, we show m′_{1}(λ) > m′_{2}(λ) where λ = \sqrt{4π(7 − π)}.

By Theorem 3.2.4 if we wish to determine if λ ∈ [γ]_{Γ_{i}} for noncentral γ ∈ Γ_{i}, we need only determine if λ ∈ [γ̄]Γ̄_{i}. That is, rather than looking at the lengths of closed geodesics on the three-step nilmanifolds (Γ_{i}\G, g), we instead look at the lengths of closed geodesics on the quotient two-step nilmanifolds (Γ̄_{i}\Ḡ, ḡ) for i = 1, 2.

However, for this example, 𝔤̄ ≅ 𝔥_{1} ⊕ R where 𝔥_{1} denotes the three-dimensional Heisenberg algebra. To see this, note that

𝔥_{1} ≅ {X̄_{1}, Ȳ_{1}, Z̄},  and  [X̄_{1}, Ȳ_{1}] = Z̄.

This is an ideal in 𝔤̄. And

R ≅ {Ȳ_{2}}

which is also an ideal in 𝔤̄. Let H_{1} be the three-dimensional Heisenberg group. Note that

H_{1} ≅ {exp(x_{1}X̄_{1}) exp(y_{1}Ȳ_{1}) exp(zZ̄) : x_{1}, y_{1}, z ∈ R}.

This direct sum is actually a Riemannian direct sum, as the metric may also be written as

ḡ = ḡ_{1} ⊕ ḡ_{2}

where ḡ_{1} is the left invariant metric on 𝔥_{1} given by the orthonormal basis {X̄_{1}, Ȳ_{1}, Z̄} and ḡ_{2} is the left invariant metric on R given by the unit vector {Ȳ_{2}}.

Furthermore, as Γ̄_{i} = (Γ̄_{i} ∩ H_{1}) ⊕ (Γ̄_{i} ∩ R), we also have the Riemannian direct sum

(Γ̄_{i}\Ḡ, ḡ) ≅ ((Γ̄_{i} ∩ H_{1})\H_{1}, ḡ_{1}) ⊕ ((Γ̄_{i} ∩ R)\R, ḡ_{2}).

Using rescaling of geodesics, it is not difficult to show that λ ∈ [γ̄]Γ̄_{i} if and only if

λ^{2} = λ_{1}^{2} + λ_{2}^{2}

where λ_{1} ∈ [γ̄_{1}]_{Γ̄_{i}∩H_{1}} and λ_{2} ∈ [γ̄_{2}]_{Γ̄_{i}∩R}. Here γ̄ = (γ̄_{1}, γ̄_{2}) with respect to the direct product Γ̄_{i} = (Γ̄_{i} ∩ H_{1}) ⊕ (Γ̄_{i} ∩ R).

Now, the length spectrum of ((Γ̄_{i} ∩R)\R, ḡ_{2}) is easily seen to be | log(γ̄_{2})| for all γ̄_{2} ∈ Γ̄_{i} ∩R. Thus the length spectrum here (not counting multiplicities) is precisely the positive integers.

The length spectrum of ((Γ̄_{i} ∩ H_{1})\H_{1}, ḡ_{1}) has been calculated by both Gordon and Eberlein (see [E], [G1]) and is known to be

(i) | log(γ̄_{1})| if γ̄_{1} ∈ Γ̄_{i} ∩ H_{1}, for γ̄_{1} ∉ Z(H_{1}).

(ii) \{|log(γ̄_{1})|, \sqrt{(4πk)(| log(γ̄_{1})| − πk)} : 1 ≤ k < (\frac{1}{2π} | log(γ̄_{1})|), k ∈ Z\}, for γ̄_{2} ∈ Γ̄_{i} ∩ Z(H_{1}).

Nonintegral lengths occur in (Γ̄_{i} ∩ H_{1})\H_{1} only when | log(γ̄_{1})| ≥ 2π > 6.

Also note that \sqrt{4π(7 − π)} ∈ [γ̄_{1}]_{Γ̄_{i}∩H_{1}} if and only if γ̄_{1} = exp(±7Z̄) ∈ Γ̄_{i} ∩ H_{1}. This is the smallest possible nonintegral length.

Thus

4π(7 − π) = λ^{2} = λ_{1}^{2} + λ_{2}^{2}

if and only if λ_{2}^{2} = 0 and λ_{1}^{2} = 4π(7 − π) if and only if γ̄ = exp(±7Z̄) ∈ Γ̄_{i}. By lifting to (Γ_{i}\G, g), we see \sqrt{4π(7 − π)} ∈ [γ]_{Γ_{i}} if and only if

γ = exp(±7Z)exp(jW) ∈ Γ_{i}.

We now count the number of distinct free homotopy classes represented by a γ of this form.

Let γ′ = exp(±7Z)exp(j′W).

Now γ′ is conjugate to γ in Γ_{1} if and only if there exists integer n̄_{1} such that

j′ = j ± 14n̄_{1}.

However, γ′ is conjugate to γ in Γ_{2} if and only if there exists integer n̄_{1} such that

j′ = j ± 7n̄_{1}.

Thus there are 14 choices for j in Γ_{1} and there are 7 choices for j in Γ_{2}. So the multiplicity of the length \sqrt{4π(7 − π)} in (Γ_{1}\G, g) is 28, (14 for each of exp(+7Z)exp(jW) and exp(−7Z)exp(jW)), and likewise the multiplicity in (Γ_{2}\G, g) is 14.

Thus the multiplicities of \sqrt{4π(7 − π)} are not equal here, as claimed.

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Ruth Gornet: Texas Tech University; Department of Mathematics; Lubbock, Texas 79409-1042; May 1995