Spectra of Sol-manifolds: arithmetic and geometry

Alexander Veselov

Department of Mathematics, University of Loughborough

1 Classical aspects - geometry

1.1 Geometric structure on manifolds

A Riemannian manifold (M^{n}, g_{ij}) is called locally homogeneous if for any two points x, y ∈ M^{n} there exist open U ∋ x, V ∋ y and an isometry (U, x) → (V, y). The universal covering X^{n} of M^{n} is homogeneous in the sense that it has a transitive group G of isometries (Singer (1960)). The pair (X^{n}, G) is a geometry (in Klein’s sense). We say that M^{n} has a geometric structure modelled on X^{n}.

In the case of 2-dimensional surfaces of constant curvature there are three types of examples. The first is a round sphere X^{2} = M^{2} = S^{2}. The second is X^{2} = E^{2} (2-dimensional Euclidean space) and M^{2} = T^{2} = ℝ^{2}/ℤ^{2} is a flat torus. The third is X^{2} = H^{2} (hyperbolic plane) and M^{2} = Σ_{g} (genus-g Riemann surface). On each 2-dimensional orientable surface one can introduce only one type of geometry, because the curvature determines the genus:

Theorem 1.1 (Gauss-Bonnet Theorem)

∫_{M^{2}} Kdσ = 2πχ(M^{2}) = 4π(1 − g).

According to W. Thurston, there are 8 geometries of dimension 3, corresponding to X^{3} = S^{3}, E^{3}, H^{3}, S^{2} × ℝ^{1}, H^{2} × ℝ^{1}, { \atop SL_{2}(ℝ)}, Nil and Sol (≈ Iso(ℝ^{1,1})). The famous Geometrisation Conjecture due to Thurston says that any M^{3} can be cut by 2-spheres and tori into pieces admitting geometric structures.

Which M^{3} admits a geometric structure? Not all, of course. For all but Sol and H^{3}-geometries the corresponding manifolds must be Seifert fibre spaces (manifolds foliated by circles). Hyperbolic case H^{3} is known to be as the most complicated. We consider here the Sol-case, which is in a way ”next” to the hyperbolic one.

1.2 Sol-geometry and Sol-manifolds

Sol-manifolds are torus bundles over circles with hyperbolic gluing maps A ∈ SL_{2}(ℤ):

M_{A}^{3} = T^{2} × ℝ^{1}/ ⟨T_{A}⟩ ,

where

T_{A} : ( \begin{matrix}x \\ y \\ z \end{matrix} ) ↦ ( \begin{matrix}A ( \begin{matrix}x \\ y \end{matrix} ) \\ z + 1 \end{matrix} ) .

Let λ and λ^{−1} be eigenvalues of A and e_{u} and e_{v} eigenvectors with respect to λ and λ^{−1}. Using this basis {e_{u}, e_{v}}, we can define new coordinates (u, v, z). Then the gluing map in (u, v, z) is as follows:

T_{A} : ( \begin{matrix}u \\ v \\ z \end{matrix} ) ↦ ( \begin{matrix}λu \\ λ^{−1}v \\ z + 1 \end{matrix} ) .

Note that the lattice L in (u, v)-coordinates looks non-trivial.

Example For the “cat-map”

A = ( \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} )

λ = \frac{3+\sqrt{5}}{2} and the new coordinates are

\{ \begin{matrix}u = \frac{1 + \sqrt{5}}{2}x + y, \\ v = \frac{1 − \sqrt{5}}{2}x + y. \end{matrix}

We introduce a class of Riemannian metrics on M_{A}, which in the coordinates (u, v, z) is given by

ds^{2} = e^{−2μz}α_{0}du^{2} + 2β_{0}dudv + e^{2μz}γ_{0}dv^{2} + dz^{2},

where μ = log λ, α_{0}, β_{0}, γ_{0} are parameters. These metrics comes from the right-invariant metrics on the 3-dimensional solvable group Sol, which is a universal covering of the Sol-manifolds. Topologically this group is ℝ^{3} with the multiplication of the form

(u, v, z) ∗ (u′, v′, z′) = (u + e^{z}u′, v + e^{−z}v′, z + z′).

1.3 Geodesics on Sol-manifolds

Let H be the Hamiltonian function of the geodesic flow:

H = \frac{1}{2} (Ee^{2μz}p_{u}^{2} + 2F p_{u}p_{v} + Ge^{−2μz}p_{v}^{2}) + \frac{1}{2}p_{z}^{2}.

H is an integral, but in spite of the absence of u and v in the Hamiltonian p_{u} and p_{v} are not integrals because p_{u} and p_{v} are not defined on T^{∗}M_{A}^{3}. Indeed they are not invariant under the following action T_{A}^{∗} on the cotangent bundle T^{∗}M_{A}^{3}:

p_{u} ↦ λ^{−1}p_{u}, p_{v} ↦ λp_{v}, p_{z} ↦ p_{z}.

However, we can take the invariant function Q = p_{u}p_{v} as the second integral. But what is the third integral?

Let

α = \frac{log (\sqrt{\frac{E}{G}} |\frac{p_{u}}{p_{v}} |)}{2 log λ},

then T_{A}(α) = α − 1, so we have the integrals

\{ \begin{matrix}f_{1} = R(Q) cos 2πα, \\ f_{2} = R(Q) sin 2πα, \end{matrix}

where R(Q) = \sqrt{|Q|} exp (−\frac{1}{Q^{2}} ). These integrals are smooth but not ana-

lytic.

Theorem 1.2 (Bolsinov-Taimanov (2000)) The geodesic flow on M_{A}^{3} is not integrable in the analytic category (no third analytic integral). Moreover, the geodesic flow has positive topological entropy. The fundamental group π_{1}(M_{A}^{3}) has exponential growth.

Theorem 1.3 (Taimanov (1988)) Analytic integrability implies that π_{1}(M_{A}^{3}) is almost commutative (contains a commutative subgroup of finite index).

If Q = p_{u}p_{v} ≠ 0 then the corresponding geodesics rotate many times along the base S^{1}, then turn back, go into the opposite direction, turn back and so on.

If p_{u} = p_{v} = 0 we have a very simple family of geodesics:

u = const., v = const., z = t,

but the corresponding flow is chaotic and has positive entropy. Indeed, the time-one map is a hyperbolic torus automorphism A (Anosov map) with entropy log λ > 0. Note that in this case chaos has a purely topological origin.

2 Quantum aspects - geometry and arithmetic

2.1 SL_{2}(ℤ) and binary quadratic forms

Let

A = ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ) ∈ SL_{2}(ℤ), A : ℤ^{2} → ℤ^{2}.

Define quadratic form Q_{A} by the formula

Av ∧ v = Q_{A}(v)e_{1} ∧ e_{2}.

Explicitly,

Q_{A}(x, y) = det ( \begin{matrix}a_{11}x + a_{12}y & x \\ a_{21}x + a_{22}y & y \end{matrix} ) = −a_{21}x^{2} + (a_{11} − a_{22})xy + a_{12}y^{2}.

Q_{A}(v) is invariant under A:

Q_{A}(Av) = Q_{A}(v).

The discriminant of Q_{A} is

(a_{11} − a_{22})^{2} + 4a_{21}a_{12} = (a_{11} + a_{22})^{2} − 4(a_{11}a_{22} − a_{12}a_{21}) = (trA)^{2} − 4.

This coincides with the discriminant of the characteristic equation for A:

λ^{2} − (a_{11} + a_{22})λ + 1 = 0.

Since A is hyperbolic, Q_{A} is indefinite. Let Q_{A} = kQ_{A}^{∗}, where

Q_{A}^{∗} = ax^{2} + bxy + cy^{2}

is a primitive form: gcd(a, b, c) = 1. Thus we have a map which send A to the primitive form Q_{A}^{∗}. Conversely, suppose we have such a form

Q = ax^{2} + bxy + cy^{2}.

Those A ∈ SL_{2}(ℤ) which preserve Q are called automorphs of Q (AutQ).

Let d = b^{2} − 4ac be not a total square (which is the case in our situation) and consider Pell’s equation

X^{2} − dY^{2} = 4.

Then the group of automorphs consists of

A = ± ( \begin{matrix}\frac{X − bY}{2} & −cY \\ aY & \frac{X + bY}{2} \end{matrix} ) ,

where (X, Y) is a solution of the above equation.

Pell’s equation always has a solution if d is not a total square. Let (X_{0}, Y_{0}) ( X_{0}, Y_{0} > 0) be a solution such that X_{0} + \sqrt{d}Y_{0} is minimal. It is called a fundamental solution. The other solutions are given by

\frac{X + \sqrt{d}Y}{2} = ± (\frac{X_{0} + \sqrt{d}Y_{0}}{2} )^{n} .

The matrix

A_{0} = ( \begin{matrix}\frac{X_{0} − bY_{0}}{2} & −cY_{0} \\ aY_{0} & \frac{X_{0} + bY_{0}}{2} \end{matrix} )

is the generator of the group of the automorphs. One can find X_{0}, Y_{0} from the continued fraction of \sqrt{d}. Example Consider the “cat-map” A case. The discriminant is d = 5 and

Q_{A}(x, y) = −(x^{2} − xy − y^{2}).

We must consider Pell’s equation:

X^{2} − 5Y^{2} = 4.

The fundamental solution of the above equation is (3, 1). Thus the group of automorphs generated by

A_{0} = ( \begin{matrix}5 & 3 \\ 3 & 2 \end{matrix} ) .

In particular,

(\frac{3 + \sqrt{5}}{2})^{2} = \frac{9 + \sqrt{5} + 5}{4} = \frac{7 + 3\sqrt{5}}{2},

so we get a new solution (7, 3).

Definition 2.1 A ∈ SL_{2}(ℤ) is called primitive if it can not be represented as a power of any B ∈ SL_{2}(ℤ):

A ≠ B^{n}.

Thus we have a correspondence between primitive elements A ∈ SL_{2}(ℤ) and primitive binary forms Q.

2.2 Spectrum of M_{A}^{3} (Bolsinov-Dullin-Veselov (2002))

Consider the Schrödinger equation:

−Δψ = Eψ,

where

Δ = Ee^{2μz} \frac{∂^{2}}{∂u^{2}} + 2F \frac{∂^{2}}{∂u}∂v + Ge^{−2μz} \frac{∂^{2}}{∂v^{2}} + \frac{∂^{2}}{∂z^{2}}

is the Laplace-Beltrami operator on M_{A}^{3}.

Look for the solutions of the form

Ψ_{γ}(u, v, z) = e^{2πi(γ,ω)}f(z)

where ω = (u, v), γ ∈ L^{∗}, where L^{∗} is the lattice dual to L. We are led to the modified Mathieu equation:

−\frac{d^{2}}{dz^{2}} f + |ν(γ)| cosh 2μ(z + α(γ))f(z) = Ef(z)

where

α(γ) = \frac{log (\sqrt{\frac{E}{G}} |\frac{(γ, e_{u})}{(γ, e_{v})}|)}{2 log λ}, ν(γ) = 8π^{2}cQ_{A}^{∗}(γ), c = \frac{Area(π(e_{1}^{∗}, e_{2}^{∗}))}{\sqrt{D} sin θ},

θ is the angle between e_{1}^{∗} and e_{2}^{∗}

Let E = E+ν(γ) cos θ, where E = E_{k}(|ν|) and f_{|ν|,k}(z) are the corresponding eigenvalues and eigenfunctions. The problem is that the corresponding functions Ψ_{γ,k} are not defined on M_{A}^{3} since they are not invariant under the T_{A} action.

Let us average this:

Φ_{γ,k} = ∑_{n∈ℤ}Ψ_{γ,k}(λ^{n}uλ^{−n}v, z + n) = ∑_{n∈ℤ}Ψ_{(A^{∗})^{n}γ,k}(u, v, z.)

This series converges well since f_{k}(z) decay fast. We see that

Φ_{γ,k} = Φ_{[γ],k}

where [γ] is the orbit of γ under the action of A^{∗} on L^{∗}: γ ∈ L^{∗}/ ⟨A^{∗}⟩. The case γ = 0 is a special one:

−f′′ = Ef ⇒ E_{k} = (2πk)^{2}, Φ_{0,k} = e^{2πikz}, k ∈ ℤ, Φ_{0,0} = 1.

Theorem 2.2 The functions Ψ_{[γ],k}, Ψ_{0,k} form a complete basis in L_{2}(M_{A}^{3}).

Thus the spectrum of M_{A}^{3} consists of two parts:

E = E_{k} = 4k^{2}π^{2}

and

E = E_{l,[γ]} = E_{l}(|ν|) + ν(γ) cos θ, γ ∈ L^{∗}/ ⟨A^{∗}⟩ .

For E = E_{l,[γ]} we have a classical problem of number theory of how many solutions (modulo automorphs) has the equation

ax^{2} + bxy + cy^{2} = n

for given n. The answer depends on the discriminant d = b^{2} − 4ac.

Let us say that discriminant is good if all the forms with this discriminant are equivalent under SL_{2}(ℤ). In general there are h(d) classes of such forms (class number). The first good square-free discriminants are:

5, 13, 17, 29, 37, 41, 53, 61, 65, 73, 85, 89, · · · .

Gauss conjectured that for positive definite forms only −3, −4, −7, −8, −11, −19, −43, −67, −163 are good and for indefinite forms there are infinitely many of them. For positive forms this was proved, but for indefinite forms it is still open.

If h(d) = 1 then for (n, d) = 1

N_{Q}(n) = N_{d}(n) = ∑_{k|n} (\frac{d}{k})

where (\frac{d}{k}) is the Kronecker’s symbol.

Example Let A be the cat-map. The discriminant of A is good and Q_{A} is primitive. The multiplicity of E_{l,[γ]} is

m_{γ} = 2 (  ∑_{k|n, k=±1 (mod 5)} 1 − ∑_{k|n, k=±2 (mod 5)} 1)

where n = Q_{A}(γ).

Let n = 11^{M} then m = 2(M + 1) since all the divisors are 1 mod 5. So m_{γ} can be arbitrarily large.

For positive definite forms there is a result of E. Landau that the average is

m(n) ∼ K\sqrt{log n.}

Probably the same is true in the indefinite case (P. Sarnak).

Conclusion: The multiplicity of E_{l,[γ]} does not depend on the metric parameters E, F, G of the metrics (rigidity). At least for good discriminants they can be arbitrarily large.

Let’s compare this with the flat torus case T^{2}:

Δ = E \frac{∂^{2}}{∂x^{2}} + 2F \frac{∂^{2}}{∂x∂y} + G\frac{∂^{2}}{∂y^{2}} ,

when the eigenfunctions are

ψ = e^{2πi(kx+ly)}, k, l ∈ ℤ.

The spectrum in this case is

E = E_{k,l} = (2π)^{2}(Ek^{2} + 2F kl + Gl^{2}) = Q(k, l).

For the square case E = G = 1 and F = 0. Here (k^{2} + l^{2}) = n and the multiplicity is given by the Gauss formula

m(n) = 4 (  ∑_{q|n, q≡1 (mod 4)} 1 − ∑_{q|n, q≡3 (mod 4)} 1) ∼ \sqrt{log n} (Landau).

But for generic E, F and G all multiplicities will drop to 2 and the degeneracy disappears. The rigidity of the Sol case reflects the hyperbolicity hidden in the topology.