j. differential geometry 58 (2001) 457-465

ISOSPECTRALITY OF FLAT LORENTZ 3-MANIFOLDS

TODD A. DRUMM & WILLIAM M. GOLDMAN

Abstract

For isometric actions on flat Lorentz (2+1)-space whose linear part is a purely hyperbolic subgroup of O(2, 1), Margulis defined a marked signed Lorentzian length spectrum invariant closely related to properness and freeness of the action. In this paper we show that, for fixed linear part, this invariant completely determines the conjugacy class of the action. We also extend this result to groups containing parabolics.

Complete flat Lorentz 3-manifolds with nonamenable fundamental group bear a striking resemblance to hyperbolic Riemann surfaces. For example, every nonparabolic closed curve is freely homotopic to a unique closed geodesic, which is necessarily spacelike. In his seminal papers [17, 18] on the subject, Margulis introduced a function α : π_{1}(M) ⟶ ℝ which associates the signed Lorentzian length of this geodesic to a conjugacy class in π_{1}(M). In this paper we show that the conjugacy class of the linear holonomy representation π_{1}(M) ⟶ SO(2, 1) and Margulis’s invariant completely determine M up to isometry. Recently Inkang Kim [15] has extended this result to higher dimensional affine actions.

1. Preliminaries

In this paper we consider actions of groups of isometries of Minkowski (2 + 1)-space 𝔼. Minkowski space is a complete simply-connected flat Lorentzian manifold, which identifies with an affine space whose underlying vector space is a 3-dimensional real vector space ℝ^{2,1} with a nondegenerate symmetric bilinear form of index 1. Explicitly we take ℝ^{2,1} to be ℝ^{3} with inner product:

The first author was partially supported by a Swarthmore College Faculty Research Grant. The second author was partially supported by NSF grants DMS-9803518 and DMS-0103889.

Received June 25, 2001.

𝔹(x, y) := x_{1}y_{1} + x_{2}y_{2} − x_{3}y_{3}

so that 𝔼 identifies with ℝ^{3} with Lorentzian metric tensor

(dx_{1})^{2} + (dx_{2})^{2} − (dx_{3})^{2}.

The automorphism group of ℝ^{2,1} is the orthogonal group O(2, 1) consisting of linear isometries of 𝔼. In general, an isometry of 𝔼 is an affine transformation

h : 𝔼 ⟶ 𝔼
x ⟼ g(x) + u

with linear part g = 𝕃(h) ∈ O(2, 1) a linear isometry, and translational part u ∈ ℝ^{2,1}. The intersection SO(2, 1) = O(2, 1) ∩ SL(3, ℝ) consists of orientation-preserving linear isometries. The nullcone

𝔑 := {x ∈ ℝ^{2,1} | 𝔹(x, x) = 0}

is invariant under O(2, 1). The complement 𝔑 − {0} consists of two components (the future and the past)

𝔑_{+} := {x ∈ 𝔑 | x_{3} > 0}, 𝔑_{−} := {x ∈ 𝔑 | x_{3} < 0}.

The subgroup SO(2, 1)^{0} of SO(2, 1) stabilizing either 𝔑_{+} or 𝔑_{−} is the identity component of the Lie group O(2, 1). The group Isom(𝔼) of affine isometries of 𝔼 equals the semidirect product O(2, 1) ⋉ ℝ^{2,1} and the quotient projection

𝕃 : Isom(𝔼) ⟶ O(2, 1)

assigns to an affine isometry h ∈ Isom(𝔼) its linear part g = 𝕃(h) ∈ O(2, 1), so now h can be written:

h(x) = g(x) + u.

An element of O(2, 1) is hyperbolic if and only if it has three distinct real eigenvalues. Since an isometry’s eigenvalues occur in reciprocal pairs, a hyperbolic element of SO(2, 1) must have 1 as an eigenvalue. If g ∈ SO(2, 1)^{0} is hyperbolic, then the other two eigenvalues are necessarily positive. Margulis associated to a hyperbolic element g ∈ SO(2, 1)^{0} a canonical basis as follows. Let the eigenvalues of g be λ^{−1} < 1 < λ. Then there exist unique eigenvectors x^{−}(g), x^{0}(g), x^{+}(g) such that

• gx^{±}(g) = λ^{±1}x^{±}(g) and gx^{0}(g) = x^{0}(g);

• x^{±}(g) ∈ 𝔑_{+} and ∥x^{±}(g)∥ = 1;

• {x^{−}(g), x^{+}(g), x^{0}(g)} is a right-handed basis for ℝ^{2,1}.

Since x^{0}(g) is fixed under the orthogonal linear transformation g,

𝔹(gu − u, x^{0}(g)) = 0 (1)

for all u ∈ ℝ^{2,1}. An affine isometry h of E is hyperbolic if its linear part g = 𝕃(h) is hyperbolic.

2. The Margulis invariant of hyperbolic affine isometries

Suppose that h ∈ Isom^{0}(𝔼) is a hyperbolic affine isometry. Following Margulis [17, 18], define

α(h; x) = 𝔹(hx − x, x^{0}(g)) (2)

for any x ∈ 𝔼. For any y ∈ 𝔼, let u = y − x. Then (1) implies

α(h; x) − α(h; y) = 𝔹((g − 𝕀)u, x^{0}(g)) = 0

so that α(h; x) = α(h) is independent of x. The foliation of 𝔼 by lines parallel to x^{0}(g) is invariant under h and therefore there is an induced affine transformation h′ on the leaf space 𝔼′ = 𝔼/(ℝ · x^{0}(g)). Since the action induced by g on 𝔼′ has no fixed vectors, h′ has a unique fixed point in 𝔼′. Therefore h leaves invariant a unique line C_{h} parallel to x^{0}(g). The restriction of h to C_{h} is translation τ by α(h)x^{0}(g). In particular α(h) = 0 if and only if h fixes a point x ∈ 𝔼. In this case the set of fixed points is exactly the line C_{h}. In general the planes parallel to the orthogonal complement x^{0}(g)^{⊥} (which is spanned by x^{±}(g)) define a foliation whose leaf space identifies to C_{h} under the quotient map Π : 𝔼 ⟶ C_{h}. The diagram

𝔼 ⟶^{h} 𝔼
Π↓  ↓^{Π}
C_{h} ⟶_{τ} C_{h}

commutes. Suppose that ⟨h⟩ acts freely on 𝔼. In this case, the projection 𝔼 ⟶ C_{h} is equivariant and C_{h} projects to the unique closed geodesic in 𝔼/⟨h⟩. Because x^{0}(g) has unit (Lorentzian) length, |α(h)| is the Lorentzian length of the unique closed geodesic in 𝔼/⟨h⟩. Let Γ_{0} be a subgroup of SO(2, 1)^{0}. An affine deformation of Γ_{0} is a representation

ϕ : Γ_{0} ⟶ Isom(𝔼) ≅ SO(2, 1)^{0} ⋉ ℝ^{2,1}

such that 𝕃 ∘ ϕ is the identity map of Γ_{0}. For γ ∈ Γ_{0}, write

ϕ(γ)(x) = 𝕃(γ)x + u(γ)

where 𝕃(γ) ∈ Γ_{0} and u(γ) ∈ ℝ^{2,1}. (When there is no danger of confusion, the symbol ϕ will be omitted.) Then u is a cocycle of Γ_{0} with coefficients in the Γ_{0}-module ℝ^{2,1} corresponding to the linear action of 𝕃 : Γ_{0} ⟶ SO(2, 1)^{0}. In this way affine deformations of Γ_{0} correspond to cocycles in Z^{1}(Γ_{0}, ℝ^{2,1}) and translational conjugacy classes of affine deformations correspond to cohomology classes in H^{1}(Γ_{0}, ℝ^{2,1}).

Lemma 1. α is a class function on π.

Proof. Let γ, η ∈ π. Then x^{0}(ηγη^{−1}) = 𝕃(η)x^{0}(γ) and

u(ηγη^{−1}) = 𝕃(η)u(γ) + (I − 𝕃(ηγη^{−1}))u(η).

Therefore

α(ηγη^{−1}) = 𝔹(u(ηγη^{−1}), x^{0}(ηγη^{−1}))
= 𝔹(𝕃(η)u(γ), 𝕃(η)x^{0}(γ))
+ 𝔹((I − 𝕃(ηγη^{−1}))u(η), 𝕃(η)x^{0}(γ))
= 𝔹(u(γ), x^{0}(γ)) = α(γ)

by (1). q.e.d.

3. Main theorem

The purpose of this note is to prove:

Theorem 1. Suppose that Γ_{0} is a discrete purely hyperbolic free subgroup of SO(2, 1)^{0} and that u, v ∈ Z^{1}(Γ_{0}, ℝ^{2,1}) define affine deformations with α(u) = α(v). Then [u] = [v].

Thus the classification of affine deformations reduces from ℝ^{2,1}- valued cohomology classes [u] of Γ to ordinary ℝ-valued class functions α(u) on Γ. The invariant α(u) depends linearly on u. That is, α(u) = α(v) if and only if α(u − v) = α(u) − α(v) = 0. Also [u] = [v] if and only if [u − v] = [0] is the null affine deformation.

Therefore to prove Theorem 1, it suffices to show that if α(u) = 0 then [u] = [0]. Equivalently, we need to show that the cohomology class [u] ∈ H^{1}(Γ, ℝ^{2,1}) corresponding to an affine deformation Γ_{u} with α_{u} = 0 must vanish. In Lemma 2 below, we show that if α is zero on just two generic elements then the deformation is null.

In the case where α_{u} = 0 for Γ_{u}, we say that Γ_{u} is radiant, that is, there exists a point x ∈ 𝔼 fixed by Γ. (The terminology [10] arises since an affine transformation is radiant if and only if it preserves a radiant vector field

∑_{i=1}^{n}(x_{i} − p_{i})\frac{∂}{∂x_{i}}

“radiating” from p ∈ 𝔼.) We shall in fact show a much stronger statement:

Lemma 2. Let h_{1}, h_{2} ∈ Isom^{0}(𝔼) be hyperbolic whose linear parts g_{1}, g_{2} generate a nonsolvable subgroup Γ_{0} of SO(2, 1)^{0}. Suppose that h_{1}, h_{2} and their product h_{2}h_{1} are radiant. Then Γ = ⟨h_{1}, h_{2}⟩ is radiant.

An alternative statement is that if α(h_{1}) = α(h_{2})α(h_{2}h_{1}) = 0, then α(w(h_{1}, h_{2})) = 0 for any word w ∈ 𝔽_{2}.

Proof. Since h_{1}, h_{2} are radiant, their invariant lines consist of their respective fixed points. For hyperbolic h ∈ Isom^{0}(𝔼), let E^{±}(h) denote the affine subspace containing C_{h} and parallel to the linear subspace spanned by x^{±}(h) and x^{o}(h). The transformations h_{1} and h_{2} are assumed to be hyperbolic and generate a nonsolvable subgroup. Thus, h_{1} and h_{2} are transversal hyperbolic transformations, that is, the four vectors {x^{±}(h_{1}), x^{±}(h_{2})} are all distinct. Since the line C_{h_{1}} is transverse to the plane E^{+}(h_{2}), the affine subspaces C_{h_{1}} and E^{+}(h_{2}) intersect at a point q. Furthermore, since h_{1} and h_{2} share no fixed points, q ∉ C_{h_{2}}. Since q ∈ E^{+}(h_{2}) − C_{h_{2}}, there exists c ≠ 0 such that

h_{2}(q) − q = cx^{+}(g_{2}).

Since g_{2}g_{1} and g_{2} share no eigenspaces, 𝔹(x^{+}(g_{2}), x^{0}(g_{2}g_{1})) ≠ 0. Therefore:

α(h_{2}h_{1}) = 𝔹(h_{2}h_{1}(q) − q, x^{0}(g_{2}g_{1}))
= 𝔹(h_{2}(q) − q, x^{0}(g_{2}g_{1}))
= c𝔹(x^{+}(g_{2}), x^{0}(g_{2}g_{1})) ≠ 0

as desired. q.e.d.

The converse is not true: If g_{1}, g_{2} are hyperbolic linear isometries which share a null eigenvector, then it is easy to construct a non-radiant affine deformation such that α(h_{1}) = α(h_{2}) = α(h_{1}h_{2}) = 0. For example, choose p_{1}, p_{2} ≠ 0 and

g_{i} = [\begin{matrix}1 & 0 & 0 \\ 0 & cosh(p_{i}) & sinh(p_{i}) \\ 0 & sinh(p_{i}) & cosh(p_{i}) \end{matrix}]

for i = 1, 2, and translational parts

u_{1} = [\begin{matrix}0 \\ 0 \\ 0 \end{matrix}] , u_{2} = [\begin{matrix}0 \\ 1 \\ 1 \end{matrix}] .

It can be shown that α(γ) = 0 for any γ ∈ ⟨h_{1}, h_{2}⟩. However, the line l = {(t, 0, 0)|t ∈ ℝ} is the fixed point set for g_{1} and g_{2}, but C_{h_{1}} = l and C_{h_{2}} = (e^{p_{2}} − 1)^{−1}(u_{2}) + l. Since C_{h_{1}} ∩ C_{h_{2}} = ∅, the group ⟨h_{1}, h_{2}⟩ is nonradiant.

4. Parabolic elements

A nontrivial element g ∈ SO(2, 1)^{0} is parabolic if and only if it has a single eigenvalue. This eigenvalue must be +1, and any eigenvector is null. As for hyperbolic elements, we choose an eigenvector x^{0}(g) such that:

• The Euclidean length ∥x^{0}(g)∥ = 1;

• (v, g(v), x^{0}(g)) is a right-handed basis, for any nonzero vector v ∈ 𝔑 which is not fixed by g.

Normalizing the Euclidean length of x^{0}(g) is completely arbitrary. For any nonzero vector v ∈ 𝔑 not fixed by g, the vectors v, g(v), x^{0}(g) are linearly independent and their orientation is independent of the choice of v. Thus the direction of x^{0}(g) is well defined by the above conditions. An example of a parabolic element in O(2, 1) is given by the following matrix:

g = [\begin{matrix}1 & s & s \\ −s & 1 − s^{2}/2 & −s^{2}/2 \\ s & s^{2}/2 & 1 + s^{2}/2 \end{matrix}]

for s ≠ 0, and the matrix has fixed eigenvector

u = \frac{1}{\sqrt{2}} [\begin{matrix}0 \\ −1 \\ 1 \end{matrix}] .

Taking

v = [\begin{matrix}0 \\ 1 \\ 1 \end{matrix}] ,

the orientation of the basis by g, the vectors v, g(v), x^{0}(g) are linearly independent and since

det(v, g(v), u) = −4s

x^{0}(g) = ∓u depending on the sign of s ≠ 0.

Following [4], we extend the definition of α to parabolic elements. Let F (g) ⊂ ℝ^{2,1} be the linear subspace of fixed vectors of g and Λ(g) be the set of linear real valued functions on F (g) . The orientation on F (g) is given by the above conditions. For any h ∈ Isom^{0}(𝔼) such that 𝕃(h) is hyperbolic or parabolic, we define α′(h) ∈ Λ(h) such that

α′(h)(v) = 𝔹(hx − x, v) (3)

We say that the sign of α′ for an element h is positive if α′(h)(v) > 0 for a positively oriented vector v ∈ F (g).

The properties of α′ are investigated more fully in [4]. For our purposes, we note that:

• α′(h) = 0 if and only if h has a fixed point.

• α′(h)(x^{0}(g)) = α(h) for a hyperbolic h.

• Margulis’s Opposite Sign Lemma [17], [18] holds for nonelliptic elements: If the linear parts of nonelliptic elements h_{1}, h_{2} do not commute and the signs of α′(h_{1}) and α′(h_{1}) are opposite then ⟨h_{1}, h_{2}⟩ does not act properly on 𝔼.

Thus α is a normalized version of α′.

Corollary 1. Suppose that Γ_{0} is a discrete free subgroup of SO(2, 1)^{0} and that u, v ∈ Z^{1}(Γ_{0}, ℝ^{2,1}) define affine deformations with α′(u) = α′(v). Then [u] = [v].

Note that any discrete free subgroup of SO(2, 1)^{0} contains only parabolic and hyperbolic nonidentity elements.

The proof of Lemma 2 can be adapted for non-hyperbolic elements (and this corollary) easily. In particular, given that α′(h) = 0 for a parabolic transformation h, there still is a unique pointwise fixed line which we can call C_{h}. The planes E^{±}(h) in the proof should be replaced with a single plane tangent to the null cone containing C_{h}.

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