ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 1, Issue 1, 1995

CROOKED PLANES

TODD A. DRUMM AND WILLIAM M. GOLDMAN

(Communicated by Gregory Margulis)

Abstract. Crooked planes are polyhedra used to construct fundamental polyhedra for discrete groups of Lorentz isometries acting properly on Minkowski (2+1)-space. These fundamental polyhedra are regions bounded by disjoint crooked planes. We develop criteria for the intersection of crooked planes and apply these criteria to proper discontinuity of discrete isometry groups.

1. Introduction

Let Γ be a group of isometries of Minkowski (2+1)-space 𝔼 acting freely and properly discontinuously on 𝔼. The quotient space 𝔼/Γ is complete flat Lorentz space-time. In 1983, Margulis constructed nonamenable subgroups of the Lorentz isometry group which act properly discontinuously on 𝔼 ([12],[13]), realizing a suggestion of Milnor [15]. That is, there exist complete flat Lorentz 3-manifolds M whose fundamental groups are nonamenable (that is, not virtually solvable) and these manifolds will be called Margulis space-times. By [7] the fundamental group of a Margulis space-time is a free group.

In [3], complete flat Lorentz manifolds with nonabelian free fundamental group were directly constructed using Poincaré’s fundamental polyhedron technique ([3], [4]). These examples are diffeomorphic to solid handlebodies. Their fundamental groups are “affine Schottky groups” with fundamental domains derived from fundamental domains of Schottky subgroups of SO^{0}(2, 1) acting on the hyperbolic plane H^{2}. These fundamental polyhedra are bounded by certain disjoint polyhedral hypersurfaces in ℝ^{3}, called crooked planes.

Using crooked planes, it was shown in [4] that the underlying linear group of a free group of Lorentz isometries acting freely and properly discontinuously on Minkowski (2+1)-space can be any free discrete subgroup of SO(2, 1).

Conjecture 1. Every Margulis space-time has a fundamental polyhedron whose faces are crooked planes.

The classification of intersections of crooked planes is the beginning of our program to analyze the moduli space of Margulis space-times.

\begin{matrix}c \\ ◯ \end{matrix}1995 American Mathematical Society

Received by the editors March 11, 1995.

1991 Mathematics Subject Classification. 51.

Key words and phrases. Space-times, affine, fundamental polyhedra, crooked planes.

Goldman was partially supported by NSF grant DMS-8902619 and the University of Maryland Institute for Advanced Computer Studies. Drumm was partially supported by an NSF Postdoctoral fellowship, and thanks Swarthmore College’s KIVA project for their hospitality.

2. Minkowski 2 + 1-space

Define ℝ^{2,1} to be the 3-dimensional real vector space with the indefinite inner product

𝔹 (u, v) = u^{T} · I_{L} · v,

where u and v are written in terms of the usual basis {e_{1}, e_{2}, e_{3}} and I_{L} is the diagonal matrix with entries +1, +1, −1. Define the Lorentzian cross-product as

u ⊠ v = I_{L} · (u × v),

where × is the usual cross product. Then 𝔹 (x, x ⊠ y) = 0. Let O(2, 1) be the full group of linear automorphisms of ℝ^{2,1}, with unimodular subgroup SO(2, 1), and identity component SO^{0}(2, 1).

For a vector v denote its Euclidean norm by |v|, the line containing v by ℝv, and the ray containing v by ℝ_{+}v. A nonzero vector v ∈ ℝ^{2,1} is called

• spacelike if 𝔹 (v, v) > 0 (and unit-spacelike if 𝔹 (v, v) = 1);

• null (or lightlike) if 𝔹 (v, v) = 0;

• timelike if 𝔹 (v, v) < 0;

The light cone 𝕎 is the set of all null vectors and its upper nappe 𝕎^{+} is called the positive time-orientation. Similarly 𝕌 is the set of timelike vectors with its upper connected component 𝕌^{+} defining the positive time orientation.

For any nonzero vector v ∈ ℝ^{2,1}, its Lorentz-perpendicular plane is

P(v) = {u ∈ ℝ^{2,1} | 𝔹 (u, v) = 0}

If v is a spacelike vector, P(v)∩𝕎 is the union of 2 null lines. There exists a unique pair

x^{−}(v), x^{+}(v) ∈ P(v) ∩ 𝕎^{+}

such that both have Euclidean norm 1 and the ordered triple (v, x^{−}(v), x^{+}(v)) is a positively-oriented basis of ℝ^{2,1}.

For a lightlike w ∈ 𝕎^{+} with Euclidean norm 1, the plane P(w) is the plane tangent to the light cone containing the line ℝw. The null line ℝw divides P(w) into two half-planes with closures denoted P^{+}(w) and P^{−}(w). Choose P^{+}(w) so that v ∈ P^{+}(w) when w = x^{+}(v). Choose P^{−}(w) so that v ∈ P^{−}(w) when w = x^{−}(v).

Let 𝔼 denote Minkowski (2+1)-space, the affine space corresponding to ℝ^{2,1}. Minkowski (2+1)-space is a simply connected complete flat Lorentzian manifold of dimension 2+1. For p, q ∈ 𝔼 the vector v = q − p corresponds to the translation carrying p to q. We also write q = p + v. Identify the vector space of translations of 𝔼 with the original inner product space ℝ^{2,1}. Denote the group of isometries of 𝔼 by

K = O(2, 1) ⋉ ℝ^{2,1},

and the homomorphism assigning to h ∈ K its linear part by:

𝕃 : H → O(2, 1).

Figure 1. A crooked plane

Let H and H^{0} be the subgroups consisting of all h ∈ K whose linear parts are 𝕃(h) ∈ SO(2, 1) and 𝕃(h) ∈ SO^{0}(2, 1), respectively. For isometries h ∈ H of 𝔼, write h(x) = g(x) + v where g = 𝕃(h) ∈ SO(2, 1). The isometries g and h are called

• hyperbolic if g has 3 distinct real eigenvalues;

• parabolic if g has 1 as its only eigenvalue;

• elliptic if g has imaginary eigenvalues; If v is spacelike or timelike and l = p + ℝv is a line parallel to v, then ι_{l} denotes the inversion in line l. If v is spacelike, then ι_{l} ∉ H.

3. Crooked planes

Our strategy is to derive fundamental polyhedra for actions on Minkowski space from fundamental polygons for the corresponding actions on H^{2}. A convenient model for H^{2} is the hyperboloid defined by 𝔹 (v, v) = −1, which has an induced Riemannian metric of constant curvature -1. Let H^{2} be the component of the hyperboloid lying in 𝕌^{+}. Points of H^{2} correspond to timelike lines and geodesics in H^{2} to spacelike planes, that is to planes that are Lorentz perpendicular to spacelike vectors. Suppose that v ∈ ℝ^{2,1} is spacelike. Then the plane P(v) ⊂ ℝ^{2,1} meets 𝕌 in timelike lines determining a geodesic in H^{2}.

Two spacelike planes always intersect in a line. In particular, two ultraparallel geodesics in H^{2} do not intersect but the spacelike planes corresponding to the geodesics intersect in a spacelike line. Any translations of these spacelike planes also intersect. We now introduce crooked planes as objects in 𝔼 that correspond to geodesics in H^{2}.

Let p ∈ 𝔼 be a point and v ∈ ℝ^{2,1} a spacelike vector. Define the positively oriented crooked plane C(v, p) ⊂ 𝔼 with vertex p and direction vector v to be the union of two wings

W^{+}(v, p) = p + P^{+}(x^{+}(v)), W^{−}(v, p) = p + P^{+}(x^{−}(v))

and the stem

S(v, p) = p + {x ∈ ℝ^{2,1} | 𝔹 (v, x) = 0, 𝔹 (x, x) ≤ 0}.

Define the negatively oriented crooked plane K(v, p) ⊂ 𝔼 with vertex p and direction vector v to be the union of two wings

M^{+}(v, p) = p + P^{−}(x^{+}(v)), M^{−}(v, p) = p + P^{−}(x^{−}(v))

and the stem S(v, p) defined above.

For the crooked plane C(v, p) (or K(v, p)) there exists a unique spacelike line p + ℝv lying on it which will be called the spine of the crooked plane. Given a spacelike line ℓ and a point p ∈ ℓ, there is a unique positively (or negatively) oriented crooked plane with spine ℓ and vertex p.

We also define the positively oriented crooked half-space H(v, p) and the negatively oriented crooked half-space N (v, p) to be the components of 𝔼 − C(v, p) and 𝔼 − K(v, p), respectively, which contain

p + {u ∈ 𝕎^{+} | 𝔹 (v, u) > 0}.

4. Intersections of crooked planes

Since two nonparallel half planes of differing orientation meet in a ray, two oppositely oriented crooked planes always intersect:

Theorem 2 ([9]). Let v_{1} and v_{2} be linearly independent spacelike vectors and let p_{1}, p_{2} ∈ 𝔼. Then C(v_{1}, p_{1}) ∩ K(v_{2}, p_{2}) is nonempty.

To examine the intersection of two crooked planes of the same orientation we introduce the following terminology: Vectors v_{1}, v_{2} are said to be ultraparallel, asymptotic, or crossing if P(v_{1}) and P(v_{2}) correspond to geodesics in H^{2} which are ultraparallel, asymptotic, or crossing, respectively (compare [9]).

For crossing spacelike vectors v_{1} and v_{2}, the intersection of the stems S(v_{1}, p_{1}) and S(v_{2}, p_{2}) contains two rays. Since C(v_{2}, p_{2}) and K(v_{2}, p_{2}) share a stem,

Theorem 3 ([9]). Let v_{1} and v_{2} be crossing spacelike vectors. Then the intersection of any two crooked planes with spines parallel to v_{1} and v_{2} is nonempty.

Now consider crooked planes of the same orientation whose spines are either ultraparallel or asymptotic. The following terminology will be useful since C(v, p) = C(−v, p) (and K(v, p) = K(−v, p)). Spacelike vectors v_{1}, v_{2}, …, v_{n} are said to be consistently oriented if 𝔹 (v_{i}, v_{j}) ≤ 0 and 𝔹 (v_{i}, x^{±}(v_{j})) ≤ 0 for i ≠ j. This means that the half-planes H(v_{i}), H(v_{j}) ∈ H^{2} corresponding to H(v_{i}, p) and H(v_{j}, p) do not intersect.

Theorem 4 ([9]). Let v_{1} and v_{2} be consistently oriented, ultraparallel, unit-space-like vectors and p_{1}, p_{2} ∈ 𝔼. The positively oriented crooked planes C(v_{1}, p_{1}) and C(v_{2}, p_{2}) are disjoint if and only if

𝔹 (p_{2} − p_{1}, v_{1} ⊠ v_{2}) > |𝔹 (p_{2} − p_{1}, v_{2})| + |𝔹 (p_{2} − p_{1}, v_{1})|.  (4.0.1)

The negatively oriented crooked planes K(v_{1}, p_{1}) and K(v_{2}, p_{2}) are disjoint if and only if

𝔹 (p_{2} − p_{1}, v_{1} ⊠ v_{2}) < −(|𝔹 (p_{2} − p_{1}, v_{2})| + |𝔹 (p_{2} − p_{1}, v_{1})|).  (4.0.2)

Figure 2. Disjoint ultraparallel crooked planes

These conditions follow from explicit description of the intersections of the components of crooked planes. We rephrase them in terms of the displacement vector w = p_{2} − p_{1} between their vertices. Fix spacelike vectors v_{1} and v_{2} and a point p ∈ 𝔼. Consider crooked planes with vertex p_{1} = p and p_{2} = p + w. For any w ∈ ℝ^{2,1}, the stems S(v_{1}, p) and and S(v_{2}, p + w) are disjoint if and only if

|𝔹 (p_{2} − p_{1}, v_{1} ⊠ v_{2})| > |𝔹 (p_{2} − p_{1}, v_{2})| + |𝔹 (p_{2} − p_{1}, v_{1})|.

The set

A = {w ∈ ℝ^{2,1} | S(v_{1}, p) ∩ S(v_{2}, p + w) = ∅}

is a disjoint union of the two open pyramids

A_{+} = {w ∈ ℝ^{2,1} | C(v_{1}, p) ∩ C(v_{2}, p + w) = ∅}

and

A_{−} = {w ∈ ℝ^{2,1} | K(v_{1}, p) ∩ K(v_{2}, p + w) = ∅}

Theorem 5 ([9]). Let v_{1} and v_{2} be consistently oriented, asymptotic, unit-spacelike vectors. Let C_{i} = C(v_{i}, p_{i}) and K_{i} = K(v_{i}, p_{i}) for i = 1, 2. The positively oriented crooked planes C_{1} and C_{2} are disjoint if and only if

\{ \begin{matrix}x^{−}(v_{1}) = x^{+}(v_{2}), \\ 𝔹 (p_{2} − p_{1}, v_{1}) < 0, \\ 𝔹 (p_{2} − p_{1}, v_{2}) < 0, \\ 𝔹 (p_{2} − p_{1}, x^{+}(v_{1}) ⊠ x^{−}(v_{2})) > 0; \end{matrix} \} or \{ \begin{matrix}x^{+}(v_{1}) = x^{−}(v_{2}), \\ 𝔹 (p_{2} − p_{1}, v_{1}) > 0, \\ 𝔹 (p_{2} − p_{1}, v_{2}) > 0, \\ 𝔹 (p_{2} − p_{1}, x^{−}(v_{1}) ⊠ x^{+}(v_{2})) > 0. \end{matrix} \}

The negatively oriented crooked planes K_{1} and K_{2} are disjoint if and only if

\{ \begin{matrix}x^{−}(v_{1}) = x^{+}(v_{2}), \\ 𝔹 (p_{2} − p_{1}, v_{1}) > 0, \\ 𝔹 (p_{2} − p_{1}, v_{2}) > 0, \\ 𝔹 (p_{2} − p_{1}, x^{+}(v_{1}) ⊠ x^{−}(v_{2})) < 0; \end{matrix} \} or \{ \begin{matrix}x^{+}(v_{1}) = x^{−}(v_{2}), \\ 𝔹 (p_{2} − p_{1}, v_{1}) < 0, \\ 𝔹 (p_{2} − p_{1}, v_{2}) < 0, \\ 𝔹 (p_{2} − p_{1}, x^{−}(v_{1}) ⊠ x^{+}(v_{2})) < 0. \end{matrix} \}

Figure 3. Intersecting ultraparallel crooked planes

A pair of positively oriented ultraparallel crooked planes which are disjoint are shown in Figure 2. A pair of negatively oriented ultraparallel crooked planes with the same stems and translation vectors as in Figure 2 are shown in Figure 3 and have a nonempty intersection. These two figures illustrate the following more general fact.

Corollary 6 ([9]). Let v_{1} and v_{2} be spacelike vectors. Either C(v_{1}, p_{1}) ∩ C(v_{2}, p_{2}) or K(v_{1}, p_{1}) ∩ K(v_{2}, p_{2}) is nonempty.

5. Applications

5.1. Groups generated by inversions. For any spacelike line l and p ∈ l, the inversion ι_{l} preserves C(v, p) leaving each wing invariant, permuting the two components of W^{+}(v, p) and W^{−}(v, p) and the two components of S(v, p) − {p}. Further, ι_{l}(H(v, p)) = H(−v, p) so that E −H(±v, p) are fundamental domains for the action of the group generated by the inversion in l.

Alternatively, ι_{l} also preserves K(v, p) and E−N (±v, p) are fundamental domains for the action of the group generated by the inversion in l.

Identify ℝ^{2,1} with 𝔼 via a choice of origin o ∈ 𝔼. For l = p+ℝv, let ℓ = ℝv ⊂ ℝ^{2,1} be the line parallel to l through the origin. The inversion ι_{ℓ} of ℝ^{2,1} through ℓ equals the reflection of H^{2} through the geodesic

S(l) = P(v) ∩ H^{2} = C(v, o) ∩ H^{2} = K(v, o) ∩ H^{2}.

Let Δ ⊂ H^{2} be a convex n-gon whose pairs of adjacent sides S_{1}, …, S_{n} are ultraparallel. ι_{ℓ_{i}} is the reflection in S_{i} and let Γ ⊂ SO(2, 1) be the group generated by ι_{ℓ_{1}}, …, ι_{ℓ_{n}}. Then Γ acts properly and discretely on H^{2} with fundamental domain Δ. Choose v_{1}, …, v_{n} to be consistently oriented so that

Δ = H^{2} − (∪_{i=1}^{n}H(v_{i}))

An affine deformation of Γ is a collection l = (l_{1}, …, l_{n}) of oriented spacelike lines such that each l_{i} has direction vector v_{i}. Such an affine deformation determines an affine representation of Γ whose image Γ_{l} is generated by inversions ι_{j} in l_{j}. Lines l ⊂ 𝔼 with direction vector v are parametrized by a point p ∈ l which is determined up to translation by ℝv. Affine deformations of Γ comprise the product

𝔼/ℝv_{1} × ⋯ × 𝔼/ℝv_{n}

of the quotient affine spaces 𝔼/ℝv_{j}.

A fundamental domain for the action of Γ_{l} is built from disjoint crooked planes C_{i} = C(v_{i}, q_{i}) with spine l_{i} which bound a common region. Since a crooked plane with given spine l is determined by its vertex, the family (C_{1}, …, C_{n}) is determined by the n-tuple q = (q_{1}, …, q_{n}) ∈ l_{1} × ⋯ × l_{n}.

Theorem 7 ([9]). Let Γ ⊂ SO(2, 1) be as above with consistently oriented v_{i}’s parallel to the ℓ_{i}’s. An affine deformation Γ_{l} of Γ is properly discontinuous if there exist q ∈ l_{1} × ⋯ × l_{n} such that either (but not both) of the following are true.

• C(q_{1}, v_{1}), …, C(q_{n}, v_{n}) are disjoint and bound a common region

𝔼 − (∪_{i=1}^{n}H(q_{i}, v_{i})) ,

which is a fundamental domain for the action.

• K(q_{1}, v_{1}), …, K(q_{n}, v_{n}) are disjoint and bound a common region

𝔼 − (∪_{i=1}^{n}N (q_{i}, v_{i})) ,

which is a fundamental domain for the action.

These results follow directly from Theorem 3.5 of [4]. Conversely:

Conjecture 8. Let Γ ⊂ SO(2, 1) be as above. An affine deformation Γ_{l} of Γ is properly discontinuous if and only if there exist q ∈ l_{1} × ⋯ × l_{n} such that either (but not both) of the following are true.

1. C(q_{1}, v_{1}), …, C(q_{n}, v_{n}) are disjoint and bound a common region.

2. K(q_{1}, v_{1}), …, K(q_{n}, v_{n}) are disjoint and bound a common region.

5.2. Groups generated by hyperbolic transformations. All g ∈ SO(2, 1) have 1 as an eigenvalue. For hyperbolic g, we define x^{0}(g) to be the unique fixed unit-spacelike vector such that x^{−}(x^{0}(g)) is an eigenvector with a corresponding eigenvalue < 1, and x^{+}(x^{0}(g)) is an eigenvector with a corresponding eigenvalue > 1.

For hyperbolic h(x) = g(x) + v, Margulis ([12],[13]) defined an invariant α(h) = 𝔹 (v, x^{0}(g)). h admits a fixed point if and only if α(h) = 0. Further, Margulis has shown ([12],[13], see also [7]) that

Theorem 9 (Margulis). If h_{1} and h_{2} are hyperbolic Lorentz transformations such that α(h_{1}) and α(h_{2}) have opposite signs then any group containing h_{1} and h_{2} does not act properly on ℝ^{2,1}.

If there exist crooked planes C(v_{1}, p_{1}) and C(v_{2}, p_{2}) such that h(C(v_{1}, p_{1})) = C(v_{2}, p_{2}) and v_{1} and x^{0}(g) cross (equivalently, v_{2} and x^{0}(g) cross) then the region bounded by C(v_{1}, p_{1}) and C(v_{2}, p_{2}) is a fundamental domain for the action of ⟨h⟩ on ℝ^{2,1}. The orientation of the crooked planes is related to α(h) by the following corollary of Theorems 4 and 5:

Corollary 10. Let h ∈ H be hyperbolic. If α(h) < 0 (α(h) > 0) then there is no fundamental domain of the action of the group generated by h on 𝔼 bounded by disjoint positively (negatively) oriented crooked planes.

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Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 (Drumm), Department of Mathematics, University of Maryland, College Park, MD 20742 (Goldman)

  E-mail address: tad@math.upenn.edu (Drumm), wmg@math.umd.edu (Goldman)