COMPLETE FLAT LORENTZIAN 3-MANIFOLDS WITH FREE FUNDAMENTAL GROUP

TODD A. DRUMM AND WILLIAM M. GOLDMAN

Contents

Introduction 1 1. The Geometry of ℝ^{2,1}. 2 2. Schottky groups in SO(2, 1) 10 3. Drumm’s geometric conditions for proper discontinuity 13 References 14

Introduction

In 1983, G.A.Margulis proved the following startling result, answering a question posed by Milnor [5] (see also [2]):

Theorem (Margulis). There exists a nonabelian free discrete group Γ of affine transformations which acts properly and freely on ℝ^{3}.

In this paper we give a new proof of this result, using ideas from Margulis’ original proof [3, 4] as well as geometric conditions due to the first author [1]. Let ℝ^{2,1} denote a 3-dimensional real vector space with Lorentzian inner product

𝔹(u, v) = u_{1}v_{1} + u_{2}v_{2} − u_{3}v_{3}

and SO(2, 1) its group of unimodular linear automorphisms. A Schottky subgroup of SO(2, 1) is a discrete free subgroup of SO(2, 1) such that every element other than the identity has distinct real eigenvalues. (The usual definition is more geometric, although it is equivalent to this one.) Let 𝔼 denote the affine space corresponding to ℝ^{2,1}; as (flat) Minkowski space, it is a simply connected complete flat Loretzian manifold and as such is unique up to Lorentz isometry. We denote the group of translations of 𝔼 by V; as a vector space is naturally identified with ℝ^{2,1}. Let G be the identity component of SO(2, 1) and H = G ⋊ V denote the identity component in the group of isometries of 𝔼 and 𝕃 : H ⟶ G be the homomorphism assigning to h ∈ H its linear part. If Λ ⊂ G is a subgroup, an affine deformation of Λ is a representation ρ : Λ ⟶ H such that 𝕃(ρ(g)) = g for all g ∈ Λ. Since such a homomorphism ρ is necessarily an embedding, we shall also refer to its image ρ(Λ) as an affine deformation of Λ.

Both authors gratefully acknowledge partial support from National Science Foundation grant DMS-8613576 as well as the Alfred P. Sloan Foundation for the second author. This is a revised version of a paper which appeared in International Journal of Mathematics, 1 (1990), 149–161.

Theorem. Every Schottky group of genus k > 1 inside SO(2, 1) possesses an affine deformation which acts freely and properly discontinuously on 𝔼.

(This answers a question raised in Margulis [3].) The set of affine deformations of Λ is the space Z^{1}(Λ, ℝ^{2,1}) of 1-cocycles and our proof provides a nonempty open cone of 1-cocycles determining properly discontinuous affine deformations.

In [5] it is shown that a connected group of affine transformations acts properly only if it is amenable; thus it seemed natural to conjecture that a closed subgroup (in particular a discrete subgroup) of the group of affine transformations which acts properly must also be amenable. In [5] Milnor showed this question is equivalent to whether a free group of rank two can act properly discontinuously by affine transformations. In [2] it is shown that a discrete group Γ of affine transformations acting properly on ℝ^{3} must be amenable, provided that the quotient space is compact; indeed it is shown there that unless Γ is amenable it preserves a parallel Lorentz metric. In [1] fundamental polyhedra for these actions are constructed explicitly from which it is shown that the quotient manifolds are solid handlebodies.

We are grateful to G.A.Margulis and G.Soifer for several helpful suggestions.

1. The Geometry of ℝ^{2,1}.

The set {x ∈ ℝ^{2,1} | 𝔹(x, x) = 0} of null vectors forms the the light cone and its two components form the time-orientations of ℝ^{2,1}. We choose one component as the positive time-orientation and denote it:

𝕎^{+} = {x ∈ ℝ^{2,1} | 𝔹(x, x) = 0, x_{3} > 0}

If u ∈ ℝ^{2,1} we denote by ∥u∥ the usual Euclidean norm of u and the Euclidean distance by ρ(u, v) = ∥u − v∥. The Euclidean unit sphere S^{2} consists of u such that ∥u∥ = 1.

The Lorentzian unit sphere (one-sheeted hyperboloid) in ℝ^{2,1} consists of vectors u ∈ ℝ^{2,1} satisfying 𝔹(u, u) = 1 and will be denoted by 𝔖. If u ∈ ℝ^{2,1}, its Lorentzian orthogonal complement u^{⊥} is the linear subspace {v ∈ ℝ^{2,1} | 𝔹(u, v) = 0}. If u ∈ 𝔖, there are uniquely determined null vectors x^{+}(u), x^{−}(u) ∈ 𝕎^{+} ∩ S^{2} ∩ u^{⊥} such that

{u, x^{−}(u), x^{+}(u)}

is a positively oriented basis of ℝ^{2,1}. Explicitly, given u ∈ 𝔖 (so that u_{1}^{2} + u_{2}^{2} = 1 + u_{3}^{2}) we have:

x^{−}(u) = \frac{1}{\sqrt{2}} [\begin{matrix}(u_{1}u_{3} − u_{2})/(u_{1}^{2} + u_{2}^{2}) \\ (u_{2}u_{3} + u_{1})/(u_{1}^{2} + u_{2}^{2}) \\ 1 \end{matrix}] ,

x^{+}(u) = \frac{1}{\sqrt{2}} [\begin{matrix}(u_{1}u_{3} + u_{2})/(u_{1}^{2} + u_{2}^{2}) \\ (u_{2}u_{3} − u_{1})/(u_{1}^{2} + u_{2}^{2}) \\ 1 \end{matrix}] .

The two rulings of the hyperboloid 𝔖 consist of the lines

{u ∈ 𝔖 | x^{+}(u) = p} (respectively{u ∈ 𝔖 | x^{−}(u) = p})

as p varies over the circle ℙ = 𝕎^{+} ∩ S^{2} of (positively time-oriented null directions). Alternatively, there exist unique polar coordinates r ≥ 1, 0 ≤ θ < 2π and a sign such that

u = [\begin{matrix}r cos(θ) \\ r sin(θ) \\ ±\sqrt{r^{2} − 1} \end{matrix}] = R_{θ} [\begin{matrix}r \\ 0 \\ ±\sqrt{r^{2} − 1} \end{matrix}]

where

R_{θ} = [\begin{matrix}cos(θ) & − sin(θ) & 0 \\ sin(θ) & cos(θ) & 0 \\ 0 & 0 & 1 \end{matrix}] ∈ SO(2, 1) ∩ O(3)

is an isometry of ℝ^{2,1} preserving the Euclidean geometry as well. Then

S^{2} ∩ 𝕎^{+} ∩ u^{⊥} = {x^{−}(u), x^{+}(u)}

where

x^{+}(u) = R_{θ} \frac{1}{\sqrt{2}r} [\begin{matrix}±\sqrt{r^{2} − 1} \\ −1 \\ r \end{matrix}] ,

x^{−}(u) = R_{θ} \frac{1}{\sqrt{2}r} [\begin{matrix}±\sqrt{r^{2} − 1} \\ +1 \\ r \end{matrix}] .

and these two vectors lie at distance \sqrt{2}/r from each other.

Let ϵ > 0. We say that u ∈ 𝔖 is ϵ-spacelike if

ρ(x^{+}(u), x^{−}(u)) ≥ ϵ.

We denote the space of ϵ-spacelike vectors by 𝔖_{ϵ}. If ϵ > ϵ′ > 0, then 𝔖_{ϵ} ⊂ 𝔖_{ϵ′}. If ϵ > \sqrt{2} then 𝔖_{ϵ} is empty and if ϵ < \sqrt{2} then the above calculation shows that 𝔖_{ϵ} is a compact annulus and 𝔖 may be written as the increasing union:

𝔖 = ⋃_{\sqrt{2}>ϵ>0} 𝔖_{ϵ}

Lemma 1.1. Let J^{+} and J^{−} are disjoint closed nondegenerate intervals in the circle ℙ of null directions. Then the set

J = {u ∈ 𝔖 | x^{+}(u) ∈ J^{+}, x^{−}(u) ∈ J^{−}}

is a nondegenerate quadrilateral embedded in 𝔖 (whose edges are straight lines, segments of the rulings of 𝔖). which lies in an open half-space.

In particular the “dual” of J, the set of all v ∈ ℝ^{2,1} such that 𝔹(v, u) > 0 for all u ∈ J, is a compact convex cone in ℝ^{2,1} having nonempty interior.

Proof. That J is a quadrilateral follows from the definition of x^{+} and x^{−} in terms of the rulings. For the other assertions, note that the “Lorentzian cross-product”

⊠ : ℝ^{2,1} × ℝ^{2,1} ⟶ ℝ^{2,1}
([\begin{matrix}a_{1} \\ a_{2} \\ a_{3} \end{matrix}] , [\begin{matrix}b_{1} \\ b_{2} \\ b_{3} \end{matrix}]) ↦ [\begin{matrix}a_{2}b_{3} − a_{3}b_{2} \\ a_{3}b_{1} − a_{1}b_{3} \\ a_{2}b_{1} − a_{1}b_{2} \end{matrix}]

is a bilinear mapping defined by:

𝔹(a ⊠ b, c) = det(a, b, c)

where a ∧ b ∧ c ↦ det(a, b, c) is the isomorphism ∧^{3}ℝ^{2,1} ⟶ ℝ corresponding to the fixed orientation of ℝ^{2,1}. Elementary consequences of the definition are:

• 𝔹(a, a ⊠ b) = 𝔹(b, a ⊠ b) = 0;

• a ⊠ b = −b ⊠ a;

• If a, b ∈ ℙ are linearly independent, then a⊠b is spacelike, x^{−}(a⊠ b) = a, and x^{−}(a ⊠ b) = b;

• 𝔹(a ⊠ b, a ⊠ b) = 𝔹(a, a)𝔹(b, b) − 𝔹(a, b)^{2} In terms of this operation, J can be described as the image J^{−} ⊠ J^{+}. We can find two points which divide ℙ into two open intervals I^{−} and I^{+} containing J^{−} and J^{+} respectively and by applying an element of G may assume these intervals consist of vectors

[\begin{matrix}sin(θ) \\ cos(θ) \\ 1 \end{matrix}]

where sin(θ) > 0 and sin(θ) < 0 respectively. Now

𝔹 ([\begin{matrix}0 \\ −1 \\ 0 \end{matrix}] , I^{−} ⊠ I^{+}) > 0

whence J = J^{−} ⊠ J^{+} ⊂ I^{−} ⊠ I^{+} lies in a half-space. The proof of the lemma is complete.

Lemma 1.2. Let u ∈ 𝔖 be ϵ-spacelike. Then for any vector v ∈ ℝ^{2,1},

|𝔹(v, u)| ≤ 2ϵ^{−1}∥v∥

Proof. By applying R_{θ} we may assume that

u = [\begin{matrix}r \\ 0 \\ ±\sqrt{r^{2} − 1} \end{matrix}]

and since u ∈ 𝔖_{ϵ},

r ≤ \frac{\sqrt{2}}{ϵ}.

Now

|𝔹(v, u)| = |rv_{1} − \sqrt{r^{2} − 1} v_{3}|

≤ r|v_{1} + v_{3}|

≤ \frac{\sqrt{2}}{ϵ}\sqrt{2}∥v∥

as claimed.

An element g ∈ G is hyperbolic if it has three distinct real eigenvalues; in this case the eigenvalues are positive and form the sequence

λ(g) < 1 < λ(g)^{−1}.

We denote the set of hyperbolic elements of G by H. Associated to every hyperbolic element g ∈ H is its characteristic frame x(g) defined as follows. Let A^{−}(g) denote the λ(g)-eigenspace of g and A^{+}(g) denote the λ(g)^{−1}-eigenspace of g. Then there exists a unique x^{0}(g) ∈ 𝔖 such that

x^{+}(g) = x^{+}(x^{0}(g)),
x^{−}(g) = x^{−}(x^{0}(g)).

In summary, given g ∈ H, x(g) = {x^{0}(g), x^{−}(g), x^{+}(g)} is the unique positively oriented basis of ℝ^{2,1} such that:

• gx^{0}(g) = x^{0}(g);

• gx^{−}(g) = λ(g)x^{−}(g) and x^{−}(g) ∈ S^{2} ∩ 𝕎^{+};

• gx^{+}(g) = λ(g)^{−1}x^{+}(g) and x^{+}(g) ∈ S^{2} ∩ 𝕎^{+}.

Lemma 1.3. Let g ∈ G, h ∈ H and ξ = ±1. Then

𝔹(g(ξv), x^{0}(h)) = 𝔹(v, x^{0}(g^{−1}h^{ξ}g)).

Proof. Since x^{±}(h^{−1}) = x^{∓}(h), it follows that x^{0}(h^{ξ}) = ξx^{0}(h). Furthermore uniqueness of x^{±}(h) implies x^{+}(g^{−1}hg) = g^{−1}x^{+}(h) and x^{−}(g^{−1}hg) = g^{−1}x^{−}(h). Thus x^{0}(g^{−1}hg) = g^{−1}x^{0}(h) and

𝔹(g(ξv), x^{0}(h)) = ξ𝔹(v, g^{−1}x^{0}(h))
= 𝔹(v, ξg^{−1}x^{0}(h))
= 𝔹(v, x^{0}(g^{−1}h^{ξ}g))

as desired.

Let ϵ > 0. A hyperbolic element g ∈ H is said to be ϵ-hyperbolic if x^{0}(g) is ϵ-spacelike. Equivalently g is ϵ-hyperbolic if

ρ(x^{+}(g), x^{−}(g)) ≥ ϵ.

We denote the set of ϵ-hyperbolic elements by H_{ϵ}. For each 0 < ϵ < \sqrt{2} the function

λ : H_{ϵ} ⟶ (0, 1)

is a proper map and decomposes H_{ϵ} into a fibration over the open interval with fiber a compact annulus. Furthermore

H = ⋃_{\sqrt{2}>ϵ>0} H_{ϵ}.

Now let h ∈ H be an isometry of 𝔼. We say that h is hyperbolic (respectively ϵ-hyperbolic) if its linear part 𝕃(h) is hyperbolic (respectively ϵ-hyperbolic). We shall write

x^{0}(h) = x^{0}(𝕃(h)),
x^{+}(h) = x^{+}(𝕃(h)),

x^{−}(h) = x^{−}(𝕃(h)).

If h ∈ H is hyperbolic and u ∈ 𝔼 is arbitrary, then (following Margulis) consider the following real number:

α(h, u) = 𝔹(h(u) − u, x^{0}(h))

The following facts are established in Margulis[3, 4]:

Lemma 1.4. Let h ∈ H be hyperbolic. Then:

• α(h, u) is independent of u;

• There exists a line ℭ(h) invariant under h, parallel to x^{0}(h), such that the restriction of h is translation by α(h)x^{0}(h);

• h acts freely on 𝔼 ⇔ α(h) ≠ 0, in which case ℭ(h) is the unique h-invariant line;

• If n ∈ ℤ, α(h^{n}) = |n|α(h).

Since α(h, u) is independent of u, henceforth we write α(h) = α(h, u).

Proof. Let λ = λ(𝕃(h)). Every y ∈ ℝ^{2,1} may be decomposed in terms of the characteristic frame of h as follows:

y = 𝔹(y, x^{+}(h))x^{+}(h)
+ 𝔹(y, x^{−}(h))x^{−}(h)
+ 𝔹(y, x^{0}(h))x^{0}(h)

Then

h(y) = 𝔹(λy + h(0), x^{−}(h))x^{−}(h)
+ 𝔹(λ^{−1}y + h(0), x^{+}(h))x^{+}(h)
+ 𝔹(y + h(0), x^{0}(h))x^{0}(h)

whence

α(h, y) = 𝔹(h(y) − y, x^{0}(h))
= 𝔹(h(0), x^{0}(h))
= α(h, 0)

is independent of y. Let

y_{0} = \frac{1}{1 − λ}𝔹(h(0), x^{−}(h))x^{−}(h)

+ \frac{1}{1 − λ^{−1}} 𝔹(h(0), x^{+}(h))x^{+}(h)

then for y = y_{0} + tx^{0}(h),

h(y) = (\frac{λ}{1 − λ} + 1) 𝔹(h(0), x^{−}(h))x^{−}(h)

+ (\frac{λ^{−1}}{1 − λ^{−1}} + 1) 𝔹(h(0), x^{+}(h))x^{+}(h)

+ (t + 𝔹(h(0), x^{0}(h))x^{0}(h)

= y + α(h)x^{0}(h)

Thus the line

ℭ(h) = y_{0} + ℝx^{0}(h)

is invariant under h, is parallel to x^{0}(h), and h acts on it by translation by α(h)x^{0}(h). Furthermore h^{n} acts on ℭ(h) by translation by

nα(h)x^{0}(h);

since x^{0}(h^{−1}) = −x^{0}(h) and x^{0}(h^{n}) = x^{0}(h) for positive n, it follows that α(h^{n}) = |n|α(h).

Suppose that h does not act freely. For any y fixed under h, α(h) = α(h, y) = 0. Conversely, if α(h) = 0, then every point on ℭ(h) is fixed under h so that h does not act freely. To see that ℭ(h) is the unique h-invariant line if h acts freely, observe that any h-invariant line is parallel to one of the eigenvectors of the linear part 𝕃(h) and the restriction will be an affine transformation whose linear part is the corresponding eigenvalue. An affine transformation of a line which acts freely must have linear part equal to 1. Thus every h-invariant line must be parallel to x^{0}(h); however if there were more than one such line, there would be a plane upon which h acts by translations, contradicting the one-dimensionality of the 1-eigenspace of 𝕃(h).

One can concretely see the dynamics of such a cyclic group as follows. After a change of coordinates, we may write

h = [\begin{matrix}1 & 0 & 0 \\ 0 & (λ^{−1} + λ)/2 & (λ^{−1} − λ)/2 \\ 0 & (λ^{−1} − λ)/2 & (λ^{−1} + λ)/2 \end{matrix}] [\begin{matrix}t \\ 0 \\ 0 \end{matrix}]

with characteristic frame

x(h) = \{[\begin{matrix}1 \\ 0 \\ 0 \end{matrix}] , [\begin{matrix}0 \\ 1/\sqrt{2} \\ 1/\sqrt{2} \end{matrix}] , [\begin{matrix}0 \\ −1/\sqrt{2} \\ 1/\sqrt{2} \end{matrix}]\}

and ℭ(h) is the x-axis. In particular if h acts freely, the quotient 𝔼/⟨h⟩ is a complete flat Lorentz 3-manifold diffeomorphic to a solid torus having ℭ(h)/⟨h⟩ as a core (spacelike) closed geodesic. Margulis’ invariant α(h) = t represents a kind of “normalized signed Lorentzian length” of the core closed geodesic on the cylinder 𝔼/⟨h⟩.

We denote the free group of rank k by 𝔽_{k} and we denote the length of the word w ∈ 𝔽_{k} by ℓ(w).

Lemma 1.5 (Margulis). Suppose that h_{1}, …, h_{k} ∈ H satisfy the property that there exists ϵ > 0, g ∈ G, and a constant C > 0 such that for each w ∈ 𝔽_{k},

• gw(h_{1}, …, h_{k}) is ϵ-hyperbolic;

• |α(gw(h_{1}, …, h_{k}))| ≥ Cℓ(w). Then the subgroup Γ of G generated by h_{1}, …, h_{k} acts properly discontinuously on E.

Proof. It suffices to show that for each compact set K ⊂ 𝔼 the set

Γ(K) = {γ ∈ Γ | γK_{∩}K ≠ ∅}

is finite. If w ∈ 𝔽_{k}, let γ = w(h_{1}, …, h_{k}). Now for any u ∈ 𝔼 and w ∈ 𝔽_{k},

Cℓ(w) ≤ |α(gγ)|

= |𝔹(gγ(u) − u, x^{0}(gγ))| ≤ \frac{2}{ϵ}∥gγ(u) − u∥

= \frac{2}{ϵ}ρ(u, gγ(u))

by Lemma 1.2 Let K ⊂ 𝔼 be a compact set and suppose that γ = w(h_{1}, …, h_{k}) ∈ Γ satisfies K ∩ γ^{−1}K ≠ ∅. Let u ∈ K ∩ γ^{−1}K. Then gγ(u) ∈ gK and ρ(gγ(u), u) ≤ diam(K ∪ gK), and by the previous assertion,

ℓ(w) ≤ \frac{2}{ϵC} ρ(gγ(u), u)

≤ \frac{2}{ϵC} diam(K ∪ gK).

Therefore

Γ(K) ⊂ \{w(h_{1}, …, h_{k}) ∈ Γ | w ∈ 𝔽_{k}, ℓ(w) ≤ \frac{2}{ϵC} diam(K ∪ gK)\}

is finite as desired.

We observe that in general we cannot expect that w(h_{1}, …, h_{k}) itself is ϵ-hyperbolic. Consider a sequence of words γ_{n} = w_{1}^{n}w_{2}w_{1}^{−n} where w_{1} is hyperbolic; then both x^{+}(γ_{n}) = w_{1}^{n}x^{+}(w_{2}) and x^{−}(γ_{n}) = w_{1}^{n}x^{−}(w_{2}) converge to the same null vector x^{+}(w_{1}). For sufficiently large n, x^{+}(γ_{n}) and x^{−}(γ_{n}) will be arbitrarily close and γ_{n} will not be ϵ-hyperbolic. The following discussion will illustrate the importance of considering the coset gΓ rather than Γ itself. If γ corresponds to a reduced word it will be possible to determine the locations of the null eigenvectors x^{±}(γ) from its initial and terminal letters (Lemma 2.1). Although γ_{n} itself may correspond to a reduced word, its “rearrangement”

γ′_{n} = w_{1}^{n−1}w_{2}w_{1}^{−n}w_{1}

will generally not be reduced if the initial and terminal letters of γ_{n} happen to be inverses.

2. Schottky groups in SO(2, 1)

Let ℙ denote the set of rays in the positive nappe 𝕎^{+} of the light cone; ℙ is diffeomorphic to a circle upon which the group G acts by projective transformations. We realize ℙ concretely as the cross-section S^{2} ∩ 𝕎^{+}; then the action of g ∈ G on ℙ is given by

g : u ↦ \frac{g(u)}{∥g(u)∥}

where g ∈ G and u ∈ S^{2} ∩ 𝕎^{+}.

Choose 2k + 2 disjoint intervals A_{0}^{±}, A_{1}^{±}, …, A_{k}^{±} ⊂ ℙ, and elements g = g_{0}, g_{1}, …, g_{k} ∈ G such that

g_{i}(A_{i}^{−}) = ℙ − Ā_{i}^{+}

and choose ϵ > 0 less than the distance between any of the endpoints of these intervals. It follows that g_{i}^{−1}(A_{i}^{+}) = ℙ − Ā_{i}^{−} and the Brouwer fixed point theorem applied to the restriction of g_{i} to ℙ − Ā_{i}^{−} implies that

x^{+}(g_{i}) ∈ A_{i}^{+}

(since g_{i}(ℙ − Ā_{i}^{−}) ⊂ A_{i}^{+}). Similarly the Brouwer fixed point theorem applied to the restriction of g_{i}^{−1} to ℙ − Ā_{i}^{+} implies that

x^{−}(g_{i}) ∈ A_{i}^{−}.

Let w = w(x_{0}, …, x_{k}) ∈ 𝔽_{k+1}, then we write

w(x_{0}, …, x_{k}) = ∏_{i=1}^{l} x_{m_{i}}^{ξ_{i}}

where ξ_{i} = ±1 and m_{i} = 0, 1, …, k for i = 1, …, l = ℓ(w). Then w is a reduced word if and only if

m_{i} = m_{i+1} ⟹ ξ_{i} = ξ_{i+1}.

If 1 ≤ j ≤ l, the left-, right-subwords are defined by

w_{j}^{L}(x_{0}, …, x_{k}) = ∏_{i=1}^{j−1} x_{m_{i}}^{ξ_{i}}

w_{j}^{R}(x_{0}, …, x_{k}) = ∏_{i=j+1}^{l} x_{m_{i}}^{ξ_{i}}

so that

w(x_{0}, …, x_{k}) = w_{j}^{L}(x_{0}, …, x_{k})x_{m_{j}}^{ξ_{j}}w_{j}^{R}(x_{0}, x_{1}, …, x_{k}).

Lemma 2.1. Let w be a reduced word as above. Then

w(g_{0}, …, g_{k})(ℙ − A_{m_{l}}^{−ξ_{l}}) ⊂ Ā_{m_{1}}^{ξ_{1}}
w(g_{0}, …, g_{k})^{−1}(ℙ − A_{m_{1}}^{ξ_{1}}) ⊂ Ā_{m_{l}}^{−ξ_{l}}

and

x^{+}(w(g_{0}, …, g_{k})) ∈ Ā_{m_{1}}^{ξ_{1}},
x^{−}(w(g_{0}, …, g_{k})) ∈ Ā_{m_{l}}^{−ξ_{l}}.

Proof. Induction on l. If l = 1, then w(g_{0}, …, g_{k}) = g_{m_{1}}^{ξ_{1}} and

w(g_{0}, …, g_{k})(ℙ − A_{m_{1}}^{−ξ_{1}}) = Ā_{m_{1}}^{ξ_{1}}
w(g_{0}, …, g_{k})^{−1}(ℙ − A_{m_{1}}^{ξ_{1}}) = Ā_{m_{1}}^{−ξ_{1}}

by hypothesis. Inductively we assume that

w_{1}^{R}(g_{0}, …, g_{k})(ℙ − A_{m_{l}}^{−ξ_{l}}) ⊂ Ā_{m_{2}}^{ξ_{2}}.

Now Ā_{m_{2}}^{ξ_{2}} ⊂ ℙ − A_{m_{1}}^{−ξ_{1}} (since w is a reduced word) and

g_{m_{1}}^{ξ_{1}}(ℙ − A_{m_{1}}^{−ξ_{1}}) = Ā_{m_{1}}^{ξ_{1}}

imply that

w(g_{0}, …, g_{k})(ℙ − A_{m_{l}}^{−ξ_{l}}) = g_{m_{1}}^{ξ_{1}}w_{1}^{R}(g_{0}, g_{1}, …, g_{k})(ℙ − A_{m_{l}}^{−ξ_{l}})
⊂ g_{m_{1}}^{ξ_{1}} Ā_{m_{2}}^{ξ_{2}}
⊂ g_{m_{1}}^{ξ_{1}}(ℙ − A_{m_{1}}^{−ξ_{1}})
= Ā_{m_{1}}^{ξ_{1}}

as desired. Replacing w by w^{−1} the second assertion follows. By applying the Brouwer fixed point theorem to the restriction of w(g_{0}, …, g_{k}) to ℙ − A_{m_{l}}^{−ξ_{l}} we see that w(g_{0}, …, g_{k}) has a fixed point in ℙ − A_{m_{l}}^{−ξ_{l}}; this fixed point must lie in w(g_{0}, …, g_{k})(ℙ − A_{m_{l}}^{−ξ_{l}}) ⊂ A_{m_{1}}^{ξ_{1}}. The final assertion also follows upon replacing w by w^{−1}.

Applying this lemma to a word of the form x_{0}w(x_{1}, …, x_{k}) we obtain:

x^{+}(gw(g_{1}, …, g_{k})) ∈ Ā_{0}^{+}
(2.1)  x^{−}(gw(g_{1}, …, g_{k})) ∈ Ā_{m_{l}}^{−ξ_{l}}

and we conclude:

Corollary 2.2. Let w = w(x_{1}, …, x_{k}) ∈ 𝔽_{k}. Then gw(g_{1}, …, g_{k}) is ϵ-hyperbolic.

Lemma 2.3. Let

δ_{i} = \frac{ξ_{i} − 1}{2} = \{ \begin{matrix}0 & if ξ_{j} = +1 \\ −1 & if ξ_{j} = −1 \end{matrix}

and write

h_{i}(x) = g_{i}(x) + v_{i}

for v_{i} ∈ 𝔼. Then

h_{i}^{ξ_{j}}(x) = g_{i}^{ξ_{j}}(x) + ξ_{j}g_{i}^{δ_{j}}v_{i}

and if w = w(x_{1}, …, x_{k}) ∈ 𝔽_{k}, then

w(h_{1}, …, h_{k})(0) = ∑_{j=1}^{l} w_{j}^{L}(g_{1}, …, g_{k})g_{m_{j}}^{δ_{j}}(ξ_{j}v_{m_{j}}).

Proof. The first assertion follows from

h_{i}^{−1}(x) = g_{i}^{−1}(x) − g_{i}^{−1}v_{i}

which is easily checked. The second assertion is proved inductively on l. If l = 0, the sum is empty and there is nothing to prove. Inductively we assume that

w_{1}^{R}(h_{1}, …, h_{k})(0) = ∑_{j=2}^{l} w̄_{j}^{L}(g_{1}, …, g_{k})g_{m_{j}}^{δ_{j}}(ξ_{j}v_{m_{j}}).

where

w̄_{j}^{L}(x_{1}, …, x_{k}) = ∏_{i=2}^{j−1} x_{m_{i}}^{ξ_{i}}.

is the j-th left-subword of w_{1}^{R}. Now

w(h_{1}, …, h_{k})(0) = h_{m_{1}}^{ξ_{1}}w_{1}^{R}(h_{1}, …, h_{k})(0)

= g_{m_{1}}^{ξ_{1}}(∑_{j=2}^{l} w̄_{j}^{L}(g_{1}, …, g_{k})g_{m_{j}}^{δ_{j}}(ξ_{j}v_{m_{j}})) + ξ_{1}g_{m_{1}}^{δ_{1}}(v_{m_{1}})

= ∑_{j=2}^{l} w_{j}^{L}(g_{1}, …, g_{k})g_{m_{j}}^{δ_{j}}(ξ_{j}v_{m_{j}}) + ξ_{1}g_{m_{1}}^{δ_{1}}(v_{m_{1}})

as desired.

3. Drumm’s geometric conditions for proper discontinuity

Consider g_{0}, g_{1}, …, g_{k} ∈ G as above. Then the set defined by

𝔘_{i} = {u ∈ 𝔖 | x^{+}(u) ∈ Ā_{i}^{+}, x^{−}(u) ∈ Ā_{i}^{−} ∪ ⋃_{j≠i} Ā_{j}^{±} ∪ Ā_{0}^{±}}

is compact and is contained in an open half-space (Lemma 1.1). Thus there exists v_{i} ∈ ℝ^{2,1} such that 𝔹(v_{i}, u) > 0 for all u ∈ 𝔘_{i}. Furthermore (for this choice (v_{1}, …, v_{k})) there exists C > 0 such that 𝔹(v_{i}, u) ≥ C for u ∈ 𝔘_{i}. In particular if f ∈ H satisfies x^{0}(f) ∈ 𝔘_{i}, then

𝔹(v_{i}, x^{0}(f)) ≥ C.

Letting h_{i}(x) = g_{i}(x) + v_{i} as above, we have:

Theorem 3.1. The group Γ generated by h_{1}, …, h_{k} acts properly on 𝔼.

Proof. Let γ = w(h_{1}, …, h_{k}) for w ∈ 𝔽_{k}. We compute α(gγ) using Lemma 2.3:

α(gγ) = 𝔹(gγ(0), x^{0}(gγ))

= ∑_{j=1}^{l} 𝔹(gw_{j}^{L}(g_{1}, …, g_{k})g_{m_{j}}^{δ_{j}}(ξ_{j}v_{m_{j}}), x^{0}(gw(g_{1}, …, g_{k})))

= ∑_{j=1}^{l} 𝔹(v_{m_{j}}, x^{0}(f_{j}))

where

f_{j} = (gw_{j}^{L}(g_{1}, …, g_{k})g_{m_{j}}^{δ_{j}})^{−1}(gw(g_{1}, …, g_{k}))^{ξ_{j}}(gw_{j}^{L}(g_{1}, …, g_{k})g_{m_{j}}^{δ_{j}}) by Lemma 1.3. Now

f_{j}^{ξ_{j}} = g_{m_{j}}^{−δ_{j}}w_{j}^{L}(g_{1}, …, g_{k})^{−1}g^{−1}gw(g_{1}, …, g_{k})gw_{j}^{L}(g_{1}, …, g_{k})g_{m_{j}}^{δ_{j}}
= g_{m_{j}}^{−δ_{j}}g_{m_{j}}^{ξ_{j}}w_{j}^{R}(g_{1}, …, g_{k})gw_{j}^{L}(g_{1}, …, g_{k})g_{m_{j}}^{δ_{j}}

so

f_{j} = \{ \begin{matrix}g_{m_{j}}w_{j}^{R}(g_{1}, …, g_{k})gw_{j}^{L}(g_{1}, …, g_{k}) & if ξ_{j} = 1 \\ g_{m_{j}}w_{j}^{L}(g_{1}, …, g_{k})^{−1}g^{−1}w_{j}^{R}(g_{1}, …, g_{k})^{−1} & if ξ_{j} = −1 \end{matrix}

is the rearrangement of (gw(g_{1}, …, g_{k}))^{ξ_{j}} with initial letter g_{m_{j}}.

In particular f_{j} is a reduced word since w(x_{1}, …, x_{k}) and w(x_{1}, …, x_{k})^{−1} are reduced words, and by Lemma 2.1,

x^{+}(f_{j}) ∈ A_{m_{j}}^{+}

and

x^{−}(f_{j}) ∈ \{ \begin{matrix}A_{m_{j−1}}^{−ξ_{j−1}} & if ξ_{j} = 1 and 1 < j < l \\ A_{m_{j+1}}^{ξ_{j+1}} & if ξ_{j} = −1 and 1 < j < l \\ A_{0}^{−ξ_{j}} & if j = 1 \\ A_{0}^{ξ_{j}} & if j = l \end{matrix}

again by Lemma 2.1. Thus x^{0}(f_{j}) ∈ 𝔘_{j} and

α(gγ) = ∑_{j=1}^{l} 𝔹(v_{m_{j}}, x^{0}(f_{j}))

≥ Cℓ(w)

establishing condition (2) of Lemma 1.5. The first condition of Lemma 1.5 follows from (2.1). Applying Lemma 1.5, Γ acts properly on 𝔼 as desired.

References

1. Drumm, T. A., Fundamental polyhedra for Margulis space-times, Doctoral dissertation, University of Maryland 1990

2. Fried, D. and Goldman, W., Three-dimensional affine crystallographic groups, Adv. Math. 47 (1983), 1–49.

3. Margulis, G. A., Free properly discontinuous groups of affine transformations, Dokl. Akad. Nauk SSSR 272 (1983), 937–940.

4. , Complete affine locally flat manifolds with a free fundamental group, J. Soviet Math. 134 (1987), 129–134.

5. Milnor, J. W., On fundamental groups of complete affinely flat manifolds, Adv. Math. 25 (1977), 178–187.

Department of Mathematics, Swarthmore College, Swarthmore, PA 19081 (Drumm), Department of Mathematics, University of Maryland, College Park, MD 20742 (Goldman)

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