Journal of Lie Theory Volume 9 (1999) 321–349 C ◯1999 Heldermann Verlag

Discrete groups of affine isometries

Herbert Abels

Communicated by E. B. Vinberg

Abstract. Given a finite set S of isometries of an affine Euclidean space. We ask when the group Γ generated by S is discrete. This includes as a special case the question when the group generated by a finite set of rotations is finite. This latter question is answered in an appendix. In the main part of the paper the general case is reduced to this special case. The result is phrased as a series of tests: Γ is discrete iff S passes all the tests. The testing procedure is algorithmic.

Suppose we are given a finite set S of isometries of some affine Euclidean space. We ask when the group Γ generated by S is discrete. This includes as a special case the question when the group generated by a finite set of rotations is finite. We deal with this special case in an appendix. In the main part of the paper we reduce the general question to this special case. The main result is phrased as a series of tests. Γ is discrete iff S passes all the tests.

The method is based on Bieberbach’s theorems characterizing discrete groups of affine isometries. Bieberbach’s second theorem says that a group Γ of isometries of an affine Euclidean space 𝔼 is discrete (if and) only if there is a Γ–invariant affine subspace 𝔽 of 𝔼 such that the restriction homomorphism r from Γ to the group of isometries of 𝔽 has finite kernel and a crystallographic group as image.

This theorem suggests the following procedure to check if a given group Γ of affine isometries is discrete. First find a minimal Γ–invariant affine subspace 𝔽. This is motivated by the fact that the Γ–invariant affine subspace in Bieberbach’s theorem is minimal. Then test if Γ restricted to 𝔽 is crystallographic. Finally, check if the restriction homomorphism r : Γ → Γ|𝔽 has finite kernel.

Accordingly, the contents of chapters two to four are as follows. In chapter two we describe a minimal Γ–invariant affine subspace 𝔽 for any group Γ of affine isometries. In chapter three we present a test for Γ to be crystallographic. Here we suppose that the group λ(Γ) of linear parts of Γ is finite, a necessary condition by Bieberbach’s first theorem. In chapter four we describe ways to check if r : Γ → Γ|𝔽 has finite kernel. In chapter five we give examples.

The main result is phrased as a series of tests. The group Γ is discrete iff S passes all the tests. The testing procedure is algorithmic and hence should

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag be capable of being computerized. We thus discuss the computational aspects as we go along.

This work was motivated by the following question asked by A. Dress in [6]. Suppose a finite set S of affine isometric involutions of an affine Euclidean space 𝔼 is given. When is the group Γ generated by S discrete, respectively finite? Restricting the discussion to a set of generators of this type seems to have more disadvantages than advantages for the general theory.

I thank E. B. Vinberg for helpful discussions.

1. The Bieberbach theorems

In this chapter we establish notations to be in force for the whole paper and recall the Bieberbach theorems.

Notations 1.1. Let V be a finite dimensional real vector space endowed with a positive definite inner product ⟨ , ⟩ . Let 𝔼 be the corresponding affine Euclidean space. So 𝔼 has a simply transitive action of V , denoted (v, x) ⟼ x + v, and a metric d(x, x + v) = ⟨v, v⟩^{\frac{1}{2}} . Let G = Iso(𝔼) be the group of isometries of 𝔼. Given x_{0} ∈ 𝔼, t ∈ V and U in the orthogonal group O of (V, ⟨ , ⟩) we define an isometry

(1.2  by)  g = A(x_{0}, t, U)
(1.3)  g(x_{0} + v) = x_{0} + t + Uv .

Every isometry of 𝔼 is of this form. We note for later reference the multiplication formula

(1.4)  iA(x_{0}, t_{0}, U_{0}) · A(x_{0}, t_{1}, U_{1}) = A(x_{0}, t_{0} + U_{0}t_{1} , U_{0}U_{1}) ,

and the dependence on the base point chosen:

(1.5)  A(x_{0}, t_{0}, U_{0}) = A(x_{1}, t_{1}, U_{1})

iff U_{0} = U_{1} and t_{1} = t_{0} + (U_{0} − 1)(x_{1} − x_{0}), where, of course, x_{1} − x_{0} is the unique vector v ∈ V such that x_{1} = x_{0} + v. So the translation t depends on the base point chosen (but cf. 2.1c)), whereas the linear part U does not. We thus have a homomorphism

(1.6)  λ : G → O

defined by

λ(A(x_{0}, t, U)) = U .

We call λ(g) the linear part or the differential of g ∈ G.

The translations of 𝔼 are the isometries in the kernel of λ. We identify V with the group of translations of 𝔼.

For an affine subspace 𝔽 of 𝔼 we denote by T 𝔽 the subspace of V of translations of 𝔽, so T 𝔽 = {v ∈ V ; x + v ∈ 𝔽 for every x ∈ 𝔽}. One can also think of T 𝔽 as the tangent space of 𝔽. A subgroup Γ of G = Iso(𝔼) is called crystallographic if it is discrete and Γ\𝔼 is compact. The following two theorems due to Bieberbach are basic.

Bieberbach’s First Theorem. For a crystallographic group Γ the group λ(Γ) of linear parts of Γ is finite.

It follows that Γ ∩ V = ker(λ|Γ) is of finite index in Γ and hence Γ ∩ V is a discrete subgroup of V with (Γ ∩ V) \ V compact. So there is a basis of the vector space V over the reals which is a set of generators for Γ ∩ V . With respect to this basis λ(γ) has integer entries for every γ ∈ Γ.

Bieberbach’s Second Theorem. If Γ is a discrete subgroup of G there is a Γ–invariant affine subspace 𝔽 of 𝔼 such that the restriction homomorphism r : Γ → Iso(𝔽) has finite kernel and a crystallographic subgroup of Iso(𝔽) as image.

Note that 𝔽 is a minimal Γ–invariant affine subspace of 𝔼, i.e. a minimal element in the set of Γ–invariant affine subspaces, partially ordered by inclusion, since the translations in r(Γ) span T 𝔽 by Bieberbach’s first theorem.

The following corollary of the Bieberbach theorems suggests our procedure to check if a given subgroup Γ of G is discrete.

Corollary. A subgroup Γ of G is discrete iff there is a minimal Γ–invariant affine subspace 𝔽 of 𝔼 such that for the restriction homomorphism r : Γ → Iso(𝔽) the kernel is finite and the image is a crystallographic group on 𝔽.

Proof. Necessity follows from the Bieberbach theorems, as seen above. Conversely, under the conditions of the corollary Γ acts properly discontinuously on 𝔽 hence is discrete.

2. A minimal invariant subspace

Let Γ be a subgroup of G, not necessarily discrete. We shall describe explicitly a minimal Γ–invariant affine subspace 𝔽 of 𝔼. It is known that any two such 𝔽 are translates of each other by translations which are fixed by λ(Γ) so that it suffices to answer our questions for one of them.

Of particular importance for our description of 𝔽 is the concept of the axis 𝔼_{g} of an affine isometry g, that is the largest g –invariant affine subspace of 𝔼 on which g acts by translations, see 2.1. To find 𝔽 we shall use the following two facts:

(i) Any Γ–invariant affine subspace 𝔽 of 𝔼 intersects the axis 𝔼_{g} of every element g of Γ. (ii) For certain affine subspaces 𝔼_{1}, 𝔼_{2} of 𝔼 the set 𝔼_{1,2} of feet of common perpendiculars from 𝔼_{2} to 𝔼_{1} intersects 𝔽 if both 𝔼_{1} and 𝔼_{2} do. Starting with (i) and using (ii) repeatedly we finally find a minimal Γ–invariant subspace.

We start by defining the notions of axis and translational part. Fix an element g ∈ G. If λ(g) is the linear part of g, let V^{1} = {v ∈ V | λ(g)v = v} be the +1–eigenspace of λ(g) in V and let V^{≠1} be the orthogonal complement of V^{1} in V . 2.1 Lemma and Definition

a) For g ∈ G there is a unique largest (not only maximal) g –invariant affine subspace 𝔼_{g} of 𝔼 on which g acts as a parallel translation. We call 𝔼_{g} the axis of g.

b) The corresponding vector subspace T 𝔼_{g} is V^{1} . In particular, 𝔼_{g} is not empty.

c) The translation τ(g) := g|𝔽_{g} ∈ T 𝔼_{g} = V^{1} is called the translational part of g.

d) If g = A(x_{0}, t, U) and t = t_{1} + t_{2} is the decomposition of t according to V = V^{1} ⊕ V^{≠1} , then τ(g) = t_{1} .

e) Every point x ∈ 𝔼 can be written uniquely in the form x = y + v with y ∈ 𝔼_{g} and v ∈ V^{≠1} . Then

gx = x + τ(g) + (λ(g) − 1)v

Proof. Write g = A(x_{0}, t, U), so U = λ(g). Decomposet = t_{1} + t_{2} according to V = V^{1} ⊕ V^{≠1} . Then, by (1.3), g(x_{0} + v) = x_{0} + t_{1} + t_{2} + Uv = x_{0} + v + t_{1} + (t_{2} + (U − 1)v). There is a vector v_{0} ∈ V^{≠1} such that (U − 1)v_{0} + t_{2} = 0, since U induces linear endomorphism of V^{≠1} and (U − 1)|V^{≠1} has kernel zero. So g(x_{0} + v_{0}) = x_{0} + v_{0} + t_{1} and for x_{1} = x_{0} + v_{0} we have g = A(x_{1}, t_{1}, U) by (1.5). It follows that g induces a parallel translation by t_{1} on x_{1} + V^{1} . Every point x ∈ 𝔼 can be written uniquely in the form x = x_{1} + v_{1} + v_{2} where v_{1} ∈ V^{1} , v_{2} ∈ V^{≠1} . Then

gx = x + t_{1} + (λ(g) − 1)v_{2} .

Thus, if x is contained in a g –invariant affine subspace 𝔽 such that g acts on 𝔽 by a parallel translation, by τ ∈ T 𝔽 say, then g^{n}x = x + nτ and on the other hand g^{n}x = x + nt_{1} + (λ(g) − 1)^{n}v_{2} which implies τ = t_{1} by taking lim_{n→∞} \frac{g^{n}x−x}{n} and hence v_{2} = 0. All the claims are now proved.

Corollary 2.2. An affine isometry g has a fixed point iff τ(g) = 0.

Here is our first handle on finding invariant subspaces.

Corollary 2.3. For every nonempty g -invariant affine subspace 𝔽 of 𝔼 we have τ(g) ∈ T 𝔽 and 𝔼_{g} intersects 𝔽.

Proof. We have the following more precise assertion. For the restriction g|𝔽 of g to 𝔽 the axis 𝔽_{g} is 𝔽 ∩ 𝔼_{g} for the following reasons: 𝔽_{g} ⊃ 𝔽 ∩ 𝔼_{g}, since g induces a parallel translation on 𝔽 ∩ 𝔼_{g}; and 𝔽_{g} ⊂ 𝔽 ∩ 𝔼_{g} since 𝔽_{g} is a g – invariant subspace of 𝔽 on which g induces a parallel translation.

The next proposition allows to get further information about invariant subspaces. We need the following lemma, which we state for later reference in greater generality than presently needed.

Lemma 2.4. Let V be a finite dimensional real Euclidean vector space and let U_{1} and U_{2} be two vector subspaces. Let P_{i}, i = 1, 2, be the orthogonal projections of V onto U_{i}. Let A be a finite dimensional subalgebra of End (V) over some subfield k of ℝ. Suppose A contains P_{1} and P_{2} . Then A also contains the orthogonal projection onto U_{1} ∩ U_{2} . Furthermore, A contains three operators Q_{1}, Q_{2} and Q_{3} with the following properties:

a) Q_{1} + Q_{2} + Q_{3} = 1 where 1 is the identity of V .

b) Q_{3} is the orthogonal projection onto (U_{1} + U_{2})^{⊥}.

c) The image of Q_{i} is contained in U_{i} for i = 1, 2.

For the proof let us first recall the following fact from linear algebra.

Lemma 2.5. Let V be a vector space over a field k and let T : V → V be a linear map. Let f ∈ k[t] be a polynomial such that f(T) = 0. Suppose f = f_{1} · f_{2} with relatively prime polynomials f_{i} ∈ k[t]. Then V is the direct sum of the subspaces V_{1} = ker f_{1}(T) and V_{2} = ker f_{2}(T), and the projections p_{i} of V onto V_{i} are in k[T].

The proof consists in writing 1 = h_{1}f_{1}+h_{2}f_{2} with h_{i} ∈ k[t] and checking that the maps p_{i} = (h_{j}f_{j})(T), j ≠ i, are the projections onto V_{i}.

Proof of Lemma 2.4. Put W = U_{1} ∩ U_{2} . Note that P_{i}(W^{⊥}) ⊂ W^{⊥}, since P_{i} is self adjoint and P_{i}(W) ⊂ W, in fact P_{i} | W = 1_{W} . Thus the operator S := P_{1}P_{2} maps W^{⊥} to itself. Furthermore S | W = 1_{W} and ∥S | W^{⊥}∥ < 1 for the following reason: Every orthogonal projection decreases the norm, i.e. ∥P x∥ ≤ ∥x∥ for every x, and ∥P x∥ = ∥x∥ iff P x = x. It follows that ∥Sv∥ < ∥v∥ for every 0 ≠ v ∈ W^{⊥} since such a vector v is not in U_{2} or if in U_{2} not in U_{1} . Thus the norm of S | W^{⊥} is strictly smaller than 1, since this operator norm is the maximum of ∥Sv∥ where v runs through the compact unit sphere of W^{⊥}. In particular, all the eigenvalues of S | W^{⊥} are of modulus strictly less than 1.

We now show that the orthogonal projection Q of V onto W is in A. The operator S fulfills a polynomial equation f ∈ k[t], since A is finite dimensional over k. Now write f = f_{1} · f_{2} with relatively prime polynomials f_{1}, f_{2} ∈ k[t] where f_{1} is a power of (t − 1). Note that the case f_{1} = 1 is not excluded. Then ker f_{1}(S) = W, since f_{1}(S) maps W to zero and induces an automorphism of W^{⊥}. And ker f_{2}(S) = W^{⊥} since W ⊕ f_{2}(S) = V and f_{2}(S) maps both W and W^{⊥} to itself and hence ker f_{2}(S) is the sum of its intersections with W and W^{⊥}. Note that if follows that the polynomial g = (t − 1) · f_{2} has the property g(S) = 0.

To finish the proof we now apply these results to the two projections 1 − P_{i} onto U_{i}^{⊥} and the operator T = (1 − P_{1})(1 − P_{2}). So T induces the identity on U_{1}^{⊥}∩U_{2}^{⊥} = (U_{1}+U_{2})^{⊥} and maps the orthogonal complement U_{1}+U_{2} to itself. There is a polynomial g ∈ k[t] of the form g = (t − 1) · f_{2} such that f(T) = 0 and f_{2} and t−1 are relatively prime. Then ker(T −1) = (U_{1}+U_{2})^{⊥} and ker f_{2}(T) = U_{1}+U_{2} , the corresponding projections are the orthogonal projections and belong to A. Let Q_{3} ∈ A be the orthogonal projection onto (U_{1} + U_{2})^{⊥}. By the proof of 2.5 there is an operator Q_{4} ∈ A such that (T − 1) · Q_{4} = 1 − Q_{3} , namely Q_{4} = h_{1}(T) in the notations of that proof. Now put Q_{1} = −P_{1}(1−P_{2})Q_{4} and Q_{2} = −P_{2}Q_{4} .

2.6. The common perpendicular. Let 𝔼_{1} and 𝔼_{2} be two nonempty affine subspaces of 𝔼. Let (x_{1}, x_{2}) be a pair of points with x_{i} ∈ 𝔼_{i} of minimal distance, so d(x_{1}, x_{2}) ≤ d(y_{1}, y_{2}) for any pair (y_{1}, y_{2}) of points with y_{i} ∈ 𝔼_{i}. Equivalently, x_{1} − x_{2} is orthogonal to V_{1} + V_{2} where V_{i} = T 𝔼_{i}. If x_{1}, x_{2} is one such pair all other such pairs are of the form (x_{1} + v_{1} , x_{2} + v_{2}) where v_{1} = v_{2} ∈ V_{1} ∩ V_{2} . So the difference vector x_{1} − x_{2} ∈ V for any such pair depends only on the two subspaces 𝔼_{1} , 𝔼_{2} and will be called the common perpendicular vector from 𝔼_{2} to 𝔼_{1} and will be denoted perp(𝔼_{2}, 𝔼_{1}). So perp (𝔼_{2}, 𝔼_{1}) is the unique vector of 𝔼_{1} − 𝔼_{2} = {x_{1} − x_{2}|x_{i} ∈ 𝔼_{i}} orthogonal to V_{1} + V_{2} . The first entry x_{1} of a pair (x_{1}, x_{2}) with x_{i} ∈ 𝔼_{i} and x_{1} − x_{2} = perp (𝔼_{2}, 𝔼_{1}) is called the foot of a common perpendicular from 𝔼_{2} to 𝔼_{1} . The set of feet of common perpendiculars from 𝔼_{2} to 𝔼_{1} will be denoted 𝔼_{1,2} . So 𝔼_{1,2} is an affine subspace of 𝔼_{1} with T 𝔼_{1,2} = V_{1} ∩ V_{2} .

Let us return to our subgroup Γ of G. Let A be the subalgebra ℝ[λ(Γ)] of End (V) consisting of all linear combinations of elements of λ(Γ) with real coefficients.

Proposition 2.7. Suppose 𝔽 , 𝔼_{1} and 𝔼_{2} are affine subspaces of 𝔼 with the following properties: 𝔽 is Γ–invariant, both E_{i} intersect 𝔽 and the orthogonal projections P_{i} of V onto T 𝔼_{i} are elements of A. Then 𝔼_{1,2} intersects 𝔽, the orthogonal projection of V onto T 𝔼_{1,2} is in A and perp (𝔼_{2}, 𝔼_{1}) is in T 𝔽 . Proof. Let P_{i} be the orthogonal projection of V onto V_{i} := T 𝔼_{i}. Then the orthogonal projection of V onto T 𝔼_{1,2} = V_{1} ∩ V_{2} is in A, by 2.4. Let Q_{1} , Q_{2} and Q_{3} be operators as in lemma 2.4. Suppose a_{i} ∈ 𝔽 ∩ 𝔼_{i}. Put t = a_{1} − a_{2} , b_{1} = a_{1} − Q_{1}t and b_{2} = a_{2} + Q_{2}t. Then t is in the A–module W := T 𝔽, hence Q_{1}t ∈ V_{1} ∩ W and thus b_{1} ∈ 𝔽 ∩ 𝔼_{1} , and similarly b_{2} ∈ 𝔽 ∩ 𝔼_{2} . Finally, b_{1} − b_{2} = Q_{3}t ∈ (V_{1} + V_{2})^{⊥}, so b_{1} is the foot of a common perpendicular from 𝔼_{2} to 𝔼_{1} . We have perp (𝔼_{2}, 𝔼_{1}) = b_{1} − b_{2} ∈ W.

We can apply the procedure of the proposition to axes 𝔼_{i} of single elements or generators of our group Γ and then iterate the procedure with the spaces obtained and thus produce further points in 𝔽 and further vectors in T 𝔽. We formalize the iteration as follows. 2.8. Let X be a set and let M_{X} be the free non–associative semigroup on X, called a free magma by Serre ([Lie algebras and Lie groups. Benjamin New York 1965] LA 4.1) We have M_{X} = ∐_{n=1}^{∞} X_{n} where X_{n} is inductively defined by X_{1} = X and X_{n} = ∐_{p+q=n}^{∞} X_{p} × X_{q} for n ≥ 1 and the multiplication M_{X} × M_{X} → M_{X} is defined by the natural inclusion X_{p} × X_{q} → X_{p+q}. The product of the elements m and n in M_{X} will be denoted by (m, n). We have a unique map M_{X} to the set of non–empty subsets of X , called support and denoted supp with the properties supp(x) = {x} for x ∈ X_{1} and supp(m, n) = supp(m) ∪ supp(n). 2.9. Now let Γ be the subgroup of G generated by a subset S of G. Define for every element m of the free magma M_{S} a subspace 𝔼_{m} of 𝔼 as follows. For γ ∈ X let 𝔼_{γ} be the axis of γ, and for m, n in M_{S} let 𝔼_{(m,n)} be the set 𝔼_{m,n} of feet of common perpendiculars from 𝔼_{n} to 𝔼_{m}. The family of 𝔼_{m} ’s has the following properties:

a) 𝔼_{(m,n)} ⊂ 𝔼_{m}. b) T 𝔼_{m} = V^{λ⟨supp(m)⟩}.

c) For every nonempty Γ–invariant affine subspace 𝔽 of 𝔼 we have 𝔼_{m} ∩𝔽 ≠ Ø

and perp (𝔼_{m}, 𝔼_{n}) ∈ T 𝔽 for m, n in M_{Γ} .

Here we denote by ⟨Y ⟩ the subgroup of G generated by the subset Y of G. For a subgroup H of λ(G) we denote by V^{H} the subspace of vectors of V fixed by every element of H. The properties a) – c) follow from 2.6 and 2.7 using the fact that the orthogonal projection of V onto V^{H} is contained in ℝ[H]. Here is our description of all the minimal Γ-invariant affine subspaces of 𝔼.

Theorem 2.10. Let W be the subspace of V spanned by the set of vectors τ(g), g ∈ Γ, and perp (𝔼_{m}, 𝔼_{n}), m, n in M_{Γ} . An affine subspace 𝔽 of 𝔼 is a minimal Γ–invariant affine subspace of 𝔼 iff T 𝔽 = W and 𝔽 intersects one of the subspaces 𝔼_{m} with m ∈ M_{Γ} and T 𝔼_{m} = V^{λ(Γ)} . Every Γ–invariant affine subspace of 𝔼 contains a space 𝔽 of this form.

The following consequence is well known, see [10], chapter 3, end of § 1.

Corollary 2.11. Any two minimal Γ–invariant affine subspaces 𝔽 of 𝔼 are translates of each other by a vector in V^{λ(Γ)} .

In particular, 𝔽 is unique if V^{λ(Γ)} = 0.We restate the theorem for the case that the group Γ is given by a set of generators.

Theorem 2.12. Let S be a subset of G generating the subgroup Γ of G. There is an m ∈ M_{S} such that V^{λ⟨supp(m)⟩} = V^{λ(Γ)} . Let x be a point contained in a minimal Γ–invariant affine subspace 𝔽 of 𝔼, e.g. x ∈ 𝔼_{m}. The subspace W = T 𝔽 is the smallest ℝ[λ(Γ)]–submodule of V containing any one of the following two sets of vectors

a) {gx − x; g ∈ S}.

b) {τ(g), g ∈ S} ∪ {perp (𝔼_{m}, 𝔼_{(g,m)}), g ∈ S}.

Note that in case V^{λ(Γ)} = 0 the sets 𝔼_{m} and 𝔼_{(g,m)} consist of one point only, by 2.9 b).

Proofs. Every Γ-invariant affine subspace 𝔽′ of 𝔼 intersects 𝔼_{m}, by 2.9 c). And T 𝔽′ contains τ(g) for every g ∈ Γ, by 2.3, and perp (𝔼_{m}, 𝔼_{n}) for any two m, n in M_{Γ} , by 2.9 c). Hence 𝔽′ contains a subspace 𝔽 as in 2.10. To see that such an 𝔽 is Γ-invariant, first note that for g and h in G we have

(2.13)  τ(hgh^{−1}) = λ(h)τ(g)

by 2.1 a), and

(2.14)  𝔼_{hgh^{−1}} = h𝔼_{g}

by the definition of the axes. Hence

(2.15)  𝔼_{γm} = γ𝔼_{m}

for m ∈ M_{Γ} and γ ∈ Γ, where m ⟼ γm is the unique morphism of magmas such that δ ⟼ γδγ^{−1} for δ ∈ Γ. So

(2.16)  perp (𝔼_{γm}, 𝔼_{γn}) = λ(γ) perp (𝔼_{m}, 𝔼_{n}) .

It follows that the vector space W of 2.10 is an ℝ[λ(Γ)]–module. So, to finish the proof of 2.10, it remains to show that γx_{0} ∈ x_{0} +W for γ ∈ Γ and x_{0} ∈ 𝔼_{m}.

Our hypothesis about m implies that every point x of 𝔼_{m} is a foot of a perpendicular from 𝔼_{n} to 𝔼_{m} for every n ∈ M_{Γ} , since

T 𝔼_{m,n} = V^{λ⟨supp(m)∪ supp(n)⟩} = V^{λ(Γ)} = V^{λ⟨supp(m)⟩} = T 𝔼_{m}

by 2.9 b). Hence for t = perp (𝔼_{m}, 𝔼_{γ}) we have x + t ∈ 𝔼_{γ} and thus

(2.17)  γx = x + τ(g) − (λ(g) − 1)t

by 2.1 e), so γx ∈ x + W for γ ∈ Γ.

Corollary 2.11 follows from the following facts. Any two minimal Γ– invariant subspaces are parallel since their translation subspaces are the same, namely W. They both intersect every 𝔼_{m} where m ∈ M_{Γ} and T 𝔼_{m} = V^{λ(Γ)} ,by 2.9 c), hence they are parallel translates of each other by a vector in T 𝔼_{m} = V^{λ(Γ)} .

The first claim of 2.12 is clear for dimension reasons. To prove the second one let W_{a} and W_{b} be the ℝ[λ(Γ)]–submodules of V generated by the sets given in a) and b), respectively. Clearly, x + W_{a} is Γ–invariant and every Γ–invariant affine subspace of 𝔼 containing x contains x+W_{a}, thus 𝔽 = x+W_{a} is aminimal Γ–invariant affine subspace of 𝔼. Concerning b), we have W_{b} ⊂ W by the definition of W in 2.10. For x_{0} ∈ 𝔼_{m} we have γx_{0} ∈ x_{0} + W_{b} for γ ∈ S by 2.17.

Note here that perp (𝔼_{1}, 𝔼_{2}) = perp (𝔼_{1}, 𝔼_{2,1}). It follows that x_{0} + W_{b} is γ –invariant for every γ ∈ S and hence for any γ ∈ Γ. This implies that x_{0} + W_{b} is a minimal Γ–invariant affine subspace, in particular W_{b} = W. Hence x+W_{b} is a minimal Γ–invariant affine subspace of 𝔼 for every point x contained in a minimal Γ–invariant affine subspace of 𝔼.

Remark 2.18. W is not generated by one of the two subsets in 2.12 b) alone. E.g. if Γ consists of translations only, then 𝔼_{m} = 𝔼 for every m ∈ M_{Γ} and hence perp (𝔼_{m}, 𝔼_{n}) = 0 always. On the other hand, there are examples of subgroups Γ of G such that every element γ of Γ has a fixed point, but there is no common fixed point. So τ(γ) = 0 for every γ ∈ Γ but W ≠ 0. An example of such a group Γ can be obtained as follows. Take two elements g_{1}, g_{2} of G for dim 𝔼 = 4 such that for every reduced word w different from the empty word all the eigenvalues of λ(w(g_{1}, g_{2})) are different from 1 and the points Fix (g_{1}) and Fix (g_{2}) are different. Then every element of the group Γ generated by g_{1} and g_{2} has a fixed point, since τ(g) ∈ V^{λ(g)} = {0}. The existence of such a pair (g_{1}, g_{2}) can be seen as follows. The group ℍ_{1} of quaternions of norm 1 acts on ℍ by left multiplication. An element h ∈ ℍ_{1} has one eigenvalue equal to 1 iff h = 1. The set of pairs (h_{1}, h_{2}) ∈ ℍ_{1} × ℍ_{1} which freely generate a free subgroup of ℍ_{1} hasa complement of measure zero. Hence there are many pairs (g_{1}, g_{2}) as above. By contrast, if Γ is discrete, the set of translational parts τ(γ), γ ∈ Γ generates W, by the Bieberbach theorems.

2.19. Computational aspects. For an element g of G the axis 𝔼_{g} can be computed as follows. Suppose g is given in the form A(x_{0}, t, U). Decompose t = t_{0} + t_{1} according to V = V^{1} ⊕ V^{≠1} , where V^{1} = ker(U − 1) and V^{≠1} is the orthogonal complement of V^{1} . Then 𝔼_{g} consists of the points x_{0} + v where t_{2} + (U − 1)v = 0, by the proof of 2.1. So 𝔼_{g} can be computed by solving systems of linear equations. If 𝔼_{1} and 𝔼_{2} are two affine subspaces of 𝔼 both perp (𝔼_{1}, 𝔼_{2}) and 𝔼_{1,2} can be computed by solving systems of linear equations, see 2.6. After at most dim V such steps we arrive at 𝔼_{m} with 𝔼_{m} = V^{λ(Γ)} . To determine 𝔽 take a point x ∈ 𝔼_{m} and compute W = T 𝔽 as follows. Let D_{0} = {γx − x; γ ∈ S}. Start with a basis B_{0} ⊂ D_{0} of the ℝ–vector subspace < D_{0} >_{ℝ} of V spanned by D_{0} . Suppose B_{j} has been defined. Check for every pair (γ, b), γ ∈ S, b ∈ B_{j}, if (γ)b ∈< B_{j} >_{ℝ}. If not define B_{j+1} = B_{j} ∪ {γb}. By numbering the elements of S and of ⋃ B_{j} starting with those of B_{0} one sees that one arrives at a basis of W after at most #S · dim V such checks.

3. When is a group crystallographic?

In the last chapter we described a minimal Γ–invariant affine subspace 𝔽 of 𝔼 for every subgroup Γ of G. According to the Bieberbach theorems our first test for discreteness of Γ will be to check whether the restriction of Γ to 𝔽 is crystallographic. Here is our test.

Theorem 3.1. The group r(Γ) is crystallographic for 𝔽 iff there is a point x in 𝔼 such that the abelian subgroup D(x) generated by {γx − x; γ ∈ Γ} is discrete.

In order to apply the theorem for the case that Γ is given by a set S of generators, note that D(x) is the smallest ℤ[λ(Γ)]–submodule of V containing {γx − x; γ ∈ S}. The group D(x) is not discrete for arbitrary points x ∈ 𝔼, nor even for arbitrary points x ∈ 𝔽, even if r(Γ) is crystallographic. E.g. let 𝔽 = 𝔼 = ℝ and Γ = {x ↦ ±x + n; n ∈ ℤ}. Then D(x) is discrete iff x ∈ ℚ. This is a typical behavior, see 3.3 e). So an obvious question – essential for computations – is how to find a point x as in the theorem. Incidentally, the answer uses the concepts of the preceding chapter. We thus use the notations of chapter two, in particular 2.9.

Addendum 3.2. Suppose r(Γ) is crystallographic. Let m ∈ M_{Γ} be such that T 𝔼_{m} = V^{λ(Γ)} . Then D(x) is discrete for x ∈ 𝔼_{m}, in fact Δ = Γ ∩ V is of finite index in D(x) and T 𝔽 = D(x)_{ℝ} = Δ_{ℝ}.

The properties of r(Γ) are independent of the particular choice of a minimal Γ–invariant affine subspace 𝔽 of 𝔼: Any two such subspaces 𝔽_{1} and 𝔽_{2} are translates of each other by a vector in V^{λ(Γ)} , according to 2.11, and this translation commutes with Γ, induces an isometry 𝔽_{1} → 𝔽_{2} and yields an isomorphism r_{1}(Γ) → r_{2}(Γ).

In the rest of this chapter we give a proof of 3.1 and 3.2. Let us start by establishing a few elementary and basic facts about D(x).

Lemma 3.3. Suppose λ(Γ) = F is a finite group of order f. Let Δ = ker(λ|Γ) = Γ ∩ V and let x be a point of 𝔼.

a) D(x) is a ℤ[λ(Γ)]–module containing Δ.

b) D(x) spans T 𝔽 over ℝ for every x ∈ 𝔽.

c) There is a point x ∈ 𝔼 with D(x) ⊂ \frac{1}{f} Δ.

d) For every point x with D(x) ⊂ Δ_{ℝ} the affine space x + Δ_{ℝ} is a minimal Γ–invariant affine subspace of 𝔼.

e) Suppose D(x) ⊂ Δ_{ℚ} then

{y; D(y) ⊂ Δ_{ℚ}} = x + Δ_{ℚ} + V^{F} .

Proof. a) is clear, b) follows from 2.12 a). c): For x ∈ 𝔼 put \overline{x} = \frac{1}{f} ∑ γ_{i}x, where γ_{i} runs through a set of representatives of Γ modulo Δ. Note that affine combinations ∑ λ_{i}x_{i} of points x_{i} ∈ 𝔼, that is combinations with ∑ λ_{i} = 1, are well defined in the affine space 𝔼, namely x_{0} + ∑ λ_{i}(x_{i} − x_{0}) is independent of a chosen point x_{0} ∈ 𝔼. For another set γ′_{i} of representatives of Γ modulo Δ we have γ′_{i} = γ_{i}δ_{i} with δ_{i} ∈ Δ and thus the point

\overline{y} = \frac{1}{f} ∑ γ′_{i}x = \frac{1}{f} (∑ λ(γ_{i})(δ_{i}x − x) + ∑ γ_{i}x) ∈ \overline{x} + \frac{1}{f} Δ

is in the \frac{1}{f} Δ–orbit of \overline{x}. In particular, for γ ∈ Γ the point γ\overline{x} = \frac{1}{f} ∑ γγ_{i}x is in the \frac{1}{f} Δ–orbit of \overline{x}.

d) The affine space x + Δ_{ℝ} is Γ–invariant since it contains the orbit of x and Δ_{ℝ} is an ℝ[λ(Γ)]–module. It is minimal since Δ ⊂ {γx − x; γ ∈ Γ} for every x ∈ 𝔼.

e) That the right hand side is a subset of the left hand side, is clear. The converse inclusion is a consequence of the following lemma.

Lemma 3.4. Let Γ_{1} be a subgroup of Γ. Suppose F_{1} = λ(Γ_{1}) is finite of order f_{1} , say. Let D and D′ be subgroups of V such that x + D and x′ + D′ are Γ_{1} –invariant. Then

f_{1} · (x − x′) ∈ D + D′ + V^{F_{1}} .

Proof. Note that D and D′ are F_{1} –modules since D contains γ(x+d)−x = (γx−x)+λ(γ)d for d ∈ D and γ ∈ Γ_{1} , and similarly for D′. For any two points x and x′ in 𝔼 we have for γ ∈ Γ

γx′ − x′ = γx − x + (λ(γ) − 1)(x′ − x) .

Take x and x′ as in the lemma and put w = x′ − x. Our hypothesis γx − x ∈ D and γx′ − x′ ∈ D′ for γ ∈ Γ_{1} imply by summing over a set of representatives of F_{1} in Γ_{1}

∑_{h∈F_{1}} hw − f_{1} · w ∈ D + D′

which implies the lemma in view of ∑_{h∈F_{1}}hw ∈ V^{F_{1}} .

We are now ready to prove theorem 3.1.

3.5. Proof of Theorem 3.1. To prove necessity we may assume that 𝔽 = 𝔼 and that Γ is crystallographic. Then, by Bieberbach’s first theorem, λ(Γ) = F is finite and Δ is discrete. Hence \frac{1}{f} Δ is a discrete subgroup of V containing γx − x for every γ ∈ Γ if x is as in 3.3 c). Conversely, let x be a point of 𝔼 such that D := D(x) is discrete. Note that we do not have x ∈ 𝔽 nor D ⊂ Δ_{ℝ}, in general, e.g. if 𝔼 = 𝔽 × U, where Γ acts on 𝔽 by translations and as a finite rotation group on the vector space U,and x is any point not in 𝔽. The smallest Γ–invariant affine subspace of 𝔼 containing x is x + W, where W = D_{ℝ}. We may suppose that 𝔼 = x + W by restricting Γ to x + W if necessary, since it suffices to prove that r(Γ) is crystallographic for some minimal Γ–invariant affine subspace 𝔽 of 𝔼, by the remark following 3.2. The subgroup λ(Γ) of the (compact) orthogonal group of W leaves the discrete spanning subset D of W invariant, hence is discrete and thus finite. So we can apply 3.3. The group Δ = ker(λ|Γ) = Γ ∩ V is discrete since Δ ⊂ D. Let x be a point with D(x) ⊂ \frac{1}{f} Δ. Such a point exists by 3.3c). Then 𝔽 = y + Δ_{ℝ} is a minimal Γ–invariant affine subspace of 𝔼, by 3.3d), and r(Γ) is crystallographic on 𝔽, since Γ contains Δ of finite index.

We are now heading for a proof of 3.2. The proof consists in redoing the proof of the last chapter, this time over the rationals rather than the reals, using the hypothesis that λ(Γ) = F is finite. To prove 3.2 we could actually assume that 𝔼 = 𝔽 and that Γ is discrete but we need only the following Assumption. λ(Γ) is finite. Again, we put λ(Γ) = F, #F = f, Δ = ker(λ|Γ) = Γ ∩ V . We do not suppose that Γ is discrete nor that 𝔽 = 𝔼. Our first handle towards finding a point x with D(x) discrete is analogous to 2.3:

Corollary 3.6. Let D be a ℚ[F]–submodule of V . Any Γ–invariant D–orbit x + D ⊂ 𝔼 intersects the axis 𝔼_{γ} for every γ ∈ Γ.

This applies in particular for D = Δ_{ℚ} and any Γ–invariant subset of the form x + Δ_{ℚ} of 𝔼, e.g. for x as in 3.3 c) Proof. Apply lemma 3.4 to the following situation: Let Γ_{1} be the subgroup of Γ generated by the element γ, let x + D be Γ–invariant, x′ ∈ E_{γ} and D′ = ℤτ(γ). We have D′ ⊂ D in view of the following facts: Δ is contained in D by Γ–invariance of x + D, the power γ^{f} of γ ∈ Γ is contained in Δ and γ^{f} is the translation by f · τ(γ). So x − x′ ∈ D + V^{λ(γ)} by lemma 3.4. Now use T 𝔼_{γ} = V^{λ(γ)} .

Again, as in chapter 2, we shall prove that every Γ–invariant subset of the form x + Δ_{ℚ} intersects 𝔼_{m} for every m ∈ M_{Γ} , thus proving 3.2.

Proposition 3.7. Suppose 𝔼_{1} and 𝔼_{2} are affine subspaces of 𝔼 such that the orthogonal projections of V onto T 𝔼_{i}, i = 1, 2, are in ℚ[F]. Let D be a ℚ[F]–submodule of V and let x + D be Γ–invariant. Suppose x + D intersects both 𝔼_{1} and 𝔼_{2} then x + D intersects 𝔼_{1,2} , the orthogonal projection of V onto T 𝔼_{1,2} is in ℚ[F] and perp (𝔼_{1}, 𝔼_{2}) is in D.

Proof. We apply lemma 2.4 for P_{i} the orthogonal projections of V onto T 𝔼_{i}, i = 1, 2, and A = ℚ[F]. Thus A contains the orthogonal projection of V onto T 𝔼_{1}∩T 𝔼_{2} and contains operators Q_{1}, Q_{2} and Q_{3} such that Q_{1}+Q_{2}+Q_{3} = 1, Q_{3} is the orthogonal projection onto (T 𝔼_{1}+T 𝔼_{2})^{⊥} and the image of Q_{i} is contained in T 𝔼_{i} for i = 1, 2. Now let a_{i} ∈ 𝔼_{i} ∩ (x + D) then b_{1} := a_{1} − Q_{1}(a_{1} − a_{2}) ∈ 𝔼_{1} ∩ (x + D), b_{2} := a_{2} + Q_{2}(a_{1} − a_{2}) ∈ 𝔼_{2} ∩ (x + D), b_{1} − b_{2} = Q_{3}(a_{1} − a_{2}) = perp (𝔼_{2}, 𝔼_{1}) ∈ D, b_{1} ∈ 𝔼_{1,2} ∩ (x + D) and b_{2} ∈ 𝔼_{2,1} ∩ (x + D).

We obtain as in 2.9:

Corollary 3.8. Every Γ–invariant Δ_{ℚ} –orbit intersects 𝔼_{m} for every m ∈ M_{Γ} and perp (𝔼_{m}, 𝔼_{n}) ∈ Δ_{ℚ} for every pair m, n of elements of M_{Γ} .

3.9. We are now ready to prove 3.2. There is a point x ∈ 𝔼 such that x + \frac{1}{f} Δ is Γ–invariant, by 3.3 c). For every point y ∈ x + \frac{1}{k} Δ we have

γy − y = (γx − x) + (λ(γ) − 1)(y − x) ∈ \frac{1}{kf} Δ,

hence D(y) ⊂ \frac{1}{kf} Δ for y ∈ x+ \frac{1}{k} Δ where D(y) is as always the abelian subgroup of V generated by {γy − y; γ ∈ Γ}. This holds in particular for a point y ∈ 𝔼_{m} with T 𝔼_{m} = V^{F} for an appropriate integer k depending on y, by 3.8. Finally, D(y) = D(y′) for every point y′ ∈ 𝔼_{m} since y − y′ is fixed by F. The last claim of 3.2 follows from 3.3d).

3.10. Computational aspects. Let S be a finite set of elements of G generating a subgroup Γ of G, let m ∈ M_{S} be such that T 𝔼_{m} = V^{λ(Γ)} , let x be a point of 𝔼_{m} and let B ⊂ D(x) be a basis of the vector space W = T 𝔽 where 𝔽 = x + W is a minimal Γ–invariant affine subspace of 𝔼. In 2.18 it was described how all these things can be computed. With these notations we have

Corollary 3.11. r(Γ) is crystallographic iff the following three conditions hold

(i) γx − x is a rational linear combination of B for every γ ∈ S. (ii) With respect to the basis B of W the linear map λ(γ) | W = λ ∘ r(γ) is

represented by a matrix in GL(d; ℚ), d = dim W, for every γ ∈ S. (iii) The subgroup of O(W) generated by λ(γ) | W = λ ∘ r(γ), γ ∈ S, is finite.

Proof. If r(Γ) is crystallographic then Δ_{ℝ} = W ⊃ D(x) ⊃ Δ and D(x) contains the subgroup Δ of finite index, hence D(x)_{ℚ} = Δ_{ℚ} which implies (i) and (ii). Bieberbach’s theorems imply (iii). To prove sufficiency let D_{0} be the ℤ–submodule of W generated by B and {γx − x; γ ∈ S}. Then D_{0} is a lattice in W, by (i). The group λ(γ)D_{0} is commensurable with D_{0} by (ii) for every γ ∈ S and hence for every γ ∈ Γ. Then D_{1} = ∑_{γ∈Γ} λ(γ)D_{0} is commensurable with D_{0} , by (iii), and hence a lattice in W in particular the subgroup D(x) of D_{1} generated as a ℤ[λ(Γ)]–module by {γx − x; γ ∈ S} is discrete.

Remark 3.12. Concerning computations, it is easy to check (i) and (ii). As for (iii), the proof above shows that a finite subgroup of GL(d; ℚ) is conjugate in GL(d, ℚ) to a subgroup of GL(d; ℤ) – a basis transformation from a ℤ–basis of D_{0} to a ℤ–basis of D_{1} will give the desired conjugation. But the order of a finite subgroup of GL(d; ℤ) is bounded by a number depending only on d, by a theorem of Minkowski. We want to point out the following consequences of what was developed in the last two chapters. These results give necessary conditions for a given subgroup Γ of G to be discrete.

Notations will be as above: Γ is a subgroup of G, 𝔽 a minimal Γ– invariant affine subspace, r : Γ → Iso(𝔽) the restriction homomorphism and Δ = r(Γ) ∩ T 𝔽 the subgroup of translations of r(Γ). We will assume that r(Γ) is crystallographic. So this applies in particular when Γ is discrete.

Proposition 3.13. The following elements are contained in Δ_{ℚ}:

a) the translational part τ(γ) for every γ ∈ Γ

b) perp (𝔼_{m}, 𝔼_{n}) for m, n in M_{Γ}

c) γx − x for γ ∈ Γ and x ∈ 𝔼_{m}, where m ∈ M_{Γ} and T (𝔼_{m} ∩ 𝔽) = (T 𝔽)^{λ(Γ)} .

It is not true in general that all these elements are contained in a lattice in Δ_{ℚ}, see example 5.11. They are though, if we restrict the length of the elements m ∈ M_{Γ} . Here the length of an element m in the free non-associative semigroup M_{X} generated by X is defined in the obvious way, i.e. l(x) = 1 for x ∈ X and l((m, n)) = l(m) + l(n).

Proposition 3.14. For every N there is a lattice Δ_{N} in Δ_{ℚ} which contains the following elements:

a) τ(γ) for every γ ∈ Γ

b) perp (𝔼_{m}, 𝔼_{n}) for m, n ∈ M_{Γ} and l(m) ≤ N, l(n) ≤ N

c) γx − x for γ ∈ Γ and x ∈ 𝔼_{m}, where m ∈ M_{Γ} , T (𝔼_{m} ∩ 𝔽) = (T 𝔽)^{λ(Γ)} and l(m) ≤ N.

A geometric consequence is the following

Corollary 3.15. Suppose Γ is crystallographic. Then for a given number N the set {𝔼_{m}, l(m) ≤ N} is locally finite. In particular, the set {𝔼_{γ}, γ ∈ Γ} is locally finite.

This holds in particular for the set 𝔽_{r(γ)} = 𝔼_{γ} ∩𝔽, γ ∈ Γ, and the derived set 𝔽_{m} = 𝔼_{m} ∩ 𝔽 if Γ is discrete. But it is not true in general for a discrete subgroup Γ of G that the set{𝔼_{γ}, γ ∈ Γ} is locally finite. For an example see 5.3.

Proof. It suffices to prove 3.14, since 3.13 is an immediate consequence and 3.15 follows in view of the following facts: The distance of 𝔼_{m} and 𝔼_{n} is the length of perp (𝔼_{m}, 𝔼_{n}) and thus equal to zero or bounded a way from zero by 3.14, if the lengths of m and n are bounded. But for a point x ∈ 𝔼_{m} an affine subspace 𝔼_{n}, n ∈ M_{Γ} , containing x is characterized by the vector subspace T 𝔼_{n} = V^{λ⟨supp(n)⟩}, see 2.9 b), which is one of a finite collection of subspaces V^{H}, H a subgroup of the finite group F = λ(Γ).

To prove 3.14 we may assume that 𝔽 = 𝔼 and thus that Γ is crystallographic and Δ is a lattice in V = T 𝔼, since all our vectors in a)–c) are in T 𝔽 by 2.3 and 2.9 c). Again, we put F = λ(Γ), f = #F. Then for γ ∈ Γ we have τ(γ^{f}) = fτ(g) and γ^{f} is a translation, i.e. γ^{f} ∈ Δ, so τ(γ) ∈ \frac{1}{f} Δ.

Let m_{0} ∈ M_{Γ} be such that λ⟨supp(m_{0})⟩ = F and hence T 𝔼_{m_{0}} = V^{F} . Then for x_{0} ∈ 𝔼_{m_{0}} the set γx − x, γ ∈ Γ, is contained in the lattice D(x_{0}), and D(x_{0}) ⊃ Δ, by 3.2. Let w = perp (𝔼_{m_{0}}, 𝔼_{γ}). Then

γ(x_{0} + w) = γx_{0} + λ(γ)w

and

γ(x_{0} + w) = x_{0} + w + τ(γ)

since x_{0} + w ∈ 𝔼_{γ}, hence

(3.16)  γx_{0} − x_{0} = τ(γ) − (λ(γ) − 1)w.

so (λ(γ) − 1)w is contained in the lattice Δ_{0} = \frac{1}{f} Δ + D(x_{0}).

For every λ(γ) ∈ F there is a polynomial h_{1} ∈ ℚ[t] such that

h_{1}(λ(γ)) · (λ(γ) − 1)w = w

for every w ∈ (V^{F})^{⊥}, e.g. if we decompose the polynomial t^{f} − 1 into the relatively prime polynomials t^{f} − 1 = (t − 1) · f_{2} , where f_{2} = ∑_{i=0}^{f−1} t^{i}, and write 1 = h_{1} · (t − 1) + h_{2} · f_{2} with h_{i} ∈ ℚ[t], see the proof of lemma 2.5. It follows that there is a lattice Δ′ ⊃ Δ_{0} which contains perp (𝔼_{m_{0}}, 𝔼_{γ}) for every γ ∈ Γ. We may assume that Δ′ is a ℤ[F]–module.

Now choose for every pair H_{1}, H_{2} of subgroups of F operators Q_{1}, Q_{2} and Q_{3} in ℚ[F] satisfying the conditions of lemma 2.4 for U_{i} = V^{H_{i}}, i = 1, 2, that means Q_{1} +Q_{2} +Q_{3} = 1, Q_{i}(V) ⊂ U_{i} for i = 1, 2 and Q_{3} is the orthogonal projection of V onto (U_{1} + U_{2})^{⊥}. Suppose M is a positive integer such that M times all these operators for all pairs H_{1}, H_{2} are in ℤ[F]. Then for γ_{1}, γ_{2} in Γ and x_{0}+w_{i} ∈ 𝔼_{γ_{i}}, w_{i} = perp (𝔼_{m_{0}}, 𝔼_{γ_{i}}), we have perp (𝔼_{γ_{1}}, 𝔼_{γ_{2}}) = Q_{3}(w_{2}−w_{1}) for Q_{3} as above for the two subgroups H_{i} = ⟨λ(γ_{i})⟩, see the proof of 3.7. So the lattice \frac{1}{M} · Δ′ contains perp (𝔼_{γ_{1}}, 𝔼_{γ_{2}}) for every pair γ_{1}, γ_{2} of elements of Γ.

The same argument shows that if we have a_{i} = x+w_{i} ∈ 𝔼_{m_{i}} for some x ∈ 𝔼 and w_{i} is contained in the ℤ[F]–submodule Δ_{i} of V then perp (𝔼_{m_{1}}, 𝔼_{m_{2}}) = Q_{3}(w_{2} − w_{1}) ∈ \frac{1}{M} (Δ_{1} + Δ_{2}), x + w_{1} − Q_{1}(w_{1} − w_{2}) ∈ 𝔼_{(m_{1},m_{2})} and x + w_{2} + Q_{2}(w_{1} − w_{2}) ∈ 𝔼_{(m_{2},m_{1})} , see the proof of 3.7. It thus follows that for m_{0} as above, firstly,x_{0} + \frac{1}{M^{l(m)}} Δ′ intersects 𝔼_{m_{0}} , and secondly,perp(𝔼_{m_{1}}, 𝔼_{m_{2}}) ∈ \frac{1}{M^{l(m_{1})+l(m_{2})}} Δ′. Finally, if x ∈ 𝔼_{m}, w = perp (𝔼_{m}, 𝔼_{γ}),x + w ∈ 𝔼_{γ}, we have

γ(x + w) = γx + λ(γ)w
= x + w + τ(γ)

since x + w ∈ 𝔼_{γ} and hence γx − x = τ(γ) − (λ(γ) − 1)w ∈ \frac{1}{M^{l(m)+1}} Δ′,which proves the claim concerning 3.14 c) since if T 𝔼_{m} = V^{F} , every point x of 𝔼_{m} is the foot of a perpendicular to 𝔼_{γ}, i.e. 𝔼_{m} = 𝔼_{(m,γ)} by 2.9 a) and b).

4. When is a group discrete?

4.1. Hypotheses and notations for this whole chapter. Γ = ⟨S⟩ is a subgroup of G = Iso(𝔼)given by a finite set S of generators, 𝔽 is a minimal Γ–invariant affine subspace of 𝔼 and r: Γ → Iso(𝔽) is the restriction homomorphism. We suppose that r(Γ) is a crystallographic group on 𝔽.

Then Γ is discrete iff the kernel of r is finite. But this seems difficult to check in terms of the generating set S of Γ. We thus propose a different procedure, which is algorithmic. It is based on two ingredients, firstly the observation that the finiteness of the kernel of r is easy to check if B is an abelian group of a certain type and secondly, that Γ contains such a subgroup B of finite index if Γ is discrete.

We start with the following well known fact (s. [11], [1]). We give a short proof based on the Bieberbach theorems which also gives additional information we shall use in the algorithmic procedure.

Proposition 4.2. If Γ is discrete then Γ contains a subgroup B of finite index such that its group λ(B)of linear parts is contained in a torus and r(B) consists of translations of 𝔽 only.

Proof. Let Γ_{1} be the normal subgroup of those γ ∈ Γ for which r(γ) is a translation of 𝔽. Then Γ_{1} is of finite index in Γ by the corollary to the Bieberbach theorems and Bieberbach’s first theorem, and the commutator subgroup of Γ_{1} is finite by Bieberbach’s second theorem. Hence the commutator subgroups of the following groups are finite: λ(Γ_{1}), the closure L_{1} of λ(Γ_{1}), the connected component L_{0} of the identity in L_{1} . So the Lie algebra of L_{0} is abelian. Both the closure L of λ(Γ) and L_{1} are compact, being closed subgroups of the orthogonal group of T 𝔼. Furthermore, L/L_{1} and L_{1}/L_{0} are finite, so L_{0} is the connected component of L and is a torus. Nowr(γγ_{1}γ^{−1}) is the translation λ(γ)r(γ_{1}) for γ_{1} ∈ Γ_{1} . So the torus L_{0} acts on the lattice r(Γ_{1}) in T 𝔽, hence fixes T 𝔽. It follows that the normal subgroupB = {γ ∈ Γ; r(γ) ∈ T 𝔽, λ(γ) ∈ L_{0}} of Γ is abelian and of finite index in Γ.

Note that the proof has shown that the torus L_{0} is the connected component of (λ(Γ))^{−}.

Corollary 4.3. If Γ is discrete then L_{0} := \overline{(λ(Γ))}_{0} is a torus and fixes T 𝔽. So B = {γ ∈ Γ ; r(γ) ∈ T 𝔽, λ(γ) ∈ L_{0}} is an abelian normal subgroup of Γ of finite index.

If our group Γ is of the special form as B in 4.2 then it is easy to check if r has finite kernel, as follows.

Proposition 4.4. Suppose a subgroup B of G leaves the affine subspace 𝔽 invariant and r(B) consists of translations only. Suppose furthermore that λ(B) is contained in a torus T. Then the homomorphism (λ, r) : B → T × T 𝔽 is injective. Let 𝔱 be the Lie algebra of T and let exp : 𝔱 → T be its exponential map. Let A be the inverse image of the subgroup (λ, r)B of T × T 𝔽 under exp ×id : 𝔱 × T 𝔽 → T × T 𝔽. Then

rank A = rank ker(r) + rank r(B) + dim T.

In particular, if B is finitely generated then ker(r) is finite iff rank A = rank r(B) + dim T.

Here and in the proof the rank of an abelian group D is dim_{ℚ} D ⊗ ℚ.

Proof. Put W = T 𝔽 and e = exp ×id : 𝔱 × W → T × W. For a subgroup A of T × W the surjection e : e^{−1}(D) → D has as its kernel the lattice ker(e) in 𝔱. Thus rank D = rank(e^{−1}D)−dim T. Hence for our group D = (λ, r)B we have rank(ker(r)) = rank D − rank(r(D)) = rank(e^{−1}(D)) − dim T − rank(r(D)).

4.5. Here is now the procedure to determine if Γ is discrete.

Step 1. Find a finite subset S_{1} of Γ with the following properties

a) r(γ) is a translation of 𝔽 for every γ ∈ S_{1} .

b) The subgroup of T 𝔽 generated by {r(γ) ; γ ∈ S_{1}} is a lattice in T 𝔽

c) ⟨λ(γ)⟩^{−} is connected for every γ ∈ S_{1} . We describe below how to find such a set S_{1} , see 4.13. Concerning c) we

use the following notation. For an element U of the orthogonal group O = O(V) of V = T 𝔼 let C(U) be the closure of the cyclic group generated by U, so C(U) = ⟨U⟩^{−}. Let C_{0}(U) be the connected component of 1 in C(U). The order m of the finite cyclic group C(U)/C_{0}(U) can be determined (see 4.8). For a non zero multiple n of m we have C(U^{n}) = C_{0}(U) = C_{0}(U^{m}).

The remaining steps of our test if Γ is discrete are as follows.

4.6. Suppose S_{1} is a finite subset of Γ with the properties a)–c) of 4.5.

Test (i) The tori C_{0}(λ(γ)), γ ∈ S_{1} , commute.

If (i) holds, let T be the subtorus ∏_{γ∈S_{1}} C_{0}(λ(γ)) of O(V) generated by the C_{0}(λ(γ)), γ ∈ S_{1} . Let 𝔱 be its Lie algebra and let exp : 𝔱 → T be its exponential map. Put e = exp × id : 𝔱 × T 𝔽 → T × T 𝔽.

Let Θ be the subgroup of 𝔱 × T 𝔽 generated by {e^{−1}(γ) , γ ∈ S_{1}}.

Test (ii) rank_{ℚ} Θ = dim 𝔽 + dim T.

The group Γ acts on the Lie algebra 𝔬 of O(V) via Ad ∘ λ. Also, Γ acts on T 𝔽 by γ · t = λ(γ)t = γtγ^{−1} , hence on the product 𝔬 × T 𝔽. So the following condition makes sense.

Test (iii) Every γ ∈ Γ maps Θ_{ℚ} to itself.

Now comes the last test which is the hardest to perform. Put B_{1} = ⟨S_{1}⟩. Note that r(B_{1}) is a subgroup of Δ of finite index, by 4.5 a) and b). Let 𝔼_{m} be as in 3.2, i.e. m ∈ M_{S} and T 𝔼_{m} = V^{λ(Γ)} . Let x_{0} ∈ 𝔼_{m} ∩ 𝔽. We have r(B_{1})_{ℚ} = Δ_{ℚ} = D(x_{0})_{ℚ} by 3.2. So we can pick an element β = β_{γ} ∈ e(Θ_{ℚ}) for every γ ∈ S such that

β(x_{0}) = γx_{0}

Here is our last test:

Test (iv) The group ⟨γ^{−1}β_{γ} ; γ ∈ S⟩ is finite.

As always, let S be a finite subset of G, Γ = ⟨S⟩, 𝔽 a minimal Γ– invariant affine subspace of 𝔼.

Theorem 4.7. Γ is discrete iff r(Γ) is crystallographic and one (any) S_{1} as in 4.5 passes the Tests (i)–(iv).

The rest of this chapter is devoted to questions of the type how to find the relevant objects (S_{1}, C_{0}(U), β_{γ}, …), how to perform the tests, possible variations and, of course, a proof of 4.7.

4.8. The first question we deal with is how to find for an element U in O(V) an integer m ≠ 0 such that C(U^{m}) = C_{0}(U), where C(U) = ⟨U⟩^{−} and C_{0}(U) is the connected component of C(U). Note that m ≠ 0 has this properties iff m is a multiple of the order r of the finite cyclic group C(U)/C_{0}(U).

To actually compute r one can proceed as follows. Choose a torus T in the special orthogonal group of V which contains U^{2} , for instance by decomposing V into mutually orthogonal one and two dimensional U –invariant subspaces and taking for T the corresponding product of SO_{2} ’s, one for every 2–dimensional U –invariant subspace.

Let 𝔱 be the Lie algebra of T and let exp : 𝔱 → T be its exponential map. The lattice Λ := ker exp gives 𝔱 a ℤ–structure. We use this structure to define the following notion: A linear form ℓ on 𝔱 is said to be defined over ℤ or ℚ, respectively, if ℓ(Λ) ⊂ ℤ or ℚ, respectively. Let 𝔱_{ℤ}^{∗} and 𝔱_{ℚ}^{∗} be the set of linear forms defined over ℤ and ℚ, respectively. Here is a description of C(U) and C_{0}(U). Choose an element X ∈ 𝔱 with exp X = U. Put D(X) = {ℓ ∈ 𝔱_{ℚ}^{∗} | ℓ(X) ∈ ℚ}.

4.9. Then

a) exp^{−1} C(U) = {Y ∈ 𝔱 | ℓ(Y) ∈ ℤ · ℓ(X) + ℤℓ(Λ)for every ℓ ∈ D(X)}

b) For the Lie algebra LC(U) = LC_{0}(U) we have

LC(U) = ⋂_{ℓ∈D(X)} ker ℓ

c) If we write X = n ; ∑_{i=1}ξ_{i}e_{i} with respect to a basis e_{1}, …, e_{n} of Λ, then

1 + dim C_{0}(U) is equal to the dimension of the ℚ–vector subspace of ℝ spanned by 1, ξ_{1}, …, ξ_{n}.

d) The order r of the cyclic group C(U)/C_{0}(U) is the maximal denominator of the numbers ℓ(X), ℓ ∈ D(X) ∩ 𝔱_{ℤ}^{∗}. Hence if ℓ_{1}, …, ℓ_{d} is a basis of the lattice D(X)∩𝔱_{ℤ}^{∗} in D(X), then r is the least common multiple of the denominators of the numbers ℓ_{1}(X), …, ℓ_{d}(X).

Proof. These claims are clear for the torus ℝ/ℤ with exponential map ℝ → ℝ/ℤ and follow in general from Pontryagin duality: Every closed subgroup of a torus T is the intersection of kernels of homomorphisms T → ℝ/ℤ. These homomorphisms correspond on the level of Lie algebras to linear maps in 𝔱_{ℤ}^{∗}. Concerning c) look at the ℚ–linear map F : ℚ^{n+1} → ℝ , F (a_{0}, …, a_{n}) = −a_{0} + a_{1}ξ_{1} + ⋯ + a_{n}ξ_{n}. For (a_{1}, …, a_{n}) ∈ ℚ^{n} we have ∑_{i=1}^{n} a_{i}e_{i}^{∗} ∈ D(X) iff (a_{0} = ∑_{i=1}^{n} a_{1}ξ_{i}, a_{1}, …, a_{n}) ∈ ℚ^{n−1} ∩ ker F where e_{1}^{∗}, …, e_{n}^{∗} is a dual basis of the basis e_{1}, …, e_{n} in c) of the ℚ–vector space 𝔱_{ℚ}. Hence dim D(X) = dim ker F and thus dim C(U) = n − dim D(X) = n − (n + 1 − rank_{ℚ}{1, ξ_{1}, …, ξ_{n}}).

4.10. Necessity of (i)–(iv) in 4.6. If Γ is discrete then (λ(Γ)^{−})_{0} is a torus, hence the tori C(λ(γ)), γ ∈ S_{1} , commute. Furthermore, the restriction homomorphism r : Γ → Iso(𝔽) has finite kernel, hence so does r|B_{1} , which is equivalent to (ii) by 4.4. Furthermore, B_{1} is of finite index in Γ, since r has finite kernel and r(B_{1}) is a lattice in T 𝔽 contained in r(Γ) ∩ T 𝔽, hence of finite index in r(Γ). It follows that there is a normal subgroup B_{2} of Γ contained in B_{1} and of finite index in B_{1} . Then Θ_{2} := e^{−1}(B_{2}) is a subgroup of Θ of finite index, hence Θ_{2,ℚ} = Θ_{ℚ} and thus every γ ∈ Γ maps Θ_{ℚ} to itself.

Concerning (iv), there is a non–zero integer m such that for Θ_{3} := \frac{1}{m}Θ we have Θ_{3} ⊃ Θ and e(Θ_{3}) contains all the elements β_{γ} we have picked, one for every γ ∈ S. So the group Γ_{3} = Γ · e(Θ_{3}) contains Γ of finite index and thus is discrete. It follows that the group generated by {γ^{−1}β_{γ} ; γ ∈ S} which fixes x_{0} is finite.

4.11. Sufficiency of the conditions (i)–(iv) in 4.6. The torus C(λ(γ)) acts on the lattice r(B_{1}) in T 𝔽, so the action of C(λ(γ)) on T 𝔽 is trivial. Thus B_{1} is abelian, by (i), and r|B_{1} has finite kernel by (ii), in view of 4.4. So B_{1} is discrete. The group Θ = e^{−1}(B_{1}) is finitely generated, since B_{1} and the kernel of exp : 𝔱 → T are so. It follows from (ii) that Θ is a lattice in 𝔱 × T 𝔽.

Let O_{1} be the subgroup of those elements of the orthogonal group of V which fix T 𝔽. Embed O_{1} × T 𝔽 into G by i(U, t) = A(x, t, U) where x ∈ 𝔽, see 1.2. This embedding does not depend on the chosen point x of 𝔽, by 1.5. Hence if γ ∈ G maps 𝔽 to itself we have

γi(t, U)γ^{−1} = i(λ(γ)Uλ(γ)^{−1} , λ(γ)t) .

It follows from (iii) that the differential of the conjugation by γ ∈ Γ maps the Lie algebra Θ_{ℝ} of T × T 𝔽 to itself and thus Γ normalizes D := T × T 𝔽. The group D · Γ acts on D by conjugation and hence on its Lie algebra Θ_{ℝ}. Here D acts trivially. For the corresponding homomorphism Γ·D → Aut(Θ_{ℝ}) the image is finite, by (iv), since β_{γ} ∈ D. Notice that the homomorphism takes values in Aut (Θ_{ℚ}) by (iii). It follows that there is a lattice Θ_{2} in Θ_{ℚ} with the following properties: e(Θ_{2}) contains β_{γ} for every γ ∈ S, Θ_{2} contains Θ of finite index and Θ_{2} is Γ-invariant. So B_{2} = e(Θ_{2}) contains β_{γ} for every γ ∈ S, contains B_{1} of finite index and is normalized by Γ. So the group Γ_{2} = Γ · B_{2} contains Γ of finite index. On the other hand, Γ_{2}/B_{2} is a quotient of the group in (iv), hence finite. So Γ is commensurable to B_{1} and hence discrete.

Remark 4.12. Note the following variation of condition (iv). Pick for every γ ∈ S a pair of elements β_{i}(γ) ∈ e(Θ_{ℚ}) such that β_{2}(γ) ∘ γ ∘ β_{1}(γ) fixes the point x_{0} ∈ 𝔽_{m}. Then theorem 4.7 holds with (iv) replaced by

Test (iv′) The group generated by {β_{2}(γ) ∘ γ ∘ β_{1}(γ) ; γ ∈ S} is finite.

For the proof note that γ ∘ β_{1}(γ) ∘ γ^{−1} ∈ e(Θ_{ℚ}) by (iii), hence (iv) is equivalent to (iv′).

4.13. It remains to describe a procedure how to find a subset of Γ with the properties of 4.5. The first step is to 4.13a. Find an element γ ∈ Γ such that r(γ) is a translation. Form a list of elements of Γ by taking successive generations of S, as follows. Let l_{S}(γ) = min{q; γ = γ_{1}^{ε_{1}} · . . . · γ_{q}^{ε_{q}} , γ_{i} ∈ S , ε_{i} ∈ {±1}} be the length of γ ∈ Γ with respect to the generating set S. Put B_{m} = {γ ∈ Γ; l_{S}(γ) ≤ m}

and S_{m} = {γ ∈ Γ; l_{S}(γ) = m}. The sets B_{m} = {e} ∪^{▪} S_{1} ∪^{▪} ⋯ ∪^{▪} S_{m} can be

computed inductively. Recall that there is an upper bound on the orders of the finite subgroups of SL(d, ℤ), d = dim 𝔽, by a theorem of Minkowski, see 3.12. Hence, if #B_{m} is larger than this bound there are two different elements g and h in B_{m} such that λ(g)|T 𝔽 = λ(h)|T 𝔽 and hence g^{−1}h restricts to a translation on 𝔽. 4.13b. Take elements γ_{1}, …, γ_{t} ∈ Γsuch that r(γ_{1}), …, r(γ_{t}) are linearly independent elements of T 𝔽 and span a λ(Γ)–module. This can be done starting from elements α ∈ Γ with r(α) ∈ T 𝔽 by taking conjugates γαγ^{−1} , γ ∈ Γ. Linear independence and spanning can be taken over ℚ or ℝ, this leads to the same conditions, since r(Γ) is crystallographic on 𝔽 and hence r(Γ) ∩ T 𝔽 a lattice in T 𝔽. At this point one can perform easy finiteness checks: If when looking for linearly independent elements in T 𝔽 one finds elements γ_{1}, …, γ_{t} in Γ such that r(γ_{1}), …, r(γ_{t}) are linearly dependent, say ∑ n_{i}r(γ_{i}) = 0, n_{i} ∈ ℤ, then the products of γ_{1}^{n_{1}}, …, γ_{t}^{n_{t}} in any order must be contained in a finite group, in particular of finite exponent. 4.13c. If we have a situation as in 4.13b we can use induction on dim 𝔽 to find more elements γ ∈ Γ with τ(γ) ≠ 0, as follows. More precisely, let γ_{1}, …, γ_{t} ∈ Γ be such that r(γ_{1}), …, r(γ_{t}) are in T 𝔽 and their ℝ–span W := ⟨r(γ_{1}), …, r(γ_{t})⟩_{ℝ} is an ℝ[λ(Γ)]–module. Then Γ acts on 𝔼/W and 𝔽/W by affine isometries and the image of Γ in the affine group of 𝔽/W is a crystallographic group on 𝔽/W, since r(Γ) is crystallographic on 𝔽. So we can apply the procedure described in 4.13a on the space 𝔽/W of smaller dimension than dim 𝔽 and we find an element γ ∈ Γ which induces a translation on 𝔽/W, so τ(γ) ∉ W, and hence γ^{q} | 𝔽 is a translation in T 𝔽 not in W if q is a multiple of the order of λ(γ)|W.

Note that the image of Γ on 𝔼/W may not be discrete even if Γ is discrete on 𝔼, e.g. if W = T 𝔽.

4.14. It would be nice to have an easier procedure to produce an element γ ∈ Γ with τ(γ) ≠ 0, easier than 4.13a which uses Minkowski’s theorem. If τ(γ) = 0 for every γ ∈ S and ⋂_{γ∈S} Fix (γ) = Ø then Γ = ⟨S⟩ is infinite, but the example

2.18 of an infinite group Γ such that Fix (γ) ≠ 0 for every γ ∈ Γ shows that one has to use the fact that the relevant group is crystallographic or at least has a finite group of linear parts. So there is no general procedure, i.e. one which does not use the arithmeticity or at least finiteness of the relevant linear parts of Γ, to find a non–commutative analogue of perp(𝔼_{g}, 𝔼_{h}),i.e. an element f ∈ ⟨g, h⟩ with non–zeroτ(f) ∈ ℚ · perp(𝔼_{g}, 𝔼_{h}).

5. Examples

In this chapter we give various examples and counterexamples. We also present a complete answer to the question when a finite set S of affine isometries of a Euclidean affine space 𝔼 generates a discrete group for dimensions of 𝔼 up to 3.

We use the following notation. If 𝔽 is an affine subspace of our Euclidean affine space then the reflection σ_{𝔽} in 𝔽 is the affine map defined by

σ_{𝔽}(x + v) = x − v

where x is a point of 𝔽 and v is a vector orthogonal to T 𝔽. Every involution σ, i.e. element of order ≤ 2, in G is the reflection in the affine subspace Fix (σ) = {x ∈ 𝔼 | σ(x) = x}.

5.1. Every finite non–abelian simple group Γ when faithfully represented is generated by linear involutions σ_{𝔽_{i}} where all the 𝔽_{i} have the same dimension.

By the theorem of Feit–Thompson Γ has even order, so contains an involution σ by the Sylow theorems. Then Γ is generated by the set of involutions γσγ^{−1} , γ ∈ Γ. Note that γσγ^{−1} is the reflection in the linear subspace γ𝔽, where 𝔽 is the linear subspace of fixed points of σ.

5.2. Let σ_{1} = σ_{𝔼_{1}} and σ_{2} = σ_{𝔼_{2}} be two involutions in G. Then Γ = ⟨σ_{1}, σ_{2}⟩ is discrete iff

a) 𝔼_{1} ∩ 𝔼_{2} = Ø or

b) σ_{1} · σ_{2} is of finite order.

Proof. Necessity: If 𝔼_{1} and 𝔼_{2} intersect, say x ∈ 𝔼_{1} ∩ 𝔼_{2} , then Γ fixes x hence must be finite if Γ is discrete. Sufficiency: The subgroup Γ_{1} = ⟨σ_{1} · σ_{2}⟩ is of index 2 in Γ. We thus may assume that 𝔼_{1} ∩𝔼_{2} = Ø. An affine subspace 𝔽 of 𝔼 is minimal Γ–invariant iff 𝔽 is a line perpendicular to and intersecting both 𝔼_{1} and 𝔼_{2} . Then σ_{1}σ_{2} induces the translation by perp(𝔼_{2}, 𝔼_{2}) on 𝔽, which implies that Γ is discrete.

Here is the counterexample promised in the paragraph following Corollary 3.15.

5.3. There is a discrete group Γ for which the set of axes 𝔼_{γ}, γ ∈ Γ, is not locally finite.

Let V = V_{1} ⊕ V_{2} , dim V_{1} = dim V_{2} = 2 and embed O(V_{2}) into O(V) as {1_{V_{1}}} × O(V_{2}). Let γ_{1} = A(t_{i}, U_{i}), i = 1, 2 with t_{i} ∈ V_{1} , U_{1} ∈ O(V_{2}) such that det U_{1} = −1, U_{2} ∈ SO(V_{2}) of finite order and {t_{1}, t_{2}} is a basis of V_{1} . Then the minimal Γ–invariant subspace is 𝔽 = x_{0} + V_{1} and for γ = γ_{2}^{n} · γ_{1} · γ_{2}^{−n} we have 𝔼_{γ} = 𝔽 + U_{2}^{n} Fix (U_{1}), an infinite set of different subspaces of 𝔼, all containing 𝔽.

5.4. For a finite set of points 𝔼_{1}, …, 𝔼_{m} in 𝔼 the subgroup Γ generated by σ_{i} := σ_{𝔼_{i}} is discrete iff for the family of difference vectors t_{ij} = 𝔼_{i} − 𝔼_{j}, 1 ≤ i, j ≤ m we have

rank_{ℚ}{t_{ij}; 1 ≤ i, j ≤ m} = rank_{ℝ}{t_{ij}; 1 ≤ i, j ≤ m} .

The group Γ generated by the σ_{i} ’s contains the group Δ of translations generated by 2t_{ij}, 1 ≤ i, j ≤ m, as a subgroup of index 2.

5.5. We proceed to give a complete discussion of our problem for affine spaces 𝔼 of small dimension, i.e. dim 𝔼 ≤ 3.

Let S be a finite subset of the affine group G of isometries of the affine Euclidean space 𝔼. Let Γ = ⟨S⟩ and let 𝔽 be a minimal Γ–invariant affine subspace of 𝔼.

5.6. dim 𝔼 = 1. The group G has two connected components, one consisting of the translations by elements t ∈ V = T 𝔼 and one consisting of the reflections σ_{ℙ} in points ℙ ∈ 𝔼.

The group Γ = ⟨S⟩ for S = ⟨t_{1}, …, t_{m}; σ_{ℙ_{1}}, …, σ_{ℙ_{r}}⟩ is discrete iff rank_{ℚ}{t_{1}, …, t_{m}, ℙ_{i} − ℙ_{j}, 1 ≤ i, j ≤ r} ≤ 1.

The subgroup Δ of translations in Γ is generated by {t_{1}, …, t_{m}; 2t_{ℙ_{i}−ℙ_{j}}, 1 ≤ i, j ≤ r} and [Γ : Δ] ≤ 2.

5.7. dim 𝔼 = 2. The group G = Iso(𝔼) has three types of elements:

a) λ(g) = 1. Then g is a translation by a vector t ∈ T 𝔼.

b) λ(g) ∈ SO(V), λ(g) ≠ 1. Then g has a unique fixed point, ℙ say, and is a rotation by a unique angle φ ∈ ℝ/2πℤ, φ ∉ 2πℤ, around ℙ. We write g = D(ℙ, φ). Here we suppose that 𝔼 is oriented.

c) λ(g) ∉ SO(V). Then g has a one–dimensional axis, 𝕃 say, and g is of the form t ∘ σ_{𝕃} = σ_{𝕃} ∘ t where t ∈ T 𝕃 and σ_{𝕃} is the orthogonal reflection in 𝕃. Then g is called a glide reflection. Both t and 𝕃 are uniquely determined by g.

We distinguish cases according to the dimension of invariant subspaces.

Fixed point.

Clearly, all the elements of S, and hence of Γ, have the point 𝔽 as a common fixed point iff S contains no non–trivial translation, for all the rotations D(ℙ, φ) in S we have ℙ = 𝔽 and for all the glide reflections t ∘ σ_{𝕃} in S we have 𝔽 ∈ 𝕃 and the translational part t is equal to zero.

If this is the case, then Γ is discrete iff Γ is finite iff the following two conditions hold:

a) For every rotation D(ℙ, φ) in S the rotation angle φ is commensurable with π, i.e. φ ∈ ℚ · π.

b) For any two reflections σ_{𝕃_{i}}, σ_{𝕃_{j}} in S the angle ∡(𝕃_{i}, 𝕃_{j}) is commensurable

with π.

The subgroup Γ ∩ λ^{−1}(SO) is of index ≤ 2 in Γ and is generated by the rotations D(ℙ, φ) in S and the products σ_{𝕃_{1}} ∘ σ_{𝕃_{j}} = D(ℙ, 2α_{i,j}) where σ_{𝕃_{i}}, σ_{𝕃_{j}} are in S and α_{i,j} = ∡(𝕃_{i}, 𝕃_{j}).

Invariant line.

All the elements of S, and hence of Γ, leave the line 𝕃 invariant iff a) all the translations of S are in T 𝕃

b) for every rotation D(ℙ, φ) in S we have ℙ ∈ 𝕃 and φ = π mod 2πℤ

c) for every glide reflection t ∘ σ_{𝕃′} in S we have c1) 𝕃 = 𝕃′ or c2) 𝕃 ⊥ 𝕃′ and t = 0.

Suppose this is the case. We then have for the restriction homomorphism r : Γ → Iso(T 𝕃) in the various cases:

a) r(t) = t for the translation t in T 𝕃.

b) r(D(ℙ, π)) = σ_{ℙ} for ℙ ∈ 𝕃. c1) r(t ∘ σ_{𝕃}) = t for the glide reflection t ∘ σ_{𝕃}. c2) r(σ_{𝕃′}) = σ_{𝕃∩𝕃′} for the reflection in the line 𝕃′ orthogonal to 𝕃.

Let

T_{a} = {r(t); t ∈ S}
T_{c1} = {r(t ∘ σ_{𝕃}); t ∘ σ_{𝕃} ∈ S}
P_{b} = {ℙ; D(ℙ, π) ∈ S}
P_{c2} = {𝕃 ∩ 𝕃′; 𝕃 ⊥ 𝕃′ and σ_{𝕃′} ∈ S}

So T_{a} and T_{c1} are subsets of T 𝕃 and P_{b} and P_{c2} are subsets of 𝕃. Then Γ is discrete iff the ℚ–rank of the following subset of T 𝕃 is ≤ 1 :

T_{a} ∪ T_{c1} ∪ {ℙ − ℙ′; where ℙ and ℙ′ are in P_{b} ∪ P_{c2}} .

The proof follows from the case dim 𝔼 = 1 in view of the following facts: r(Γ) is generated by T_{a} ∪ T_{c1} ∪ {σ_{ℙ}; ℙ ∈ P_{b} ∪ P_{c2}}. An element γ of the kernel of r is uniquely determined by λ(γ) | (T 𝕃)^{⊥}, so ker(r) has at most two elements.

Note that although the line 𝕃 is invariant in the case under consideration it may not be a minimal Γ–invariant subspace in certain very special cases.

Γ crystallographic

The whole plane 𝔼 is the only Γ–invariant subspace iff none of the cases described above occurs, i.e. iff there is no fixed point and no invariant line. Then Γ is crystallographic if Γ is discrete. The following well known fact is crucial for dim 𝔼 = 2: 5.8. If Γ is crystallographic then every element of λ(Γ) has order 1, 2, 3, 4 or 6.

Proof. Every element of O(V), not in SO(V), has order 2. If λ(γ) is in SO(V), say λ(γ) is rotation by the angle φ, then trace λ(γ) = 2 cos φ.

On the other hand λ(γ) is represented by a matrix in SL(2; ℤ) with respect to a basis of Δ = Γ ∩ V , hence trace λ(γ) ∈ ℤ, which immediately implies that φ ≡ 0, ±π/3, ±π/2, ±2π/3, π mod 2π.

5.9. dim 𝔽 = 2, S finite ⊂ G = Iso(𝔼). Then Γ = ⟨S⟩ is discrete if the following two conditions A and B hold. Conversely, if Γ is crystallographic then A and B hold.

To formulate the two conditions we need the following set of notations:

T_{a} = {t ∈ T 𝔼; t ∈ S}
𝔼_{b} = {ℙ; D(ℙ, φ) ∈ S and φ ≢ 0}
λ_{b} = {φ ∈ ℝ/2πℤ; D(ℙ, φ) ∈ S and φ ≢ 0}
𝔼_{c} = {𝕃 line in 𝔼; t ∘ σ_{𝕃} ∈ S for some t ∈ T 𝕃}
λ_{c} = {σ_{𝕃}; t ∘ σ_{𝕃} ∈ S for some t ∈ T 𝕃}
T_{c} = {t ∈ T 𝔼; there is a line 𝕃 in 𝔼 such that t ∈ T 𝕃 and t ∘ σ_{𝕃} ∈ S}

A. (Angle Condition) All the angles in the following set of angles are integer multiples of π/6 or all of them are integer multiples of π/4:

λ_{b} ∪ {2 · ∡(𝕃_{i}, 𝕃_{j}); 𝕃_{i} and 𝕃_{j} in 𝔼_{c}} .

Note that λ(Γ) is generated by λ_{b} ∪ λ_{c} and hence λ(Γ) ∪ SO is generated by the set in condition A, for any S ⊂ G.

In order to formulate the translational condition we use the following notations. Let M and N be sets of affine subspaces of 𝔼. Then we define

perp(M, N) := {perp(𝔼_{1}, 𝔼_{2}); 𝔼_{1} ∈ M, 𝔼_{2} ∈ N}
perp(M) := perp(M, M)
feet(M, N) := {𝔼_{1,2}; 𝔼_{1} ∈ M; 𝔼_{2} ∈ N}
feet(M) := feet(M, M)

where as usual the set 𝔼_{1,2} is the affine subset of 𝔼_{1} consisting of the feet of perpendiculars from 𝔼_{2} to 𝔼_{1} . B. (Translational Condition) The smallest λ(Γ)–module containing the following subset of T 𝔼 is discrete:

T_{a} ∪ T_{c} ∪ perp(feet(𝔼_{b} ∪ 𝔼_{c})) .

Let us write more explicitly the last part T_{3} := perp(feet(𝔼_{b} ∪ 𝔼_{c}))

feet(𝔼_{b} ∪ 𝔼_{c}) = 𝔼_{b} ∪ 𝔼_{c} ∪ 𝔼_{c,b} ∪ 𝔼_{c,c}

where 𝔼_{c,b} = feet(𝔼_{c}, 𝔼_{b}) and 𝔼_{c,c} = feet(𝔼_{c}, 𝔼_{c}) is the disjoint union of 𝔼_{c}

and the set 𝔼^{▪}_{c,c} of points ℙ such that {P } = 𝕃_{i} ∩ 𝕃_{j} for two non parallel lines 𝕃_{i} and 𝕃_{j} in 𝔼_{c}. We thus have to consider 10 types of translations, using perp(M, N) = −perp(N, M). Things simplify considerably in the following two special cases to which we can always reduce. Namely we assume that our set S of generators of Γ contains at most one glide reflection. This can be achieved by replacing the set of glide reflections γ_{1}, …, γ_{r} in S by the set γ_{1}, γ_{1}^{−1}γ_{2}, … γ_{1}^{−1}γ which does not change the group generated but γ_{1}^{−1}γ_{i} is a translation or a rotation.

Case 0. S contains no glide reflection. Then

T_{3} = perp(𝔼_{b}, 𝔼_{b}) = {ℙ_{i} − ℙ_{j}; ℙ_{i} and ℙ_{j} in 𝔼_{b}}

Case 1. S contains exactly one glide reflection, say t ∘ σ_{𝕃} with axis 𝕃 and translational part t ∈ T 𝕃.

Then 𝔼_{c} = 𝔼_{c,c} = {𝕃}, T_{c} = {t} and 𝔼_{c,b} is the set of feet of perpendiculars from points ℙ ∈ 𝔼_{b} to the line 𝕃. It is easy to see that the set T_{b,c} ∪ T_{(c,b),(c,b)} generates the same ℤ–module as T_{3} does, where

T_{b,c} = perp(𝔼_{b}, 𝔼_{c}) ⊂ (T 𝕃)^{⊥}

and

T_{(c,b),(c,b)} = perp(𝔼_{c,b}, 𝔼_{c,b} ⊂ T 𝕃 .

Thus in case 1 the translational condition B reduces to one–dimensional conditions for T 𝕃 and (T 𝕃)^{⊥} if the angle condition A holds. So suppose A holds. Note that in the case 1 at hand the second set

{2 · ∡(𝕃_{i}, 𝕃_{j}); 𝕃_{i} and 𝕃_{j} in 𝔼_{c}}

in condition A reduces to 0, since 𝔼_{c} = {𝕃}. We distinguish the following three cases of the angle condition A. A 2: Every element of λ_{b} has oder ≤ 2. A 4: There is an element in λ_{b} of order 4. A 6: There is an element in λ_{b} of order 3 or 6.

Let p_{W} be the orthogonal projection of T 𝔼 onto the linear subspace W of T 𝔼. Put

T_{1} = p_{T 𝕃}(T_{a} ∪ T_{(c,b),(c,b)})
T_{2} = p_{(T 𝕃)^{⊥}}(T_{a} ∪ T_{c} ∪ T_{b,c})

and let U = D(0, ±π/2) be one of the two elements of SO(T 𝔼) of order 4. So U maps (T 𝕃)^{⊥} to T 𝕃. Then in case 1 under the hypothesis that the angle condition A holds the translational condition B is equivalent to B, A 2: rk_{ℚ}T_{1} ≤ 1 and rk_{ℚ}T_{2} ≤ 1 if A 2 holds. B, A 4: rk_{ℚ}(T_{1} ∪ UT_{2}) ≤ 1 if A 4 holds. B, A 6: rk_{ℚ}(T_{1} ∪ \sqrt{3}UT_{2}) ≤ 1 if A 6 holds.

Remark . If the subgroup F of O(V) contains a rotation of order 4, respectively 3, then every F –invariant lattice Θ in V = T 𝔼 is equal to the lattice generated by the roots of the root system B_{2} or BC_{2} , respectively A_{2} or G_{2} , up to homothety. If we identify V with ℂ then Θ = ℤ + ξℤ where ξ = i resp. ξ is a primitive third (or sixth) root of unity, up to a conformal linear map, i.e. multiplication by a non–zero complex number. 5.10. Proof of 5.9. With notations as in 5.9 note that for an arbitrary subset S of G the group F := λ(Γ) of O(T 𝔼) is generated by λ_{b} ∪ λ_{c} and hence F ∩ SO(T 𝔼) is generated by the set λ_{b} ∪ λ_{c}λ_{c}^{−1} mentioned in condition A. Let T be the F –module generated by the set T_{a} ∪ T_{b} ∪ T_{3} of condition B. We embed F into G by taking as center of the rotations a point of 𝔼_{b} ∪ 𝔼_{c,c}, this set is not empty, otherwise F is the trivial group. Then Γ ⊂ F ⋉T, so conditions A and B are sufficient. Conversely, if Γ is crystallographic then every element of F ∩ SO is of order 1, 2, 3, 4, or 6 and hence the set of rotations λ_{b} ∪ λ_{c}λ_{c}^{−1} generating this group cannot contain one of order 4 and of order 3 or 6. So A is necessary. That B is necessary follows from proposition 3.14.

The statements claimed under case 1 are seen as follows. That the ℤ–module T_{3,ℤ} generated by T_{3} is the same as the ℤ–module generated by its subset T_{b,c} ∪ T_{(c,b),(c,b)} is immediate. Let Θ be the ℤF –module generated by T_{3,ℤ}. Then Θ is commensurable with p_{T 𝕃}Θ ⊕ p_{(T 𝕃)^{⊥}}Θ = T_{1,ℤ} ⊕ T_{2,ℤ}, since the reflection in T 𝕃 is in F. If we are then in case Ai, i = 2, 4 or 6, the equivalence of B with B, Ai follows readily.

Example 5.11. We are now ready to give the counterexample promised after 3.13, namely that the set ofperp(𝔼_{m}, 𝔼_{n}), m, n in M_{Γ} , need not be discrete. Let 𝔼_{1}, 𝔼_{2}, 𝔼_{3} be three lines in the Euclidean plane 𝔼 any two of which intersect at an angle of π/3 and 𝔼_{1} ∩ 𝔼_{2} ∩ 𝔼_{3} = Ø, thus forming an equilateral triangle. The group Γ generated by the reflections σ_{i} := σ_{𝔼_{i}}, i = 1, 2, 3, is crystallographic, by 5.9, Case 1, B, A 6 for σ_{1}, σ_{1}σ_{2}, σ_{1}σ_{3} . It is in fact the affine Weyl group Ã_{2} . Let now 𝔼_{m_{k}} be the following sequence of iterated feet of perpendiculars, where m_{k} is defined inductively by m_{0} = (2, 3), m_{2k−1} = (1, m_{2k−2}), m_{2,k} = (3, m_{2k−1}) for k ≥ 1. Then the sequence of points 𝔼_{m_{k}} converges to the point 𝔼_{(1,3)} .

5.12. dim 𝔼 = 3.

Let us fix notations. We choose an orientation on V = T 𝔼. Every element g of SO(V) fixes some line 𝕃 and induces a rotation on the plane 𝕃^{⊥}. The line 𝕃 is unique unless g = 1. The sign of the rotation angle on 𝕃^{⊥} depends on the orientation of 𝕃^{⊥}. Every element of O(V) of determinant −1 has a line 𝕃 as −1−eigenspace and induces a rotation of 𝕃^{⊥}. The line 𝕃 is unique unless g = −1. Again the sign of the rotation angle depends on the orientation of 𝕃^{⊥}.

We have the following classification of elements g of G according to the dimension of the +1−eigenspace of λ(g), i.e. the dimension of the axis of g.

a) dim (axis) = 3: λ(g) = 1. Then g is a translation by a vector t ∈ V . c2) dim (axis) = 2: λ(g) has eigenvalues {+1, +1, −1}. Then g induces a

translation t on a unique plane 𝔽 in 𝔼. We have t ∈ T 𝔽 and g = σ_{𝔽} ∘ t = t ∘ σ_{𝔽}, where σ_{𝔽} is the orthogonal reflection in the plane 𝔽. So g is a glide reflection.

b) dim (axis) = 1: λ(g) ∈ SO(V), λ(g) ≠ 1. Then g induces a translation t on a unique line 𝕃. Then t ∈ T 𝕃 and

g(x + v) = x + t + λ(g)v

for x ∈ 𝕃 and v ∈ T 𝕃^{⊥}. We denote g = Screw(𝕃, t, φ) if φ is the rotation angle where we orient 𝕃^{⊥} so that (t, e_{1}, e_{2}) is an oriented basis of V if (e_{1}, e_{2}) is an oriented basis of 𝕃^{⊥}. This makes sense unless t = 0 in which case we write g = Screw(𝕃, 0, ±φ). This ambiguity of φ causes no problems since we are only interested in groups containing g. So in case b) g is a screw motion along the line 𝕃 for t ≠ 0 or a rotation around the line 𝕃.

c0) dim (axis)=0: the number +1 is not an eigenvalue of λ(g). Then det λ(g) = −1, g has a unique fixed point, say ℙ, and

g(ℙ + v) = ℙ + Uv for v ∈ V

where U = λ(g) ∈ O(V). We denote g = D(ℙ, U). So we can think of g as a linear orthogonal map U if we take ℙ as the origin of a linear coordinate system.

Let now S be a finite subset of G and let Γ = ⟨S⟩. Again, we discuss the cases according to the dimension of invariant subspaces. Fixed point

The point ℙ of 𝔼 is a common fixed point for Γ iff S contains only elements with translational part zero and axis containing ℙ. If this is the case we can think of ℙ as the origin of the vector space V = 𝔼 and Γ ⊂ O(V). And Γ is discrete iff Γ is finite.

Invariant line

Let 𝕃 be a line in 𝔼. The subgroup G_{𝕃} = {g ∈ G; g𝕃 = 𝕃} is isomorphic to Iso(𝕃)×O((T 𝕃)^{⊥}) via g ↦ (g | 𝕃 , λ(g) | (T 𝕃)^{⊥}). Choosing coordinates we can thus identify G_{𝕃} with G_{1} × O(2), where G_{1} is the group of (affine) isometries of the Euclidean line ℝ. The group of connected components of G_{1} × O(2) can be identified with {±1} × {±1} via Det: G_{1} × O(2) → {±1} × {±1}, Det(g, A) = (λ(g), det A).

Now suppose S is a finite subset of G_{1} × O(2) then it is easy to find a set S_{0} of generators for the intersection Γ_{0} of Γ = ⟨S⟩ with the connected component ℝ × SO(2) of G_{1} × O(2). A wasteful but short way to write down such set is

S_{0} = {γ_{1}γ_{2}γ_{3}^{−1}; γ_{i} ∈ S_{δ_{i},ε_{i}} , δ_{3} = δ_{1}δ_{2} , ε_{3} = ε_{1}ε_{2}}

where S_{δ,ε} = {γ ∈ S ∪ {e}; Det(γ) = (δ, ε)}.

Finally, let e: ℝ ⊕ ℝ → ℝ × SO(2), e(t, φ) = (t, e^{iφ}), be the exponential map of ℝ × SO(2). Then Γ is discrete iff rank_{ℚ} e^{−1}(S_{0}) ≤ 2 Γ is finite iff rank_{ℚ} e^{−1}(S_{0}) ≤ 1.

The proof follows immediately from proposition 4.4.

Let us make explicit what it means for an element g ∈ G in the classification above to leave the line 𝕃 invariant.

a) A translation t leaves 𝕃 invariant iff t ∈ T 𝕃.

b) A screw motion or rotation g leaves 𝕃 invariant iff b1) g has axis 𝕃, or b2) g is a rotation by the angle π around an axis 𝕃′, which intersects 𝕃

orthogonally. c0) An orthogonal transformation g = D(ℙ, U) with unique fixed point ℙ leaves

𝕃 invariant iff ℙ ∈ 𝕃 and T 𝕃 is contained in the −1−eigenspace of U. c2) A glide reflection g = t ∘ σ_{𝔽} leaves 𝕃 invariant iff

c2.1 𝕃 ⊂ 𝔽 and t ∈ T 𝕃, or

c2.2 t = 0 and 𝔽 intersects 𝕃 orthogonally We summarize the information in the following table.

λ(g) | T 𝕃 det U eigenvalues of U = λ(g) | (T 𝕃)^{⊥} translational part t ∈ T 𝕃 dim(axis)

{axis \atop 𝔸}

type

a  1  1  {1, 1}  3  𝔼 translation

b1 1 1

e^{±iφ} ≠ 1

1

𝕃

screw motion or rotation around 𝕃

b2

−1  −1  {+1, −1}

0 1 *) reflection inline 𝔸

c0 −1 1

e^{±iφ} ≠ 1

0 0

∈ 𝕃

“linear” orthogonal map

c2.1

1  −1  {+1, −1}

2

⊃ 𝕃 glide reflection

c2.2 −1 1

{+1,+1}

0 2 *)

reflection in plane 𝔸

*) 𝔸 ∩ 𝕃 ≠ Ø, 𝔸 ⊥ 𝕃

Invariant plane Let 𝔽 be a plane in 𝔼. Then the group G_{𝔽} = {g ∈ G; g𝔽 = 𝔽} is isomorphic to Iso(𝔽) × O(T 𝔽)^{⊥}) via g ↦ (g | 𝔽, λ(g) | T 𝔽). So a subgroup Γ of G_{𝔽} is discrete iff r(Γ) is discrete, where r : G_{𝔽} → Iso(𝔽) is the restriction homomorphism. We are thus reduced to the 2–dimensional case.

Γ crystallographic The whole space 𝔼 is the only Γ–invariant affine subspace if none of the cases above occurs, i.e. if there is no invariant point, line or plane. Then Γ is crystallographic if Γ is discrete. The following well known facts are crucial.

5.13. If Γ is crystallographic then every element of λ(Γ) has order 1, 2, 3, 4 or 6.

The proof is very similar to the 2-dimensional case. One eigenvalue of λ(γ) is det λ(γ) ∈ {±1},the other two are e^{±iφ}. We have trace λ(γ) = 2 cos φ ± 1 ∈ ℤ, hence φ ∈ ℝ/2πℤ has order p, where p is one of the numbers of the claim. So the order of λ(γ) is p, if det λ(γ) = +1, and the least common multiple of 2 and p if det λ(γ) = −1.

5.14. A finite subgroup of SO(3) is either contained in a dihedral group or an octahedral, a tetrahedral or an icosahedral group. Here the latter are the groups of elements of SO(3) preserving a regular octahedron, tetrahedron or icosahedron, respectively. The corresponding groups are isomorphic to S_{4} , A_{4} and A_{5} , respectively. These two facts viz. 5.13 and 5.14, together leave only very few cases for λ(Γ) if Γ is crystallographic. In particular, the icosahedral group is impossible. One can hence discuss the various cases according to the possible groups λ(Γ). We will not go into the details.

Remark 5.15. For hyperbolic groups of motions the following result is due to Jørgensen [9]. A non–elementary group Γ of motions of the hyperbolic space of dimension at most 3 is discrete if every two generator subgroup is discrete. This is not true for subgroups Γ of the group G of affine isometries of a Euclidean space 𝔼. This is already false for subgroups Θ of the group V of translations of 𝔼. E.g. if dim V = 2 and Θ is the ℤ–module in V generated by (e_{1}, e_{2}, α_{1}e_{1} + α_{2}e_{2}) where (e_{1}, e_{2}) is a basis of V and α_{1}, α_{2} are real numbers such that (1, α_{1}, α_{2}) are linearly independent over ℚ, then Θ is not discrete but every two generator subgroup of Θ is discrete since any two elements of Θ are either linearly dependent over ℚ or linearly independent over ℝ.

The question when two elements of PSL(2, ℝ) generate a discrete group is difficult, but it has been answered, see [8].

References

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[2] Bieberbach, L., Über die Bewegungsgruppen des n–dimensionalen euclidischen Raumes mit einem endlichen Fundamentalbereich, Nachr. Königl. Ges. Wiss. Göttingen. Math. phys. Klasse (1910), 75–84.

[3] —, Über die Bewegungsgruppen der Euklidischen Räume (Erste Abhandlung), Math. Ann. 70 (1911), 297–336.

[4] —, Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung). Die Gruppen mit einem endlichen Fundamentalbereich, Math. Ann. 72 (1912), 400–412.

[5] Brown, H., R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, “Crystallographic groups of four-dimensional space”, Wiley, New York, 1978.

[6] Dress, A., A combinatorial theory of Grünbaum’s new regular polyhedra. Part II: complete enumeration, Aequationes Math. 29 (1985), 222–243.

[7] Frobenius, G., Über die unzerlegbaren diskreten Bewegungsgruppen, Sitzber. Preuß. Akad. Wiss. Berlin (Ges. Abh. Nr. 89, Band III 507–518) (1911), 654–665.

[8] Gilman, J., “Two–generator discrete subgroups of PSL(2,R)”, Mem. Amer. Math. Soc. 117 (1995).

[9] Jørgensen, T., A note on subgroups of SL(2,C), Quart. J. Math. Oxford Ser. (2) 28 (1977), 209–211.

[10] Vinberg, E. B., and O. V. Shvartsman, “Discrete groups of motions of spaces of constant curvature”, Encyclopedia of Mathematical Sciences 29, Geometry II, Springer-Verlag 1993, English translation of Itogi nauki i tekhniki, Sovremennye problemy matematiki, Fundamental’nye napravleniya, 29, Geometriya 2, Publisher VINITI, Moscow 1988.

[11] Wolf, J. A., “Spaces of constant curvature”, Publish or Perish Press, Fourth ed. 1977.

Fakultät für Mathematik Universität Bielefeld Universitätsstrasse 25 D-33501 Bielefeld, Germany abels@mathematik.uni-bielefeld.de

Received February 29, 1999 and in final form August 13, 1998