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Fundamental Domains in Lorentzian Geometry

We consider discrete subgroups Gamma of the simply connected Lie group SU~(1,1), the universal cover of SU(1,1), of finite level, i.e. the subgroup intersects the centre of SU~(1,1) in a subgroup of finite index, this index is called the level of the group. The Killing form induces a Lorentzian metric of constant curvature on the Lie group SU~(1,1). The discrete subgroup Gamma acts on SU~(1,1) by left translations. We describe the Lorentz space form SU~(1,1)/Gamma by constructing a fundamental domain F for Gamma. We want F to be a polyhedron with totally geodesic faces. We construct such F for all Gamma satisfying the following condition: The image of Gamma in PSU(1,1) has a fixed point u in the unit disk of order larger than the index of Gamma. The construction depends on the group Gamma and on the orbit Gamma(u) of the fixed point u.


Introduction
We consider the universal cover of PSU(1, 1) ∼ = PSL(2, R), the group of orientation-preserving isometries of the hyperbolic plane. Here our model of the hyperbolic plane is the unit disk D in C.
The level of a discrete subgroup Γ ⊂ SU(1, 1) is the index of Γ ∩ Z as a subgroup of Z. There is a one-to-one correspondence between discrete subgroups of level k in SU(1, 1) and liftings of discrete subgroups in PSU(1, 1) into the k-fold covering of PSU(1, 1).
We consider a discrete subgroup Γ in SU(1, 1) of finite level k. We suppose that the image Γ of Γ in PSU(1, 1) has at least one fixed point in D of order p, i.e. a point in D, which is fixed by a nontrivial element of Γ of order p. Furthermore we assume that the order p of the fixed point is larger then the level k of the subgroup Γ. Our construction depends on the choice of the fixed point u ∈ D of Γ, or actually on its orbit Γ(u).
The Killing form on the Lie group SU(1, 1) gives rise to a Lorentz biinvariant metric of constant curvature. The quotient of SU(1, 1) by the discrete subgroup Γ is a Lorentz space form with respect to this metric, i.e. a complete Lorentz manifold of constant curvature (compare R.S. Kulkarni and F. Raymond [KR85]).
The main result of this paper is the construction of fundamental domains for the action of Γ on SU(1, 1) by left translations, applicable to any discrete subgroup Γ in SU(1, 1) as above. This fundamental domain is a polyhedron in the Lorentz manifold SU(1, 1) with totally geodesic faces. For a co-compact subgroup the corresponding fundamental domain is compact. The precise formulation of this result is contained in Theorems A and B.
The construction of fundamental domain was a part of author's Ph.D. thesis [Pra01]. It was outlined in the survey article [BPR03]. The object of this paper is to provide complete proofs for this result together with a concise and self-contained description of the construction.
Our results generalize a construction by Th. Fischer [Fis92]. He suggested how to construct a fundamental domain for the action of a discrete subgroup of PSU(1, 1) by left multiplication. His construction can be interpreted in our terms as a construction for discrete subgroup of SU(1, 1) of level 1. A less technical proof of Th. Fischer's result is given in [BKN + 98].
The study of discrete subgroups of finite level was originally motivated by some deep connections between these subgroups and quasi-homogeneous isolated singularities of complex surfaces studied by J. Milnor, I. Dolgachev and W. Neumann [Mil75,Dol83,Neu77,Neu83]. In particular the quotient SU(1, 1)/Γ is diffeomorphic to the link of some quasi-homogeneous Gorenstein singularity. For a more detailed treatment of this connection see [BPR03], §1-2.
The paper is organized as follows: We start in section 2 by discussion of lowdimensional analogues of our problem and use these examples to indicate some of the main ingredients of our construction. Section 3 contains some general remarks on the Lie groups SU(1, 1) and SU(1, 1) and their embeddings in some 4dimensional pseudo-Euclidean space resp. in a certain R + -bundle, the universal cover of a positive cone in that pseudo-Riemannian space. We describe in section 4 some elements of the construction, such as affine half-spaces and their substitutes in the R + -bundle. We also define prismatic sets Q u , certain finite intersections of half-spaces, and study their properties.
After that we are prepared to state in section 5 our main results, Theorems A and B, and to prove them. In section 6 we report on our explicit computations of fundamental domains for certain infinite series of discrete subgroups. The choice of this series is motivated by the connections between them and some series of quasi-homogeneous surface singularities. We also give some pictures of fundamental domains. Finally, in section 7 we discuss some relations to and similarities with other fundamental domain constructions and give an outlook on possible generalizations.
The results described in section 6 have been announced in [BPR03] together with computations of fundamental domains for other singularities by E. Brieskorn and F. Rothenhäusler. The synopsis of these results points at a very regular pattern for these series of singularities and their fundamental domains.
I am indebted to Egbert Brieskorn and Thomas Fischer who opened up this field of research. I am very grateful to Egbert Brieskorn for his guidance and helpful discussions. I would like to thank Ludwig Balke, Werner Ballmann, Ilya Dogolazky, Pierre Pansu, and Frank Rothenhäuser for useful conversations related to this work. I thank Ilya Dogolazky for his help in producing the figures.

Low-Dimensional Analogues
Before we start to describe the construction for SU(1, 1), we discuss in this section the corresponding problem of finding fundamental domains for an action of a discrete subgroup for two toy cases, for the one-dimensional Lie groups SO(2) and SO(1, 1). Some of the main ideas of the construction for SU(1, 1) can be seen already in the discussion of these low-dimensional examples.
We first consider the Lie group SO(2). We identify SO(2) with the circle in the Euclidean plane E 2 . We consider a discrete subgroup Γ m , the finite cyclic subgroup of order m generated by z(2π/m), and its action on SO(2) by left multiplication, which extends to an action on E 2 by isometries.
Clearly, the segment of length 2π/m with midpoint z(0) is a fundamental domain for this action, this is the Dirichlet domain with respect to the point z(0). However, for the description of Dirichlet domains we need the distance. The following description of the same fundamental domain as projection of an affine construction with tangent half-planes is more appropriate for generalizations in pseudo-Riemannian setting.
For g ∈ G let H g = {a ∈ E 2 a, g 1} be the half-plane with boundary tangent to the circle G in the point g. Then the intersection of the tangent halfplanes P = g∈Γm H g is a regular Γ m -invariant m-gon, its faces are fundamental domains for the action of Γ m on the boundary ∂P , and the projection of the faces under the contraction a → a/|a| yields to a tiling of the circle by (Dirichlet) fundamental domains with respect to Γ m . Figure 1 illustrates the construction for m = 6.
A more involved example, where the construction of Dirichlet fundamental domains can be described in the same way, is the case of the action of discrete subgroups on SU(2), in particular the construction of the 4-dimensional regular polyhedron bounded by 120 dodecahedra, the tiling of the 3-dimensional sphere by 120 spherical dodecahedra and the resulting construction of the Poincaré homology sphere using the binary icosahedral group. Here we identify SU(2) with the 3-dimensional sphere in the Euclidean space E 4 .
Our second description of the fundamental domain in SO(2) does not use any distance. It only uses the embedding of SO(2) in the (pseudo-)Euclidean space. However, the simple-minded attempt to generalize this affine construction to pseudo-Riemannian quadrics fails. Our second one-dimensional example, the hyperbola in  We now consider the Lie group SO(1, 1) and identify it with the hyperbola Let us fix d > 0 and consider a discrete subgroup Γ d , the subgroup generated by the elements z(d, 1) and z(d, −1) and isomorphic to Z × {±1}. Moreover we consider the action of Γ d on SO(1, 1) by left multiplication, which extends to an action on E 1,1 by isometries. Clearly, the segment of length d with midpoint z(0, +1) is a fundamental domain for this action. This is the Dirichlet domain with respect to the point z(0, +1).
Let L be the cone over G, i.e. L = R + · G = {a ∈ E 1,1 a, a < 0}. For g ∈ G let H g be the half-plane H g = {a ∈ E 1,1 a, g −1} with boundary tangent to the hyperbola G in the point g. The polytope P = ∩ g∈Γ d H g contains in this case only one point (0, 0), so we can not get any information about the tiling from this set. Instead we consider the set and the part of its boundary contained in L. This construction suggests some important ingredients of the construction for SU(1, 1), namely the embedding of SU(1, 1) as a quadric in a 4-dimensional pseudo-Euclidean space, appropriate decomposition of the discrete subgroup in countable  many finite subsets T (x), x ∈ X, and finally the study of the 4-dimensional polyhedron Some new ideas come in when we generalize the fundamental domain construction for SU(1, 1). We consider an embedding of SU(1, 1) as an image of a section in a (trivial) R + -bundle over SU(1, 1), namely in the universal cover of the positive cone R + · SU(1, 1), and we define appropriate substitutes for tangent spaces and half-spaces there.
We also want to point out that these two one-dimensional examples are not only examples, we meet them again in the construction for SU(1, 1). We identify SU(1, 1) with a quadric in a 4-dimensional pseudo-Euclidean space. The subgroups SO(2) ∼ = SU(1) and SO(1, 1) of SU(1, 1) can be identified with certain plane sections of this quadric, and the corresponding constructions of fundamental domains are then sections of the construction for SU(1, 1).
We may think of L as a R + -bundle over G with radial projection s : L → G as bundle map. The map L → D defined by (z, w) → z/w is principal C * -bundle, where the action of λ ∈ C * is defined by λ · (z, w) = (λ −1 z, λ −1 w). Let π :G → G be the universal covering. Henceforth we identify the Lie group SU(1, 1) with G via w z zw → (z,w), and SU(1, 1) withG. The biinvariant metrics on G andG are proportional to the Killing forms. We denote the pull-backL →G of the R + -bundle s : L → G under the covering map π :G → G also by s. The following diagram commutes G resp.G is canonically embedded in L resp.L and therefore there exist canonical trivializations L ∼ = G × R + resp.L ∼ =G × R + . The coveringL inherits canonically a pseudo-Riemannian metric from L.
We now give a brief description of the full isometry group ofG (compare sections 2.1-2.3 in [KR85]). The action ofG ×G onG via (g, h) · x = gxh −1 is by Lorentz isometries since the metric is biinvariant. The standard estimates for the dimension of the isometry group show that the identity component Isom 0 (G) is isomorphic to (G ×G)/∆ Z , where ∆ Z = {(z, z) z ∈ Z} and Z is the centre ofG. The full isometry group ofG has four components corresponding to timeand/or space-reversals. Let ε be the geodesic symmetry at the identity given by g → g −1 and η the lift of the conjugation by the matrix ( 0 1 1 0 ) in G fixing the identity. Then ε preserves the space-orientation and reverses the time-orientation, while η reverses both the space-and time-orientation. Moreover, the group Isom + (G) = Isom 0 (G), η is the full group of orientation-preserving isometries and is the full isometry group ofG.

The Elements of the Construction
For g ∈G let E g resp. I g be the connected component of π −1 (Ēḡ) resp. π −1 (Īḡ) containing g, whereḡ := π(g) is the image of g in G, is the intersection of L with the affine tangent space on G in the pointḡ and is the intersection of L with the half-space of C 2 bounded byĒḡ and not containing 0.Ēḡ andĪḡ are simply connected and even contractible, hence their pre-images under the covering map π consist of infinitely many connected components, one of them containing g.
The three-dimensional submanifold E g subdividesL in two connected components, the closure of one of them is I g , and we denote the closure of the other by H g . The boundary of I g resp. H g is equal to E g .
The subsets E g resp. I g have the analogous properties because E g = g · E e and I g = g · I e .
We make use of the following construction (compare [Mil75]). Given a base-point x ∈ D and a real number t, let ρ x (t) ∈ PSU(1, 1) denote the rotation through angle t about the point x. Thus we obtain a homomorphism ρ x : R → PSU(1, 1), which clearly lifts to the unique homomorphism r x : R → SU(1, 1) into the universal covering group. Since ρ x (2π) = Id D , it follows that the lifted element r x (2π) belongs to the central subgroup Z of SU(1, 1). Note that this element r x (2π) ∈ Z depends continuously on x, and therefore is independent of the choice of x. We easily compute r 0 (2t) = (0, −t, 1) and hence r x (2π) = r 0 (2π) = (0, −π, 1) for all x ∈ D. Moreover we obtain r 0 (2t) · (z, α, r) = (ze it , α − t, r).
Let Γ be a discrete subgroup of finite level k in SU(1, 1) and let Γ be the image of Γ in PSU(1, 1). We assume the existence of a fixed point u ∈ D of Γ. The isotropy group Γ u of u in Γ is a finite cyclic group generated by ρ u (2π/p), where p := |Γ u |. The isotropy group Γ u of u in Γ is a infinite cyclic group generated by r d := r u (2ϑ), where ϑ := πk p . We can assume without lost of generality that u = 0 ∈ D. Under this assumption it follows r d = (0, −ϑ, 1) and r d · (z, α, r) = ze iϑ , α − ϑ, r .
An important assumption for the following construction is p > k, the order of u as a fixed point of Γ is greater than the level of the group. In terms of r d this means that the argument ϑ of r d is less then π.

Figure 3:
The image X u of Q u in the (α, r)-half-plane As an example, for x = u we have T (u) = Γ u , the infinite cyclic subgroup generated by the element r d = (0, −ϑ, 1). The generator r d acts onG by left multiplication and it acts on the (α, r)-half-plane by the translation τ mapping (α, r) to (α − ϑ, r). How does the set Q u = ∩ m∈Z H r m d look like? The images of the sets H r m d = r m d · H e under the projection (z, α, r) → (α, r) are the translates τ m (X e ) of the image X e = {(α, r) ∈ R × R + r · cos α 1 or |α| π/2} of H e . The manifold Q u is a disc bundle over its image X u = m∈Z τ m (X e ) in the (α, r)-plane. The shaded area in figure 3 is X u .
The manifolds gQ u play a central role in our construction. We want to explain the geometric nature of these objects. We have described Q u as a disc bundle over the set X u in the (α, r)-half-plane R × R + . We may describe Q u ⊂L ⊂ C × R × R + as Q u = (C × X u ) ∩L.
We think of X u as a universal covering of a punctured plane polygon. Consider the following diagram of covering maps where π(α, r) = re iα and π ′ (α, r) = r 1/k e iα/k and π ′′ (z) = z k . Consider the curve π(∂X u ). It is easy to see that this is a regular star polygon 2p k when k is odd and a regular star polygon p k when k is even. (For the definition of a star polygon see for example H.S.M. Coxeter [Cox69], §2.8, pp. 36-38.) Therefore the curve π ′ (∂X u ) is a curvilinear 2p-gon covering the star polygon once or twice. Let P ′ ⊂ C and P = P u ⊂ C be the plane areas bounded by the curvilinear polygon π ′ (∂X u ) and by the star polygon π(X u ). The images of X u are the punctured plane polygons π ′ (X u ) = P ′ \ {0} and π(X u ) = P \ {0}. We think of the product C × P ′ as a 4-dimensional 2p-gonal prism. C × X u is the universal covering of the pierced prism C × (P ′ \ {0}). The product C × P ⊂ C 2 might be considered as a 4-dimensional "star prism". Its axis C × {0} does not meet L ⊂ C × C * . Therefore the universal covering π :L → L maps Q u to the intersection of L with the star prism: Remark. With the help of the figure 3 we easily prove the estimate r 1 cos ϑ 2 for points (z, α, r) ∈ Q u . From this inequality we shall obtain in lemma 1(i) some useful estimates for points of Q x .
The following properties can be easily proven for x = u from the explicit description of the set Q u . The subsets Q x have the analogous properties because Q x = a · Q u for a ∈ T (x).
Lemma 1. For a point x ∈ Γ(u) the following holds: (ii) The set Q x is a subgraph of a section in the bundleL ∼ =G × R + , while its boundary is the graph of this section. This means that for some section s :G → R + in the bundleL we have Proof. We prove that the family (π(Q x )) x∈Γ(u) is locally finite (in L). This fact implies the local finiteness of the family (Q x ) x∈Γ(u) , since if a subset U of L has an empty intersection with π(Q x ) then the intersection of the pre-image π −1 (U ) with Q x is empty too. By lemma 1(i) for any point x ∈ Γ(u) and any point (z, w) ∈ π(Q x ) the difference |w| − |z| is bounded from above by f (|x|). The values f (t) tend to zero as t tends to 1. Choosing a point (z 0 , w 0 ) ∈ L and a positive number ε < |w 0 | − |z 0 |, the neighbourhood U := {(w, z) ∈ L |w| − |z| > ε} of the point (z 0 , w 0 ) can intersect π(Q x ) only for |x| sufficiently small (so that f (|x|) > ε). But the group Γ is discrete, so there are only finitely many points x in Γ(u) with norm |x| under a given bound. This finishes the proof.
Remark. This property of Q x allows us to deal with P = ∪Q x in a similar way as with a finite union of polytopes.
Lemma 3. The family (E g ∩ Q g(u) ) g∈Γ is locally finite.
Proof. This is immediate from the local finiteness of the family (Q x ) x∈Γ(u) plus the easy observation that the family (E g ∩ Q g(u) ) g∈Γu is locally finite.
We consider inL the four-dimensional polytope Lemma 4. The projection ∂P →G is a Γ-equivariant homeomorphism.
Proof. From lemma 1(ii) we know that the set Q x is a subgraph of a section in the bundleL ∼ =G × R + . The polyhedron P = ∪Q x inherits this property of the prisms Q x as a union of a locally finite family of subgraphs. But for a subgraph of a section in the bundleL it is clear that the bundle mapL →G induces a homeomorphism from its boundary (equal to the graph of the section) ontoG. This homeomorphism is Γ-equivariant since the projectionL →G is Γ-equivariant.

Now we can state the main result
Theorem A. The boundary of P is invariant with respect to the action of Γ. The subset is a fundamental domain for the action of Γ on ∂P . The family (F g ) g∈Γ is locally finite in ∂P . The projectionL →G induces a Γ-equivariant homeomorphism ∂P → G. The image F g of F g under the projection is a fundamental domain for the action of Γ onG. The family (F g ) g∈Γ is locally finite. For every pair of elements g, h ∈ Γ with g = h the intersection F g ∩ F h lies in a totally geodesic submanifold ofG.
Proof. First let us prove that the intersection Int(F g ) ∩ F h is empty if g = h.
Suppose on the contrary that there are elements g, h ∈ Γ such that g = h and Int(F g ) ∩ F h = ∅. Since E g ∩ ∂P is closed, it follows by general topology arguments that Int F g = Int Cl Int(E g ∩ ∂P ) = Int(E g ∩ ∂P ) (in this proof all closures are taken in ∂P ). The assumption Int(F g ) ∩ F h = ∅ can be rewritten as Int(E g ∩ ∂P ) ∩ Cl Int(E h ∩ ∂P ) = ∅. It can be shown in the standard way that if A and B are closed subsets of some topological space such that Int(A) ∩ Cl Int(B) = ∅ then it follows Int(A ∩ B) = ∅. In our case this implies Int(E g ∩ E h ∩ ∂P ) = ∅. But since the totally geodesic submanifolds E g and E h intersect transversally, the intersection E g ∩ E h has no inner points in ∂P . Since F g ⊂ E g ∩ Q g(u) lemma 3 implies that the family (F g ) g∈Γ is locally finite in ∂P . Lemma 4 says that the projection ∂P →G is a Γ-equivariant homeomorphism. Now let us prove that Cl(∪ g∈Γ Int F g ) = ∂P . Since it suffices to prove that ∪ g∈Γ F g = ∂P . Consider a ∈ ∂P . From the definition of P and lemma 3 it follows that there exist elements g 1 , . . . , g n ∈ Γ and a neighbourhood U of the point a inL such that We may assume without loss of generality that the map π| U : U → π(U ) is a homeomorphism. The image of P ∩ U under this homeomorphism is an intersection of an open subset of L with a finite union of finite intersections of half-spaces H g with the property a ∈ ∂H g . Suppose that a ∈ Cl Int(E gi ∩ ∂P ) = F gi for all i ∈ {1, . . . , n}. This is only possible if for each i ∈ {1, . . . , n} the set E gi ∩ ∂P ∩ U is contained in a 2-dimensional submanifold ofL. But from lemma 4 it follows that ∂P is homeomorphic to a 3-dimensional manifoldG. This contradiction implies that a ∈ F g for some g ∈ Γ.
Remark. The family (Q x ) x∈Γ(u) is locally finite, hence it is clear that a point p is in the boundary of P = ∪ x∈Γ(u) Q x if and only if p is not an interior point of Q x for all x ∈ Γ(u) and p is a boundary point of Q x for some x ∈ Γ(u).
Lemma 5. We have Int F e ⊂ ∂Q u .
Proof. Suppose there is a point a ∈ Int F e = Int(E e ∩ ∂P ) such that a ∈ ∂Q u . Since a ∈ ∂P and a ∈ ∂Q u there exists x ∈ Γ(u)\{u} such that a ∈ ∂Q x . Then any neighbourhood of a intersects E e ∩ Int Q x ⊂ E e \∂P . The projection s :L →G is continuous and the restriction s| ∂P : ∂P →G is a homeomorphism, therefore any neighbourhood of a intersects ((s| ∂P ) −1 • s)(E e \∂P )) ⊂ ∂P \E e . This implies a ∈ Int(E e ∩ ∂P ) = Int F e . Contradiction.
Proposition 6. We have Proof. LetF := (E e ∩ ∂Q u ) − (∪ x∈Γ(u)\{u} Int Q x ). It holds F e ⊂ E e by definition, Int F e ⊂ ∂Q u by lemma 5 and F e ∩ Int Q x = ∅ because of F e ⊂ ∂P . This implies Int F e ⊂F and therefore Int F e ⊂ IntF . On the other hand fromF ⊂ E e ∩ ∂P it follows that IntF ⊂ Int(E e ∩ ∂P ) = Int F e . From IntF = Int F e it follows that F e = Cl IntF .
Lemma 7. If Γ is co-compact, then F g is compact.
Proof. Consider a sequence a k in Int F g . Let ϕ be the composition of the projection maps ∂P →G andG →G/Γ. Since the quotientG/Γ is compact we may assume without loss of generality that the sequence ϕ(a k ) tends to a limitā ∈G/Γ. Since ϕ is surjective there exists a pre-image a ∈ ∂P ofā under ϕ. Hence there is a sequence h k in Γ such that the sequence h k a k tends to a. Since the family (F g ) g∈Γ is locally finite there exists a neighbourhood U of a that intersects only finitely many fundamental domains F g , and therefore the set {h k |k ∈ N} is finite. After choosing a subsequence we may assume that the sequence h k is constant, say h k = h. Then the sequence ha k tends to a, hence the sequence a k tends to h −1 a. This implies h −1 a ∈ F g .
Theorem B. If Γ is co-compact then F g is a compact polyhedron, i.e. a finite union of finite compact intersections of half-spaces I a .
Proof. The family (Q x ) x∈Γ(u) is locally finite and the fundamental domain F e is compact by lemma 7. From this it follows that there is a finite subset E ⊂ Γ(u) such that F e ∩ Q x = ∅ for all x ∈ Γ(u)\E. By proposition 5 this implies the assertion.

Examples
We have computed the fundamental domains explicitly for those infinite series of discrete subgroups, which correspond via the construction of I. Dolgachev [Dol83] to the Arnold series E, Z and Q of quasi-homogeneous surface singularities. In particular the quotient of SU(1, 1) by one of this groups is diffeomorphic to the link of the corresponding quasi-homogeneous singularity. Whenever it is convenient, we shall denote these subgroups also by the symbols E n , Z n , Q n .
Some explanations are required to make the figures of fundamental domains comprehensible. The image π(F e ) of the fundamental domain F e for a discrete cocompact group Γ ⊂ SU(1, 1) of finite level is a compact polyhedron in su(1, 1) with flat faces. The Lie algebra su(1, 1) is a 3-dimensional flat Lorentz space of signature (n + , n − ) = (2, 1). Such a polyhedron has a distinguished rotational axis of symmetry. The direction of this axis is negative definite, and the orthogonal complement is positive definite. Changing the sign of the pseudo-metric in the direction of the rotational axis transforms Lorentz space into a well-defined Euclidean space. The image π(F e ) of the fundamental domain is then transformed into a polyhedron in Euclidean space with dihedral symmetry. Figure 4 shows the Euclidean polyhedra obtained in this way. The direction of the rotational axis is vertical. The top and bottom faces are removed.
The polyhedra in figure 4 are all scaled by the same factor to illustrate the proportions between different fundamental domains. Figure 5 illustrates the identification scheme for the cases E 10+2n . The face identification is equivariant with respect to the dihedral symmetry of the polyhedron.

Concluding Remarks
1) The construction of fundamental domains in the flat Lorentz case using crooked planes by T. Drumm and W. Goldman (see [DG95], [   therein) is the only other fundamental domain construction in the non-Riemannian pseudo-Riemannian case we are aware of, besides the construction for SU(1, 1) presented in this paper and the construction for PSU(1, 1) studied before in [Fis92], [KNRS96], [BKN + 98].
2) The idea of projection of an affine construction with half-planes onto a quadric is also used in the algorithmic construction of Voronoi diagrams for (finite) point sets in the Euclidean and hyperbolic plane, compare J.-D. Boissonnat and M. Yvinec [BY98], Part V. 3) R.S. Kulkarni and F. Raymond studied in [KR85] standard Lorentz space forms.
We can think of a discrete subgroup ofG acting by left translations as a discrete subgroup of (G × {e})/∆ Z ⊂ J 0 , so the Lorentz space forms studied in this paper are standard. We would like to generalize our fundamental domain construction for the case of other Lorentz space forms, at least for standard ones.
For instance, our fundamental domain construction can be modified to work for the subgroups of Isom 0 (G) of the form (Γ × Φ)/∆ Z , where Γ is a discrete subgroup ofG and Φ is a discrete subgroup of K. Also the work of I. Dolgachev [Dol83] can be generalized to obtain a correspondence between subgroups of this special form and quasi-homogeneous Q-Gorenstein surface singularities. These constructions will be studied in a forthcoming paper. 4) One can also think of generalizations of the fundamental domain construction in the cases of other (pseudo-)Riemannian quadrics in pseudo-Euclidean spaces. In section 2 we discussed the analogous constructions for two one-dimensional cases. It would be interesting to generalize the constructions for higher dimensional quadrics.