We assume, as already stated, that the scalar product is nondegenerate. Let be the basis dual to the basis in the scalar product ,

The ’s are called “fundamental weights”. (The fundamental weights are really defined by Equation (3.38) up to normalization, as we will see in Section 3.6 on crystallographic Coxeter groups. They thus differ from the solutions of Equation (3.38) only by a positive multiplicative factor, irrelevant for the present discussion.)Consider the vector , where the vector is such that and . This vector exists since we assume the Coxeter group to be of indefinite type. Let be the hyperplane orthogonal to . Because , the vectors ’s all lie on the positive side of , . By contrast, the vectors ’s all lie on the negative side of since . Furthermore, has negative norm squared, . Thus, in the case of Coxeter groups of indefinite type (with a nondegenerate metric), one can choose a hyperplane such that the positive roots lie on one side of it and the fundamental weights on the other side. The converse is true for Coxeter group of finite type: In that case, there exists such that is positive, implying that the positive roots and the fundamental weights are on the same side of the hyperplane .

We now consider a particular subclass of Coxeter groups of indefinite type, called Lorentzian Coxeter groups. These are Coxeter groups such that the scalar product is of Lorentzian signature . They are discrete subgroups of the orthochronous Lorentz group preserving the time orientation. Since the are spacelike, the reflection hyperplanes are timelike and thus the generating reflections preserve the time orientation. The hyperplane from the previous paragraph is spacelike. In this section, we shall adopt Lorentzian coordinates so that has equation and we shall choose the time orientation so that the positive roots have a negative time component. The fundamental weights have then a positive time component. This choice is purely conventional and is made here for convenience. Depending on the circumstances, the other time orientation might be more useful and will sometimes be adopted later (see for instance Section 4.8).

Turn now to the cone defined by Equation (3.35). This cone is clearly given by

Similarly, its closure is given by The cone is thus the convex hull of the vectors , which are on the boundary of .By definition, a hyperbolic Coxeter group is a Lorentzian Coxeter group such that the vectors in are all timelike, for all . Hyperbolic Coxeter groups are precisely the groups that emerge in the gravitational billiards of physical interest. The hyperbolicity condition forces for all , and in particular, : The fundamental weights are timelike or null. The cone then lies within the light cone. This does not occur for generic (non-hyperbolic) Lorentzian algebras.

The following theorem enables one to decide whether a Coxeter group is hyperbolic by mere inspection of its Coxeter graph.

Theorem: Let be a Coxeter group with irreducible Coxeter graph . The Coxeter group is hyperbolic if and only if the following two conditions hold:

- The bilinear form is nondegenerate but not positive definite.
- For each , the Coxeter graph obtained by removing the node from is of finite or affine type.

(Note: By removing a node, one might get a non-irreducible diagram even if the original diagram is connected. A reducible diagram defines a Coxeter group of finite type if and only if each irreducible component is of finite type, and a Coxeter group of affine type if and only if each irreducible component is of finite or affine type with at least one component of affine type.)

Proof:

- It is clear that if a Coxeter group is hyperbolic, then its bilinear form fulfills the first condition. Let be one of the vectors of the dual basis. The vectors with form a basis of the hyperplane orthogonal to . Because is non-spacelike (the group is hyperbolic), the hyperplane is spacelike or null. The Coxeter graph defined by the with (i.e., by removing the node ) is thus of finite or affine type.
- Conversely, assume that the two conditions of the theorem hold. From the first condition, it follows that the set is non-empty. Let be the hyperplane spanned by the with , i.e., orthogonal to . From the second condition, it follows that the intersection of with each is empty. Accordingly, each connected component of lies in one of the connected components of the complement of , namely, is on a definite (positive or negative) side of each of the hyperplanes . These sets are of the form with for some ’s (fixed throughout the set) and for the others. This forces the signature of to be Lorentzian since otherwise there would be at least a two-dimensional subspace of such that . Because is connected, it must lie in one of the subsets just described. But this is impossible since if , then .

We now show that . Because the signature of is Lorentzian, is the inside of the standard light cone and has two components, the “future” component and the “past” component. From the second condition of the theorem, each lies on or inside the light cone since the orthogonal hyperplane is non-timelike. Furthermore, all the ’s are future pointing, which implies that the cone lies in , as had to be shown (a positive sum of future pointing non spacelike vectors is non-spacelike). This concludes the proof of the theorem.

In particular, this theorem is useful for determining all hyperbolic Coxeter groups once one knows the list of all finite and affine ones. To illustrate its power, consider the Coxeter diagram of Figure 8, with 8 nodes on the loop and one extra node attached to it (we shall see later that it is called ).

The bilinear form is given by

and is of Lorentzian signature. If one removes the node labelled 9, one gets the affine diagram (see Figure 9). If one removes the node labelled 8, one gets the finite diagram of the direct product group (see Figure 10). Deleting the nodes labelled 1 or 7 yields the finite diagram of (see Figure 11). Removing the nodes labelled 2 or 6 gives the finite diagram of (see Figure 12). If one removes the nodes labelled 3 or 5, one obtains the finite diagram of (see Figure 13). Finally, deleting the node labelled 4 yields the affine diagram of (see Figure 14). Hence, the Coxeter group is hyperbolic.Consider now the same diagram, with one more node in the loop (). In that case, if one removes one of the middle nodes 4 or 5, one gets the Coxeter group , which is neither finite nor affine. Hence, is not hyperbolic.

Using the two conditions in the theorem, one can in fact provide the list of all irreducible hyperbolic Coxeter groups. The striking fact about this classification is that hyperbolic Coxeter groups exist only in ranks , and, moreover, for there is only a finite number. In the case, on the other hand, there exists an infinite class of hyperbolic Coxeter groups. In Figure 15 we give a general form of the Coxeter graphs corresponding to all rank 3 hyperbolic Coxeter groups, and in Tables 3 – 9 we give the complete classification for .

Note that the inverse metric , which gives the scalar products of the fundamental weights, has only negative entries in the hyperbolic case since the scalar product of two future-pointing non-spacelike vectors is strictly negative (it is zero only when the vectors are both null and parallel, which does not occur here).

One can also show [116, 107] that in the hyperbolic case, the Tits cone coincides with the future light cone. (In fact, it coincides with either the future light cone or the past light cone. We assume that the time orientation in has been chosen as in the proof of the theorem, so that the Tits cone coincides with the future light cone.) This is at the origin of an interesting connection with discrete reflection groups in hyperbolic space (which justifies the terminology). One may realize hyperbolic space as the upper sheet of the hyperboloid in . Since the Coxeter group is a subgroup of , it leaves this sheet invariant and defines a group of reflections in . The fundamental reflections are reflections through the hyperplanes in hyperbolic space obtained by taking the intersection of the Minkowskian hyperplanes with hyperbolic space. These hyperplanes bound the fundamental region, which is the domain to the positive side of each of these hyperplanes. The fundamental region is a simplex with vertices , where are the intersection points of the lines with hyperbolic space. This intersection is at infinity in hyperbolic space if is lightlike. The fundamental region has finite volume but is compact only if the are timelike.

Thus, we see that the hyperbolic Coxeter groups are the reflection groups in hyperbolic space with a fundamental domain which (i) is a simplex, and which (ii) has finite volume. The fact that the fundamental domain is a simplex ( vectors in ) follows from our geometric construction where it is assumed that the vectors form a basis of .

The group relevant to pure gravity in four dimensions is easily verified to be hyperbolic.

For general information, we point out the following facts:

- Compact hyperbolic Coxeter groups (i.e., hyperbolic Coxeter groups with a compact fundamental region) exist only for ranks 3, 4 and 5, i.e., in two, three and four-dimensional hyperbolic space. All hyperbolic Coxeter groups of rank have a fundamental region with at least one vertex at infinity. The hyperbolic Coxeter groups appearing in gravitational theories are always of the noncompact type.
- There exist reflection groups in hyperbolic space whose fundamental domains are not simplices. Amazingly enough, these exist only in hyperbolic spaces of dimension . If one imposes that the fundamental domain be compact, these exist only in hyperbolic spaces of dimension . The bound can probably be improved [164].
- Non-hyperbolic Lorentzian Coxeter groups are associated through the above construction with infinite-volume fundamental regions since some of the vectors are spacelike, which imply that the corresponding reflection hyperplanes intersect beyond hyperbolic infinity.

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