### 3.1 Preliminary example: The BKL billiard (vacuum gravity)

To illustrate the regularity of the gravitational billiards and motivate the mathematical developments through an explicit example, we first compute in detail the billiard characterizing vacuum, gravity. Since this corresponds to the case originally considered by BKL, we call it the “BKL billiard”. We show in detail that the billiard reflections in this case are governed by the “extended modular group” , which, as we shall see, is isomorphic to the hyperbolic Coxeter group .

#### 3.1.1 Billiard reflections

There are three scale factors so that after radial projection on the unit hyperboloid, we get a billiard in two-dimensional hyperbolic space. The billiard region is defined by the following relevant wall inequalities,

(symmetry walls) and
(curvature wall). The remarkable properties of this region from our point of view are:
• It is a triangle (i.e., a simplex in two dimensions) because even though we had to begin with 6 walls (3 symmetry walls and 3 curvature walls), only 3 of them are relevant.
• The walls intersect at angles that are integer submultiples of , i.e., of the form
where is an integer. The symmetry walls intersect indeed at sixty degrees () since the scalar product of the corresponding linear forms (of norm squared equal to ) is , while the gravitational wall makes angles of zero (, scalar product ) and ninety (, scalar product ) degrees with the symmetry walls.

These angles are captured in the matrix of scalar products,

Recall from the previous section that the scalar product of two linear forms and is, in a three-dimensional scale factor space,
where we have taken , and . The corresponding billiard region is drawn in Figure 1.

Because the angles between the reflecting planes are integer submultiples of , the reflections in the walls bounding the billiard region,

obey the following relations,
The product is a rotation by and hence squares to one; the product is a rotation by and hence its cube is equal to one. There is no power of the product that is equal to one, something that one conventionally writes as

The group generated by the reflections , and is denoted , for reasons that will become clear in the following, and coincides with the arithmetic group , as we will now show (see also [75116107]).

#### 3.1.2 On the group

The group is defined as the group of matrices with integer entries and determinant equal to , with the identification of and ,

Note that although elements of the real general linear group have (non-vanishing) unrestricted determinants, the discrete subgroup only allows for in order for the inverse to also be an element of .

There are two interesting realisations of in terms of transformations in two dimensions:

• One can view as the group of fractional transformations of the complex plane
with
Note that one gets the same transformation if is replaced by , as one should. It is an easy exercise to verify that the action of when defined in this way maps the complex upper half-plane,
onto itself whenever the determinant of is equal to . This is not the case, however, when .
• For this reason, it is convenient to consider alternatively the following action of ,
(), which does map the complex upper-half plane onto itself, i.e., which is such that whenever .

The transformation (3.14) is the composition of the identity with the transformation (3.11) when , and of the complex conjugation transformation, with the transformation (3.11) when . Because the coefficients , , , and are real, commutes with and furthermore the map (3.11) (3.14) is a group isomorphism, so that we can indeed either view the group as the group of fractional transformations (3.11), or as the group of transformations (3.14).

An important subgroup of the group is the group for which , also called the “modular group”. The translation and the inversion are examples of modular transformations,

It is a classical result that any modular transformation can be written as the product
but the representation is not unique [4].

Let , and be the -transformations

to which there correspond the matrices
The ’s are reflections in the straight lines , and the unit circle , respectively. These are in fact just the transformations of hyperbolic space , and described in Section 3.1.1, since the reflection lines intersect at , and degrees, respectively.

One easily verifies that and that . Since any transformation of not in can be written as a transformation of times, say, and since any transformation of can be written as a product of ’s and ’s, it follows that the group generated by the 3 reflections , and coincides with , as announced above. (Strictly speaking, could be a quotient of that group by some invariant subgroup, but one may verify that the kernel of the homomorphism is trivial (see Section 3.2.5 below).) The fundamental domains for and are drawn in Figure 2. The equivalence between and the Coxeter group has been discussed previously in [75116107].