There are three scale factors so that after radial projection on the unit hyperboloid, we get a billiard in twodimensional 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:These angles are captured in the matrix of scalar products,
which reads explicitly Recall from the previous section that the scalar product of two linear forms and is, in a threedimensional 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^{6},
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 asThe 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 [75, 116, 107]).
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 (nonvanishing) 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:
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 [75, 116, 107].

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