In this section, we develop the theory of Coxeter groups with a particular emphasis on the hyperbolic case. The importance of Coxeter groups for the BKL analysis stems from the fact that in the case of the gravitational theories that have been studied most (pure gravity, supergravities), the group generated by the reflections in the billiard walls is a Coxeter group. This follows, in turn, from the regularity of the corresponding billiards, whose walls intersect at angles that are integer submultiples of .

3.1 Preliminary example: The BKL billiard (vacuum gravity)

3.1.1 Billiard reflections

3.1.2 On the group

3.2 Coxeter groups – The general theory

3.2.1 Examples

The dihedral group

The infinite dihedral group

3.2.2 Definition

Another example:

The isomorphism problem

3.2.3 The length function

3.2.4 Geometric realization

3.2.5 Positive and negative roots

3.2.6 Fundamental domain

3.3 Finite Coxeter groups

3.4 Affine Coxeter groups

3.5 Lorentzian and hyperbolic Coxeter groups

3.6 Crystallographic Coxeter groups

On the normalization of roots and weights in the crystallographic case

3.1.1 Billiard reflections

3.1.2 On the group

3.2 Coxeter groups – The general theory

3.2.1 Examples

The dihedral group

The infinite dihedral group

3.2.2 Definition

Another example:

The isomorphism problem

3.2.3 The length function

3.2.4 Geometric realization

3.2.5 Positive and negative roots

3.2.6 Fundamental domain

3.3 Finite Coxeter groups

3.4 Affine Coxeter groups

3.5 Lorentzian and hyperbolic Coxeter groups

3.6 Crystallographic Coxeter groups

On the normalization of roots and weights in the crystallographic case

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