Hyperbolic Weyl groups and gravity

Hyperbolic Weyl groups appear as symmetries of many gravitational systems when these systems are studied in extreme limits near space-like singularities. After reviewing the origin of this appearance of arithmetic structures in gravity, the hyperbolic reflection groups will be reinterpreted as modular groups of type similar to $PSL(2,Z)$ but over other integer structures in algebras of higher dimension than the real numbers. This can be used to reformulate the fundamental equation of quantum gravity in this limit in terms of automorphic forms. Some possible physical implications will be discussed.