Erlangen Program at Large-1: Geometry of Invariants

This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of $SL_2(\mathbb R)$ group. We describe here geometries of corresponding domains. The principal rôle is played by Clifford algebras of matching types. In this paper we also generalise the Fillmore–Springer–Cnops construction which describes cycles as points in the extended space. This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach.