  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,R) group. We describe here
geometries of corresponding domains. The principal role 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. For an easy-reading
introduction see arXiv:math/0607387. An outline of the whole approach is given
in arXiv:1006.2115.
