  This paper lays down a foundation for a systematic treatment of three main
(elliptic, parabolic and hyperbolic) types of analytic function theory 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.
  Keywords: analytic function theory, semisimple groups, elliptic, parabolic,
hyperbolic, Clifford algebras
