  The aim of this paper is to give a generalization of the theory equivariant
functions, initiated in [17, 4], to arbitrary subgroups of PSL2(R). We show
that there is a deep relation between the geometry of these groups and some
analytic and algebraic properties of these functions. As an application, we
give a new proof of the classification of automorphic forms for non discrete
groups. Also, we prove the following automorphy condition: If $f$ is an
automorphic form for a Fuchsian group of the first kind $\Gamma$, then $f$ has
infinitely many non $\Gamma$-equivalent critical points.
