Equivariant functions for the Möbius subgroups and applications

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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 Γ, then f has infinitely many non Γ-equivalent critical points.