For 2-3 decades many paper's on automorphism of Riemann surfaces have referred to "Singerman's list" of pairs of finitely maximal Fuchsian groups. The list  has been especially useful in determining when a given groups  of automorphisms  is the full automorphism.  of a surface.  Singerman's list may also be interpreted as the list of Fuchsian group pairs H < G where  H and G have the same Teichmuller dimension.

In this talk we discuss the extension of  the classification to pairs of Fuchsian group pairs H < G where the Teichmuller codimension of H exceed that of G by a small integer d(H,G) called the Teichmuller codimension. The main result  is that for each fixed codimension there is a finite number of exceptional pairs and a finite number of families. This neatly generalizes Singerman's result.  Applications of the classification to extending actions of groups  and the analysis of the branch locus of moduli space will be discussed.