Establishing Equivariant Class [O] for Hyperbolic Groups

This paper aims to create a class [O] concerning the groups associated with Gromov hyperbolic over correspondence and equivalence through Fuchsian, Kleinian, Schottky when subject Laplace – Beltrami in Teichmüller space where for 3-manifold fundamental of Dehn extended any occurrence Švarc-Milnor lemma satisfies same quotient Jørgensen inequality. Thus relation (and class) Mostow Prasad Rigidity Theorem finite degree isometry structure commensurator higher order generalizations suffice CAT(k) space. The map established is shown at end paper.