Establishing Equivariant Class [O] for Hyperbolic Groups

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