Constructing Geometrically Equivalent Hyperbolic Orbifolds

TL;DR

This paper introduces a method to construct nonisometric hyperbolic orbifolds sharing identical classes of totally geodesic subspaces, extending Sunada's technique to handle multiple codimensions.

Contribution

It develops a novel variant of Sunada's method to produce hyperbolic orbifolds with geometrically equivalent subspaces, a significant advancement in geometric topology.

Findings

Constructed explicit families of hyperbolic orbifolds with identical geodesic subspace classes.

Extended Sunada's method to handle multiple codimensions simultaneously.

Demonstrated the existence of nonisometric orbifolds with the same geometric substructure.

Abstract

In this paper, we construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multiple codimensions simultaneously.