Talk:Volume of a closed hyperbolic 3-manifold (Ex)
From Manifold Atlas
Let be a hyperbolic closed -manifold. Let be another hyperbolic closed -manifold with .
Since and are both hyperbolic, they are covered by . Therefore, for , so both and are spaces. Then there is a homotopy equivalence . The Mostow Rigidity theorem says that is homotopic to an isometry. Therefore, and have the same volume.