Abstract
A compact, orientable 3-manifoldMis calledhyperbolicif intMadmits a complete hyperbolic structure (Riemannian metric of constant curvature − 1) of finite volume. Any hyperbolic 3-manifoldMis irreducible, and each component of∂Mis an incompressible torus. Letf:M→Nbe a proper, continuous map between hyperbolic 3-manifolds. By Mostow's Rigidity Theorem [8], iffis π1-isomorphic thenfis properly homotopic to a diffeomorphismg:M→Nsuch thatg| intM: intM→ intNis isometric. In particular, the topological type of intMdetermines uniquely the hyperbolic structure on intMup to isometry, so the volume vol (intM) of intMis well-defined. This Rigidity Theorem is generalized by Thurston[11, theorem 6·4] so that any proper, continuous mapf:M→Nbetween hyperbolic 3-manifolds with vol(intM) = deg(f) vol(intN) is properly homotopic to a deg(f)-fold coveringg:M→Nsuch thatg| intM: intM→ intNis locally isometric.
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