This chapter begins with basic facts about complete hyperbolic manifolds and their geometric limits. We then give a proof of rigidity for manifolds whose injectivity radius is bounded above. Mostow rigidity for closed manifolds is a special case; the more general result will be used in the construction of hyperbolic manifolds which fiber over the circle.

The proof of rigidity combines geometric limits with the Lebesgue density theorem and the a.e. differentiability of quasiconformal mappings. This well-known argument is carried further in §2.4 to show certain open hyperbolic manifolds, while not rigid, are*inflexible*— any deformation is asymptotically isometric