Focussing on the geometry of hyperbolic manifolds, the aim here is to provide an exposition of some fundamental results, while being as self-contained, complete, detailed and unified as possible. Following some classical material on the hyperbolic space and the Teichmüller space, the book centers on the two fundamental results: Mostow's rigidity theorem (including a complete proof, following Gromov and Thurston) and Margulis' lemma. These then form the basis for studying Chabauty and geometric topology; a unified exposition is given of Wang's theorem and the Jorgensen-Thurston theory; and much space is devoted to the 3D case: a complete and elementary proof of the hyperbolic surgery theorem, based on the representation of three manifolds as glued ideal tetrahedra.

Inhalt

A. Hyperbolic Space.- B. Hyperbolic Manifolds and the Compact Two-dimensional Case.- C. The Rigidity Theorem (Compact Case).- D. Margulis' Lemma and its Applications.- E. The Space of Hyperbolic Manifolds and the Volume Function.- F. Bounded Cohomology, a Rough Outline.- Notation Index.- References.