Oneofthemainthemesofthisbookistheconflictbetweenthe"flexibility'andthe"rigiditypropertiesofthehyperbolicmanifolds:thefirstradicaldifferencearisesbetweenthecaseofdimension2andthecaseofhigherdimensions(asprovedinchaptersBandC),anelementaryfeatureofthusphenomenonbeingthedifferencebetweentheRiemannmappingtheoremandLiouville'stheorem,aspointedoutinchapterA.Thuschapterisratherclementaryandmostofitsmaterialmay'betheobjectofanundergraduatecourse.
Togetherwiththerigiditytheorem,abasictoolforthestudyofhyperbolicmanifoldsisMargulis'lemma,adetailedproofofwhichwegiveinchapterD;asaconsequenceofthisresultinthesamechapterwealsogivearatheraccuratedescription,inalldimensions,ofthethin-thickdecompositionofahyperbolicmanifold(especiallyincaseoffinitevolume).