THE MATHEMATICAL PAPERS
CONTENTS CHAPTER 1 GEOMETRY AND THREE-MANIFOLDS CHAPTER 2 ELLIPTIC AND HYPERBOLIC GEOMETRY 2 1 THE POINCAR´E DISK MODEL 2 2 THE SOUTHERN HEMISPHERE 2 3 THE UPPER HALF-SPACE MODEL 2 4 THE PROJECTIVE MODEL 2 5 THE SPHERE OF IMAGINARY RADIUS 2 6 TRIGONOMETRY CHAPTER 3 GEOMETRIC STRUCTURES ON MANIFOLDS 3 1 A HYPERBOLIC STRUCTURE ON THE ?GURE-EIGHT KNOT COMPLEMENT 3 2 A HYPERBOLIC MANIFOLD WITH GEODESIC BOUNDARY 3 3 THE WHITEHEAD LINK COMPLEMENT 3 4 THE BORROMEAN RINGS COMPLEMENT 3 5 THE DEVELOPING MAP 3 8 HOROSPHERES 3 9 HYPERBOLIC SURFACES OBTAINED FROM IDEAL TRIANGLES 3 10 HYPERBOLIC MANIFOLDS OBTAINED BY GLUING IDEAL POLYHEDRA CHAPTER 4 HYPERBOLIC DEHN SURGERY 4 2 GLUING CONSISTENCY CONDITIONS 4 3 HYPERBOLIC STRUCTURE ON THE ?GURE-EIGHT KNOT COMPLEMENT 4 4 THE COMPLETION OF HYPERBOLIC 3-MANIFOLDS OBTAINED FROM IDEAL POLYHEDRA 4 5 THE GENERALIZED DEHN SURGERY INVARIANT 4 6 DEHN SURGERY ON THE ?GURE-EIGHT KNOT 4 8 DEGENERATION OF HYPERBOLIC STRUCTURES 4 10 INCOMPRESSIBLE SURFACES IN THE ?GURE-EIGHT KNOT COMPLEMENT CHAPTER 5 FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES 5 1 DEFORMATIONS OF GEOMETRIC STRUCTURE 5 2 A CRUDE DIMENSION COUNT 5 3 TEICHM¨ULLER SPACE 5 5 THE DIMENSION OF THE DEFORMATION SPACE OF A HYPERBOLIC THREE-MANIFOLD 5 7 MOSTOW’S THEOREM 5 8 GENERALIZED DEHN SURGERY AND HYPERBOLIC STRUCTURES 5 9 A PROOF OF MOSTOW’S THEOREM 5 10 A DECOMPOSITION OF COMPLETE HYPERBOLIC MANIFOLDS 5 11 COMPLETE HYPERBOLIC MANIFOLDS WITH BOUNDED VOLUME 5 12 JØRGENSEN’S THEOREM CHAPTER 6 GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD 6 1 GROMOV’S INVARIANT 6 3 GROMOV’S PROOF OF MOSTOW’S THEOREM 6 4 STRICT VERSION OF GROMOV’S THEOREM 6 5 MANIFOLDS WITH BOUNDARY 6 6 ORDINALS 6 7 COMMENSURABILITY 6 8 SOME EXAMPLES CHAPTER 7 COMPUTATION OF VOLUME 7 1 THE LOBACHEVSKY FUNCTION L(?) 7 2 VOLUMES OF SOME POLYHEDRA 7 3 SOME MANIFOLDS 7 4 ARITHMETIC EXAMPLES REFERENCES CHAPTER 8 KLEINIAN GROUPS 8 1 THE LIMIT SET 8 2 THE DOMAIN OF DISCONTINUITY 8 3 CONVEX HYPERBOLIC MANIFOLDS 8 4 GEOMETRICALLY ?NITE GROUPS 8 5 THE GEOMETRY OF THE BOUNDARY OF THE CONVEX HULL 8 6 MEASURING LAMINATIONS 8 7 QUASI-FUCHSIAN GROUPS 8 8 UNCRUMPLED SURFACES 8 9 THE STRUCTURE OF GEODESIC LAMINATIONS: TRAIN TRACKS 8 10 REALIZING LAMINATIONS IN THREE-MANIFOLDS 8 11 THE STRUCTURE OF CUSPS 8 12 HARMONIC FUNCTIONS AND ERGODICITY CHAPTER 9 ALGEBRAIC CONVERGENCE 9 1 LIMITS OF DISCRETE GROUPS 9 2 GEOMETRIC TAMENESS 9 3 THE ENDING OF AN END 9 4 TAMING THE TOPOLOGY OF AN END 9 5 INTERPOLATING NEGATIVELY CURVED SURFACES 9 6 STRONG CONVERGENCE FROM ALGEBRAIC CONVERGENCE 9 7 REALIZATIONS OF GEODESIC LAMINATIONS FOR SURFACE GROUPS WITH EXTRA CUSPS, WITH A DIGRESSION ON STEREOGRAPHIC COORDINATES 9 9 ERGODICITY OF THE GEODESIC ?OW