This electronic edition of the 1980 notes from Princeton University, distributed by Princeton University, covers the geometry and topology of three-manifolds. The text, typeset in TeX by Sheila Newbery, includes figures scanned and corrected for typos. The notes are based on a course taught in 1978-79, with contributions from Bill Floyd, Steve Kerckhoff, and John Milnor. The content aims to describe the strong connection between geometry and low-dimensional topology, particularly in the context of 3-manifolds and Kleinian groups, making it accessible to graduate students and mathematicians. The introduction highlights the intricate relationship between topology and geometry in dimensions up to 3, emphasizing the potential for systematic understanding of all three-manifolds. The notes cover various topics, including elliptic and hyperbolic geometry, geometric structures on manifolds, hyperbolic Dehn surgery, flexibility and rigidity of geometric structures, Gromov's invariant, Kleinian groups, and orbifolds. Key sections include: - **Chapter 1**: Introduction to geometry and three-manifolds. - **Chapter 2**: Elliptic and hyperbolic geometry models. - **Chapter 3**: Geometric structures on manifolds, including hyperbolic manifolds and Dehn surgery. - **Chapter 4**: Hyperbolic Dehn surgery and its applications. - **Chapter 5**: Flexibility and rigidity of geometric structures. - **Chapter 6**: Gromov's invariant and the volume of hyperbolic manifolds. - **Chapter 7**: Computation of volume. - **Chapter 8**: Kleinian groups and their properties. - **Chapter 9**: Limits of discrete groups and ergodicity. - **Chapter 11**: Deforming Kleinian manifolds by homeomorphisms. - **Chapter 13**: Orbifolds and their applications. The notes also provide examples and exercises to illustrate the concepts, such as the figure-eight knot complement and the construction of three-manifolds through various topological surgeries and identifications.This electronic edition of the 1980 notes from Princeton University, distributed by Princeton University, covers the geometry and topology of three-manifolds. The text, typeset in TeX by Sheila Newbery, includes figures scanned and corrected for typos. The notes are based on a course taught in 1978-79, with contributions from Bill Floyd, Steve Kerckhoff, and John Milnor. The content aims to describe the strong connection between geometry and low-dimensional topology, particularly in the context of 3-manifolds and Kleinian groups, making it accessible to graduate students and mathematicians. The introduction highlights the intricate relationship between topology and geometry in dimensions up to 3, emphasizing the potential for systematic understanding of all three-manifolds. The notes cover various topics, including elliptic and hyperbolic geometry, geometric structures on manifolds, hyperbolic Dehn surgery, flexibility and rigidity of geometric structures, Gromov's invariant, Kleinian groups, and orbifolds. Key sections include: - **Chapter 1**: Introduction to geometry and three-manifolds. - **Chapter 2**: Elliptic and hyperbolic geometry models. - **Chapter 3**: Geometric structures on manifolds, including hyperbolic manifolds and Dehn surgery. - **Chapter 4**: Hyperbolic Dehn surgery and its applications. - **Chapter 5**: Flexibility and rigidity of geometric structures. - **Chapter 6**: Gromov's invariant and the volume of hyperbolic manifolds. - **Chapter 7**: Computation of volume. - **Chapter 8**: Kleinian groups and their properties. - **Chapter 9**: Limits of discrete groups and ergodicity. - **Chapter 11**: Deforming Kleinian manifolds by homeomorphisms. - **Chapter 13**: Orbifolds and their applications. The notes also provide examples and exercises to illustrate the concepts, such as the figure-eight knot complement and the construction of three-manifolds through various topological surgeries and identifications.