##
**Foundations of hyperbolic manifolds.
2nd ed.**
*(English)*
Zbl 1106.51009

Graduate Texts in Mathematics 149. New York, NY: Springer (ISBN 0-387-33197-2). xii, 779 p. (2006).

Designed to be useful as both textbook and a reference, this book renders a real service to the mathematical community by putting together the tools and prerequisites needed to enter the territory of Thurston’s formidable theory of hyperbolic 3-manifolds, an area that used to be accessible before the publication of the first edition of this book only after a much more thorough and lengthy preparation. Although the author’s stated prerequisites of “a basic knowledge of algebra and topology at the first-year graduate level of an American university” should be supplemented by a very solid background in integration, preferably on manifolds, no use is being made of either algebraic topology or differential geometry.

The first four chapters deal with \(n\)-dimensional Euclidean (\(E^n\)), spherical (\(S^n\)), hyperbolic (\(H^n\)), and inversive geometry. They require a firm grounding in linear algebra and \(n\)-dimensional analysis (preferably some analysis on manifolds). For each of Euclidean, spherical, and hyperbolic geometry, the author determines the arclength, the geodesics, the element of volume, the corresponding trigonometries (laws of sines, cosines, etc.). To emphasize the similarity between spherical and hyperbolic geometry, the latter is first introduced by means of the hyperboloid model inside Lorentzian \(n\)-space. The conformal ball (\(B^n\)) and the upper half-space (\(U^n\)) models of hyperbolic geometry are introduced in the chapter on inversive geometry, which also contains characterization theorems for elliptic transformations of \(B^n\) and for parabolic and hyperbolic transformations of \(U^n\).

Chapters 5–9 are devoted to the pre-Thurston part of the story. Chapter 5 is on discrete subgroups of both the group of isometries of \(E^n\) and of the group \(M(B^n)\) of Möbius transformations of \(B^n\). Chapter 6, on the geometry of discrete groups, introduces the projective disk model of \(n\)-dimensional hyperbolic geometry, convex sets for \(E^n, S^n, H^n\), emphasizing polyhedra and polytopes, studies fundamental domains, convex fundamental polyhedra, and tessellations. Chapter 7, on classical discrete groups, studies reflection groups, simplex reflection groups for \(E^n, S^n, H^n\), generalized simplex reflection groups for \(H^n\), proves that the volume of an \(n\)-simplex in \(S^n\) or \(H^n\) is an analytic function of the dihedral angles of that simplex, the Schläfli differential formula, as well as a study of the theory of crystallographic groups, culminating with a proof of Bieberbach’s theorem. Chapter 8 contains the basic notions and theorems of geometric manifols, including Clifford-Klein space-forms, \((X, G)\)-manifolds, geodesic completeness. Chapter 9, on geometric surfaces, deals with gluing surfaces, the Gauss-Bonnet theorem for surfaces of constant curvature, moduli spaces, Teichmüller space, the Dehn-Nielsen theorem, closed Euclidean and hyperbolic surfaces, hyperbolic surfaces of finite area.

Chapters 10–13, the heart of the introduction to the results of Thurston, Gromov, and the author, are, as expected, more heavy going than the previous ones. Chapter 10 is on hyperbolic 3-manifolds, containing results on gluing of 3-manifolds, some examples of finite volume hyperbolic 3-manifolds (the Whitehead link, the Borromean rings complement), the computation of the volume of a compact orthotetrahedron and of an ideal tetrahedron, hyperbolic Dehn surgery. Chapter 11, on hyperbolic \(n\)-manifolds, deals with gluing, Poincaré’s fundamental polyhedron theorem, the Gauss-Bonnet theorem for the special case of closed spherical, Euclidean, or hyperbolic \(n\)-manifolds of constant sectional curvature, the characterization of simplices of maximal volume in \(B^n\), differential forms, the Gromov norm (with Gromov’s theorem), measure homology, the de Rham chain complex, and the Mostow rigidity theorem. Chapter 12, on geometrically finite \(n\)-manifolds, deals with limit sets of discrete groups (including classical Schottky groups), the basic properties of conical and cusped limit points of a discrete group of Möbius transformations of \(B^n\), the characterization, in terms of their convex fundamental polyhedra, of the discrete subgroups of \(M(B^n)\) that have the property that every limit point is either conical or cusped, the study of nilpotent subgroups of the group \(I(H^n)\) of \(n\)-dimensional hyperbolic isometries, the Margulis lemma, used to prove the existence of Margulis regions for discrete subgroups of \(I(H^n)\), and geometrically finite hyperbolic manifolds. Chapter 13 studies the geometry of geometric orbifolds.

Every chapter is followed by historical notes, with attributions to the relevant literature, both of the originators of the ideas presented in the chapter and of modern presentations thereof. The bibliography contains 463 entries.

The first four chapters deal with \(n\)-dimensional Euclidean (\(E^n\)), spherical (\(S^n\)), hyperbolic (\(H^n\)), and inversive geometry. They require a firm grounding in linear algebra and \(n\)-dimensional analysis (preferably some analysis on manifolds). For each of Euclidean, spherical, and hyperbolic geometry, the author determines the arclength, the geodesics, the element of volume, the corresponding trigonometries (laws of sines, cosines, etc.). To emphasize the similarity between spherical and hyperbolic geometry, the latter is first introduced by means of the hyperboloid model inside Lorentzian \(n\)-space. The conformal ball (\(B^n\)) and the upper half-space (\(U^n\)) models of hyperbolic geometry are introduced in the chapter on inversive geometry, which also contains characterization theorems for elliptic transformations of \(B^n\) and for parabolic and hyperbolic transformations of \(U^n\).

Chapters 5–9 are devoted to the pre-Thurston part of the story. Chapter 5 is on discrete subgroups of both the group of isometries of \(E^n\) and of the group \(M(B^n)\) of Möbius transformations of \(B^n\). Chapter 6, on the geometry of discrete groups, introduces the projective disk model of \(n\)-dimensional hyperbolic geometry, convex sets for \(E^n, S^n, H^n\), emphasizing polyhedra and polytopes, studies fundamental domains, convex fundamental polyhedra, and tessellations. Chapter 7, on classical discrete groups, studies reflection groups, simplex reflection groups for \(E^n, S^n, H^n\), generalized simplex reflection groups for \(H^n\), proves that the volume of an \(n\)-simplex in \(S^n\) or \(H^n\) is an analytic function of the dihedral angles of that simplex, the Schläfli differential formula, as well as a study of the theory of crystallographic groups, culminating with a proof of Bieberbach’s theorem. Chapter 8 contains the basic notions and theorems of geometric manifols, including Clifford-Klein space-forms, \((X, G)\)-manifolds, geodesic completeness. Chapter 9, on geometric surfaces, deals with gluing surfaces, the Gauss-Bonnet theorem for surfaces of constant curvature, moduli spaces, Teichmüller space, the Dehn-Nielsen theorem, closed Euclidean and hyperbolic surfaces, hyperbolic surfaces of finite area.

Chapters 10–13, the heart of the introduction to the results of Thurston, Gromov, and the author, are, as expected, more heavy going than the previous ones. Chapter 10 is on hyperbolic 3-manifolds, containing results on gluing of 3-manifolds, some examples of finite volume hyperbolic 3-manifolds (the Whitehead link, the Borromean rings complement), the computation of the volume of a compact orthotetrahedron and of an ideal tetrahedron, hyperbolic Dehn surgery. Chapter 11, on hyperbolic \(n\)-manifolds, deals with gluing, Poincaré’s fundamental polyhedron theorem, the Gauss-Bonnet theorem for the special case of closed spherical, Euclidean, or hyperbolic \(n\)-manifolds of constant sectional curvature, the characterization of simplices of maximal volume in \(B^n\), differential forms, the Gromov norm (with Gromov’s theorem), measure homology, the de Rham chain complex, and the Mostow rigidity theorem. Chapter 12, on geometrically finite \(n\)-manifolds, deals with limit sets of discrete groups (including classical Schottky groups), the basic properties of conical and cusped limit points of a discrete group of Möbius transformations of \(B^n\), the characterization, in terms of their convex fundamental polyhedra, of the discrete subgroups of \(M(B^n)\) that have the property that every limit point is either conical or cusped, the study of nilpotent subgroups of the group \(I(H^n)\) of \(n\)-dimensional hyperbolic isometries, the Margulis lemma, used to prove the existence of Margulis regions for discrete subgroups of \(I(H^n)\), and geometrically finite hyperbolic manifolds. Chapter 13 studies the geometry of geometric orbifolds.

Every chapter is followed by historical notes, with attributions to the relevant literature, both of the originators of the ideas presented in the chapter and of modern presentations thereof. The bibliography contains 463 entries.

Reviewer: Victor V. Pambuccian (Phoenix)

### MSC:

51M10 | Hyperbolic and elliptic geometries (general) and generalizations |

57M50 | General geometric structures on low-dimensional manifolds |

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

57M60 | Group actions on manifolds and cell complexes in low dimensions |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

51-02 | Research exposition (monographs, survey articles) pertaining to geometry |