*(English)*Zbl 1025.57001

After the Geometrization Theorem of Thurston see for example *W. Thurston* [The Geometry and Topology of 3-Manifolds, Mimeographed Notes, Princeton Univ. Press, Princeton, N. J. (1979)] and *W. Thurston* [Three-Dimensional Geometry and Topology, Princeton Univ. Press, Princeton, N. J. (1997; Zbl 0873.57001)], one of the main subject of 3-manifold topology is the classification of hyperbolic 3-dimensional manifolds and orbifolds (and Kleinian groups). An orientable hyperbolic 3-manifold (of finite volume) is isometric to a quotient of the hyperbolic 3-space over a torsion-free Kleinian group (of finite covolume). Missing the torsion-free condition yields the concept of hyperbolic 3-orbifold. A compact orientable 3-manifold is called hyperbolizable if its interior admits a complete hyperbolic structure. Examples are given by atoroidal Haken 3-manifolds whose fundamental groups do not contain abelian subgroups of finite index. It is known that “most” links in the standard 3-sphere have complements that admit complete hyperbolic structures. For a non-trivial prime knot $K$, the complement ${\mathbb{S}}^{3}\setminus K$ is hyperbolic with finite volume if and only if $K$ is not a torus knot or a satellite knot. The study of hyperbolic 3-manifolds and orbifolds is based on techniques from different areas of mathematics, namely topology, differential geometry, analysis and group theory. There are a lot of texts which cover these aspects of the theory of hyperbolic 3-manifolds (see for example the references of the book under review and the texts listed below). But the particular feature of the present book is the study of the arithmetic aspects of hyperbolic 3-manifolds, and their connections with algebra and number theory. For this, the starting point is to observe that the matrix entries of the elements of $\text{SL}(2,\u2102)$, representing a finite-covolume Kleinian group, can be taken to lie in a field which is a finite extension of the rationals. This follows from the Mostow Rigidity Theorem and permits to establish many connections between the geometry of arithmetic Kleinian groups and number theory.

This beautiful book, which is a fundamental work on the arithmetic theory of hyperbolic 3-manifolds, is devoted to readers who already know the basic concepts and results on hyperbolic geometry. However, it can also be used as an introduction to the theory of (arithmetic) Kleinian and Fuchsian groups. The main subject is the study and the determination of the invariant number field and the invariant quaternion algebra associated to a Kleinian group of finite covolume. These arithmetic objects are invariant with respect to the commensurability class of the group. We now examine in detail the various chapters of the book.

Chapter 0 is a reference chapter which contains terminology and background information on algebraic number theory. There are few proofs in this chapter, and the reader is assumed to be familiar with standard results on field extensions and Galois theory. Chapters 2 to 5 are devoted to the constructions of the invariant number field and the invariant quaternion algebra associated to any finite-covolume Kleinian group. Here we recall briefly these constructions. Let ${\Gamma}$ be a non-elementary subgroup of $\text{PSL}(2,\u2102)$. Let $\widehat{{\Gamma}}={P}^{-1}\left({\Gamma}\right)$, where $P:\text{SL}(2,\u2102)\to \text{PSL}(2,\u2102)$ is the canonical map. Then the trace field of ${\Gamma}$, denoted $\mathbb{Q}\left(\text{tr}{\Gamma}\right)$, is the field $\mathbb{Q}(\text{tr}\widehat{\gamma}:\widehat{\gamma}\in \widehat{{\Gamma}})$. For a finite-covolume Kleinian group ${\Gamma}$, its trace field is a number field, that is, it is a finite extension of the rationals. If $M={\mathbb{H}}^{3}/{\Gamma}$ is a hyperbolic 3-manifold of finite volume, then the trace field $\mathbb{Q}\left(\text{tr}{\Gamma}\right)$ is a topological invariant of $M$. Several methods and techniques for calculating this invariant for many significant classes of hyperbolic 3-manifolds are described in these chapters. If ${\Gamma}$ is a non-elementary subgroup of $\text{SL}(2,\u2102)$, then the set ${A}_{0}{\Gamma}$ of all the sums ${\sum}_{i}{a}_{i}{\gamma}_{i}$ (where only finitely many of the ${a}_{i}\in \mathbb{Q}\left(\text{tr}{\Gamma}\right)$ are non-zero, and ${\gamma}_{i}\in {\Gamma}$) is a quaternion algebra over $\mathbb{Q}\left(\text{tr}{\Gamma}\right)$. The trace field $\mathbb{Q}\left(\text{tr}{\Gamma}\right)$ is an invariant of the Kleinian group ${\Gamma}$, but, in general, it is not an invariant of the commensurability class of ${\Gamma}$ in $\text{PSL}(2,\u2102)$. We can get over this problem by considering the subgroup ${{\Gamma}}^{\left(2\right)}$ formed by the squares of the elements in ${\Gamma}$. Indeed, if ${\Gamma}$ is a finitely generated non-elementary subgroup of $\text{SL}(2,\u2102)$, then the trace field $\mathbb{Q}\left(\text{tr}{{\Gamma}}^{\left(2\right)}\right)$ and the quaternion algebra ${A}_{0}{{\Gamma}}^{\left(2\right)}$ are invariants of the commensurability class of ${\Gamma}$. They are called the invariant number field and the invariant quaternion algebra of ${\Gamma}$, respectively. If ${\Gamma}$ is generated by the elements ${\gamma}_{1}$,..., ${\gamma}_{n}$, then $\mathbb{Q}\left(\text{tr}{{\Gamma}}^{\left(2\right)}\right)$ is generated over $\mathbb{Q}$ by the traces of a small collection of elements. This permits to obtain another description of the invariant trace field in terms of traces. It turns out that this description is applicable to methods of characterizing arithmetic Kleinian and Fuchsian groups. Generators for invariant quaternion algebras are also described at the end of Chapter 3.

The invariant trace fields and quaternion algebras of a number of classical examples of hyperbolic 3-manifolds (as, for example, complements of hyperbolic knots or links, hyperbolic fibre bundles, once-punctured torus bundles, Fibonacci manifolds, the Weeks-Matveev-Fomenko manifold etc.) and Kleinian groups (as, for example, Bianchi groups, Coxeter groups, Polyhedral groups, triangle groups, etc.) are determined in Chapter 4. Finally, many geometric applications of the above-mentioned algebraic and arithmetic invariants are discussed in these chapters.

From Chapter 6 onward, the book treats deeply the theory of arithmetic Kleinian groups and its applications to geometry and topology of (arithmetic) hyperbolic 3-manifolds and orbifolds. The definition of an arithmetic Kleinian (resp. Fuchsian) group is based on the quaternion algebra. This facilitates the interplay between number theory and geometry. The most important question is to identify, among all Kleinian (resp. Fuchsian) groups, those that are arithmetic. This is solved in Chapter 8 and the result is known as the identification theorem. As a consequence of this theorem, the number field and the quaternion algebra used to define the arithmetic structure of a Kleinian group coincide with the invariant trace field and the invariant quaternion algebra as defined in Chapter 3. This gives techniques to determine whether or not the group is arithmetic (see Chapter 8). Geometric applications on arithmetic hyperbolic 3-manifolds and orbifolds can be found in Chapter 9.

In Chapter 10 there is a treatment of arithmetic Kleinian groups via quadratic forms. In Chapter 11 the authors present volume formulas for arithmetic Kleinian (resp. Fuchsian) groups. In Chapter 12 one can find a discussion on the structure and properties of the set of closed geodesics, particularly, in arithmetic hyperbolic 2- and 3-orbifolds. The book ends with an appendix of computations.

This is a book of great importance on the theory of hyperbolic manifolds (and Kleinian groups) since it is the first to provide a complete, precise, clearly-written and self-contained exposition of the arithmetic aspects of the theory. For this, the book fills a void in the mathematics literature concerning hyperbolic geometry. The authors are two of the most fine mathematicians in the subject, and have made fundamental and beautiful contributions to the material included. An extensive literature on the subject is listed and quoted in the references of the book under review.

Finally, the reviewer highly recommends this beautiful book to anyone interested in the arithmetic aspects of the theory of hyperbolic 3-manifolds and orbifolds (and Kleinian groups), as well as in the interplay between number theory and geometry. Here we list only some texts (without pretence of completeness) that can accomplish usefully the reading of the book under review.

(1) *J. Anderson*, Hyperbolic Geometry. Springer-Verlag, Berlin-Heidelberg-New York (1999; Zbl 0934.51012);

(2) *A. Beardon*, The Geometry of Discrete Groups. Graduate Texts in Mathematics 91, Springer-Verlag, Berlin-Heidelberg-New York (1983; Zbl 0528.30001);

(3) *R. Benedetti* and *C. Petronio*, Lectures on Hyperbolic Geometry. Springer-Verlag, Berlin-Heidelberg-New York (1992; Zbl 0768.51018);

(4) *D. Cooper, C. D. Hodgson* and *S. P. Kerckhoff*, Three-dimensional orbifolds and cone-manifolds. Math. Soc. Japan Memoirs 5, Tokyo (2000; Zbl 0955.57014);

(5) *J. Elstrodt, F. Grunewald* and *J. Mennicke*, Groups acting on hyperbolic space Monographs in mathematics. Springer-Verlag, Berlin-Heidelberg-New York (1998; Zbl 0888.11001);

(6) *W. Fenchel*, Elementary geometry in hyperbolic space. Walter de Gruyter, Berlin-New York (1989; Zbl 0674.51001);

(7) *K. Matsuzaki* and *M. Taniguchi*, Hyperbolic manifolds and Kleinian groups. Oxford University Press, Oxford (1998; Zbl 0892.30035);

(8) *J. M. Montesinos*, Classical tessellations and three-manifolds. Universitext Springer-Verlag, Berlin-Heidelberg-New York (1987; Zbl 0626.57002);

(9) *J. Ratcliffe*, Foundations of hyperbolic manifolds. Graduate Texts in Mathematics 149, Springer-Verlag, Berlin-Heidelberg-New York (1994; Zbl 0809.51001);

(10) *W. Thurston*, The geometry and topology of 3-manifolds Mimeographed notes, Princeton Univ. Press, Princeton, N. J. (1979);

(11) *W. Thurston*, Three-dimensional geometry and topology. Princeton Univ. Press, Princeton, N. J. (1997; Zbl 0873.57001).

##### MSC:

57-01 | Textbooks (manifolds) |

51H20 | Topological geometries on manifolds |

20H10 | Fuchsian groups and their generalizations (group theory) |

30F99 | Riemann surfaces (one complex variable) |

57M50 | Geometric structures on low-dimensional manifolds |

57M10 | Covering spaces (manifolds) |