The arithmetic of hyperbolic 3-manifolds / Colin MacLachlan, Alan W. Reid

Auteur: MacLachlan, Colin (....-2012) - AuteurCo-auteur: Reid, Alan W. - AuteurType de document: MonographieCollection: Graduate texts in mathematics ; 219Langue: anglaisPays: Etats UnisÉditeur: New York : Springer-Verlag, 2003Description: 1 vol. (XIII-463 p.) : ill. ; 24 cm ISBN: 9780387983868 ; rel. Note: Over the last 25 years, the Thurstonian revolution in 3-manifold topology has pushed geometric manifolds, particularly hyperbolic manifolds (those admitting a complete Riemannian metric of constant negative sectional curvature), to the front of the stage, largely displacing the Haken manifolds that we naïvely believed were so prevalent. The subtlety of these manifolds has led to the development of many new techniques, none of which (so far) has been capable of dealing completely effectively with the whole class of hyperbolic manifolds. One approach that suggests itself is restricting attention to particularly tractable subclasses of manifolds, in the hope that insights gleaned there may be at least partially applicable to the entire class. One such tractable yet richly varied subclass is that of arithmetic hyperbolic 3-manifolds. These manifolds (and their orbifold cousins) are amenable to the use of techniques from algebra and number theory, as well as the now-standard collection of topological, analytical and geometric tools. Unfortunately, these techniques lie outside the core of knowledge that is generally held by working topologists. This book is aimed at rectifying that problem by exposing readers with knowledge of hyperbolic 3-manifolds and orbifolds to the specific techniques from algebra and number theory needed to effectively study arithmetic manifolds and orbifolds. The book is fairly self-contained, provided one is willing to accept some results for which sketches and references are given instead of full proofs. The authors have struck a balance in which details which do not seem to have been particularly enlightening or useful in the topological context are omitted in favor of including more topological examples. The list of references is quite extensive, but even more useful are the Further Reading sections at the end of most chapters, which comprise a carefully annotated bibliography of the field. The appendices are also quite useful and interesting, including a table of arithmetic Fuchsian triangle groups, a variety of lists of hyperbolic Kleinian groups (tetrahedral groups, knot complements, small-volume closed and cusped manifolds) and an "Arithmetic zoo'' containing a number of examples of arithmetic Kleinian groups which have particularly interesting properties. This book fills a real void in the literature, providing working topologists and graduate students with an accessible introduction to the useful and beautiful world of arithmetic hyperbolic 3-manifolds and orbifolds. (MathSciNet)Bibliographie: Bibliogr. p. [443]-458. Index. Sujets MSC: 57-01 Manifolds and cell complexes -- Instructional exposition (textbooks, tutorial papers, etc.)
51H20 Geometry -- Topological geometry -- Topological geometries on manifolds
20H10 Group theory and generalizations -- Other groups of matrices -- Fuchsian groups and their generalizations
30Fxx Functions of a complex variable -- Riemann surfaces
57M50 Manifolds and cell complexes -- Low-dimensional topology -- Geometric structures on low-dimensional manifolds
57M10 Manifolds and cell complexes -- Low-dimensional topology -- Covering spaces
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Over the last 25 years, the Thurstonian revolution in 3-manifold topology has pushed geometric manifolds, particularly hyperbolic manifolds (those admitting a complete Riemannian metric of constant negative sectional curvature), to the front of the stage, largely displacing the Haken manifolds that we naïvely believed were so prevalent. The subtlety of these manifolds has led to the development of many new techniques, none of which (so far) has been capable of dealing completely effectively with the whole class of hyperbolic manifolds. One approach that suggests itself is restricting attention to particularly tractable subclasses of manifolds, in the hope that insights gleaned there may be at least partially applicable to the entire class.
One such tractable yet richly varied subclass is that of arithmetic hyperbolic 3-manifolds. These manifolds (and their orbifold cousins) are amenable to the use of techniques from algebra and number theory, as well as the now-standard collection of topological, analytical and geometric tools. Unfortunately, these techniques lie outside the core of knowledge that is generally held by working topologists. This book is aimed at rectifying that problem by exposing readers with knowledge of hyperbolic 3-manifolds and orbifolds to the specific techniques from algebra and number theory needed to effectively study arithmetic manifolds and orbifolds.
The book is fairly self-contained, provided one is willing to accept some results for which sketches and references are given instead of full proofs. The authors have struck a balance in which details which do not seem to have been particularly enlightening or useful in the topological context are omitted in favor of including more topological examples.
The list of references is quite extensive, but even more useful are the Further Reading sections at the end of most chapters, which comprise a carefully annotated bibliography of the field. The appendices are also quite useful and interesting, including a table of arithmetic Fuchsian triangle groups, a variety of lists of hyperbolic Kleinian groups (tetrahedral groups, knot complements, small-volume closed and cusped manifolds) and an "Arithmetic zoo'' containing a number of examples of arithmetic Kleinian groups which have particularly interesting properties.
This book fills a real void in the literature, providing working topologists and graduate students with an accessible introduction to the useful and beautiful world of arithmetic hyperbolic 3-manifolds and orbifolds. (MathSciNet)

Bibliogr. p. [443]-458. Index

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