Hyperbolic Manifolds

Author: Albert Marden

Publisher: Cambridge University Press

ISBN: 1107116740

Category: Mathematics

Page: 550

View: 4790

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Second edition of Outer circles, which has changed title to: Hyperbolic manifolds.
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Hyperbolic Manifolds and Kleinian Groups

Author: Katsuhiko Matsuzaki,Masahiko Taniguchi

Publisher: Clarendon Press

ISBN: 0191591203

Category: Mathematics

Page: 264

View: 401

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A Kleinian group is a discrete subgroup of the isometry group of hyperbolic 3-space, which is also regarded as a subgroup of Möbius transformations in the complex plane. The present book is a comprehensive guide to theories of Kleinian groups from the viewpoints of hyperbolic geometry and complex analysis. After 1960, Ahlfors and Bers were the leading researchers of Kleinian groups and helped it to become an active area of complex analysis as a branch of Teichmüller theory. Later, Thurston brought a revolution to this area with his profound investigation of hyperbolic manifolds, and at the same time complex dynamical approach was strongly developed by Sullivan. This book provides fundamental results and important theorems which are needed for access to the frontiers of the theory from a modern viewpoint.
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Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups

Author: Richard Douglas Canary,Darryl McCullough

Publisher: American Mathematical Soc.

ISBN: 0821835491

Category: Mathematics

Page: 218

View: 5252

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This text investigates a natural question arising in the topological theory of $3$-manifolds, and applies the results to give new information about the deformation theory of hyperbolic $3$-manifolds. It is well known that some compact $3$-manifolds with boundary admit homotopy equivalences that are not homotopic to homeomorphisms. We investigate when the subgroup $\mathcal{R}(M)$ of outer automorphisms of $\pi_1(M)$ which are induced by homeomorphisms of a compact $3$-manifold $M$ has finite index in the group $\operatorname{Out}(\pi_1(M))$ of all outer automorphisms. This question is completely resolved for Haken $3$-manifolds.It is also resolved for many classes of reducible $3$-manifolds and $3$-manifolds with boundary patterns, including all pared $3$-manifolds. The components of the interior $\operatorname{GF}(\pi_1(M))$ of the space $\operatorname{AH}(\pi_1(M))$ of all (marked) hyperbolic $3$-manifolds homotopy equivalent to $M$ are enumerated by the marked homeomorphism types of manifolds homotopy equivalent to $M$, so one may apply the topological results above to study the topology of this deformation space.We show that $\operatorname{GF}(\pi_1(M))$ has finitely many components if and only if either $M$ has incompressible boundary, but no 'double trouble', or $M$ has compressible boundary and is 'small'. (A hyperbolizable $3$-manifold with incompressible boundary has double trouble if and only if there is a thickened torus component of its characteristic submanifold which intersects the boundary in at least two annuli). More generally, the deformation theory of hyperbolic structures on pared manifolds is analyzed. Some expository sections detail Johannson's formulation of the Jaco-Shalen-Johannson characteristic submanifold theory, the topology of pared $3$-manifolds, and the deformation theory of hyperbolic $3$-manifolds. An epilogue discusses related open problems and recent progress in the deformation theory of hyperbolic $3$-manifolds.
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Spaces of Kleinian Groups

Author: Yair N. Minsky,Makoto Sakuma,Caroline Series

Publisher: Cambridge University Press

ISBN: 1139447211

Category: Mathematics

Page: N.A

View: 8828

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The subject of Kleinian groups and hyperbolic 3-manifolds is currently undergoing explosively fast development, the last few years having seen the resolution of many longstanding conjectures. This volume contains important expositions and original work by some of the main contributors on topics such as topology and geometry of 3-manifolds, curve complexes, classical Ahlfors-Bers theory, computer explorations and projective structures. Researchers in these and related areas will find much of interest here.
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Punctured torus groups and 2-bridge knot groups (I)

Author: Hirotaka Akiyoshi,Makoto Sakuma,Masaaki Wada

Publisher: Springer Verlag

ISBN: 9783540718062

Category: Mathematics

Page: 252

View: 6189

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This monograph is Part 1 of a book project intended to give a full account of Jorgensen's theory of punctured torus Kleinian groups and its generalization, with application to knot theory.Although Jorgensen's original work was not published in complete form, it has been a source of inspiration. In particular, it has motivated and guided Thurston's revolutionary study of low-dimensional geometric topology.In this monograph, we give an elementary and self-contained description of Jorgensen's theory with a complete proof. Through various informative illustrations, readers are naturally led to an intuitive, synthetic grasp of the theory, which clarifies how a very simple fuchsian group evolves into complicated Kleinian groups.
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