Research article fundamental group and covering properties of. Geometry and topology interact closely in the study of hyperbolic 3manifolds, a fact best manifested by mostows rigidity theorem. Hyperbolic manifolds and orbifolds examples 3manifold topology and dehn surgery rigidity volumes and ideal tetrahedra part 1. Foundations of hyperbolic manifolds john ratcliffe. The author considers also the two closed, orientable, hyperbolic manifolds of the same homotopy type have the same volume by using the gromov invariant, a quantity defined in terms. Hyperbolic manifolds with geodesic boundary which are. Suppose that there is a sideparing between the 2dimensional faces of these polyhedra i. Volumes of hyperbolic manifolds mathematics stack exchange. Hyperbolic manifolds hilary term 2000 marc lackenby geometry and topologyis, more often than not, the study of manifolds. Examples of complete hyperbolic 3 manifolds of finite volume are constructed in section 10. Ratcliffe, foundations of hyperbolic manifolds, third edition, graduate texts in. Outline geometry and analysis on hyperbolic manifolds yiannis petridis1,2 1the graduate center and lehman college city university of new york 2maxplanckinstitut fur mathematik, bonn. Ratcliffe, foundations of hyperbolic manifolds, gtm springer, 2nd edition, 2005.
Simplicial volume and fillings of hyperbolic manifolds koji fujiwara and jason fox manning abstract. Arithmetic of hyperbolic manifolds columbia university. We show that its universal cover is the hyperbolic space hn, and that its group deck transformations acts by isometries. Intuitively a hyperbolic manifold is just a space which locally looks like hyperbolic space. It is largely selfcontained in that it defines all the needed concepts and machinery and often provides proofs of facts that can be found elsewhere in the literature. Foundations of hyperbolic manifolds graduate texts in mathematics book 149 kindle edition by ratcliffe, john. Compactification and isometries of hyperbolic space. Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 for example, for all other geometries it is not hard to give an explicit enumeration of the finitevolume manifolds with this geometry, while this is far from being the case for hyperbolic manifolds. Our approach is to form an exhaustion mi of m and modify the boundary to make them 2convex. Tschantz, who constructed 1171 noncompact finite volume hyperbolic 4manifolds of minimal volume. Ratcliffe foundations of hyperbolic manifolds index of. However, fundamental questions about asymptotically hyperbolic manifolds remain unresolved. Foundation of hyperbolic manifold conical and hyperbolic paraboloid shell foundation hyperbolic foundations of hyperbolic manifolds hyperbolic manifolds ratcliffe hyperbolic functions questions and answers manifold riemann manifold differentiable manifold ocimf manifold 2017 aq140a exhaust manifold manifold gauge meter manifold gauge meter function manifold theory an introduction for mathematical physicists 2007 hyundai sonata intake manifold removal mathematics foundation course for jee. T g cp phase space space of complex projective structures back to hyperbolic manifolds.
They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3 manifolds, respectively. All the main ideas were worked out together, i have done my. Same question can be asked in the context of chernsimons theory on a handlebody, compare weitsman 91 p. Neumannt and don zagier received 11 jonuary 1984 5 1. Hyperbolic geometry and 3manifold topology david gabai 1 hyperbolic geometry and 3manifold topology 3 introduction 3 1. Since then, many consequences of this conjecture have been. Foundations of differentiable manifolds and lie groups gives a clear, detailed, and careful development of the basic facts on manifold theory and lie groups. We conjecture that for every dimension n63 there exists a noncompact hyperbolic nmanifold whose volume is smaller than the volume of any compact hyperbolic nmanifold. A new foreword for notes on notes of thurston mathematics. Introduction marden conjectured that a hyperbolic 3manifold m with. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an american university. Hyperbolic manifolds we begin by discussing a few properties of the universal cover and deck transformations of a compact hyperbolic manifold without boundary.
Poincare, fricke, and klein were among the earliest explorers of the great expanse of mathematics that related to. Silva the simplest example of a hyperbolic manifold is hyperbolic geometry itself, which we describe using the poincarediskmodel. Introduction byhyperbolic 3manifold we will mean an orientable complete hyperbolic 3manifold m of finite volume. Foundations of hyperbolic manifolds by john ratcliffe, 9781441922021, available at book depository with free delivery worldwide. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3manifolds, respectively. Introduction to hyperbolic geometry 1 topics i geometry of real and complex hyperbolic space. So the urstonjorgensentheory ofhyperbolicsurgerygivesthe following result.
Fundamental group and covering properties of hyperbolic. Taniguchi, hyperbolic manifolds and kleinian groups. Download it once and read it on your kindle device, pc, phones or tablets. Examples of complete hyperbolic 3manifolds of finite volume are constructed in section 10. Let m 1 and m 2 be ndimensional connected orientable finitevolume hyperbolic manifolds with geodesic boundary, and let. Using plmethods, we prove the mardens conjecture that a hyperbolic 3manifold m with. The general case of ndimensional hyperbolic manifolds is the subject of chapter 11, with the constructions in chapter 10 generalized to deal with high dimensions. The role of hyperbolic geometry in dimension n 4 is less clear.
In view of the initial value formulation for einsteins equations, global hyperbolicity is seen to be a very natural condition in the context of general relativity, in the sense that given arbitrary initial data, there is a unique maximal globally hyperbolic solution. The first part is concerned with hyperbolic geometry and discrete groups. Foundations of hyperbolic manifolds graduate texts in. In the closed case, one uses thurstons dehn surgery theorem to relate volumes of closed hyperbolic 3manaifolds with volumes of cusped hyperbolic 3manifolds, as for example in this 1985 paper of neumann and zagier, or a later paper of neumann, combinatorics of triangulations and the chernsimons invariant for hyperbolic 3manifolds. Foundations of hyperbolic manifolds by ratcliffe, john g. Its called hyperbolic because the fundamental condition that generates the lorentzian manifold is. This book is an exposition of the theoretical foundations of hyperbolic manifolds. This book is an exposition of the theoretical foundations of hyperbolic.
Asymptotic laws for geodesic homology on hyperbolic. On the number of hyperbolic 3manifolds of a given volume. We study a family of closed connected orientable 3manifolds obtained by dehn surgeries with rational coefficients along the oriented components of certain links. As we discuss in chapter 2, asymptotically hyperbolic manifolds arise naturally in the study of initial data sets in general relativity. For any integer 1 and for almost all pairs of surgery coe cients,and, the closed connected. Foundations of hyperbolic manifolds graduate texts in mathematics book title. Towards the end of 1 on pages 553554 sullivan sketched why closed and almostparallelizable1 hyperbolic manifolds exist in all dimensions. Hyperbolic surfaces or 3manifolds are quotients of 2 or 3dimensional hyperbolic constant negative curvature space by torsionfree discrete subgroups known as fuchsian or kleinian groups of their respective isometry groups. Hyperbolic geometry is the star of noneuclidean geometries, and gives. On the number of hyperbolic 3manifolds of a given volume hidetoshi masai joint work with craig hodgson tokyo institute of technology dc2, university of melbourne 2012, may, 31st representation spaces, twisted topological invariants and geometric structures of 3manifolds. Foundations of differentiable manifolds and lie groups. He also notes that one may combine his minimum principle with work of sullivan 145 to. Ratcliffes book rat06 is a good reference for learning hyperbolic geometry for the first. Particular emphasis has been placed on readability and completeness of ar gument.
Research article fundamental group and covering properties. In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold that is, a lorentzian manifold. The plmethods for hyperbolic 3manifolds to prove tameness suhyoung choi abstract. The invariant of hyperbolic 3manifolds tomoyoshi yoshida department of mathematics, okayama university, okayama, japan introduction let m 3 be a compact oriented riemannian manifold of dimension 3. We construct complete, open, hyperbolic 4 manifolds of smallest volume by gluing together the sides of a regular ideal 24cell in hyperbolic 4space. The problem of computing the volume of a hyperbolic 3manifold is taken up in section 10. Volume preserving surgeries on hyperbolic 3manifolds. Construction of hyperbolic 3manifolds of finite volume hyperbolic polyhedra, reflection groups. The volume spectrum of hyperbolic 4 manifolds ratcliffe, john g. Foundations of hyperbolic manifolds graduate texts in mathematics by john g. We find geometric presentations for the fundamental group of such manifolds and represent them as branched covering spaces. Foundations of hyperbolic manifolds graduate texts.
Foundations of hyperbolic manifolds graduate texts in mathematics this book is an exposition of the theoretical foundations of hyperbolic manifolds. In fm10, the authors constructed a variety of nonpositively and negatively curved spaces as \2. We study the problem whether there exists an isometry. It includes differentiable manifolds, tensors and differentiable forms. Let mbe a hyperbolic nmanifold whose cusps have torus crosssections. It is intended to be used both as a textbook and as a reference. Tschantz, who constructed 1171 noncompact finite volume hyperbolic 4 manifolds of minimal volume. Use features like bookmarks, note taking and highlighting while reading foundations of hyperbolic manifolds graduate texts in mathematics book 149. In particular, a globally hyperbolic manifold is foliated by cauchy surfaces. Hyperbolic manifolds and discrete groups is an essential text for anyone working in the topology and geometry of 3manifolds. The folding of hyperbolic manifold into itself is defined and discussed. Kirby diagrams and the ratcliffetschantz hyperbolic 4.
To fully appreciate the material we will cover you should have had. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3manifolds proved by. Foundations of hyperbolic manifolds, second edition. This heavily classtested book is an exposition of the theoretical foundations of hyperbolic manifolds. By mostow rigidity the volume of m is a topological invariant, indeed a. Introduction by a hyperbolic 3manifold we mean a complete orientable hyperbolic 3manifold of. The treatment of the material is for the most part elementary and selfcontained. A basic knowledge of algebra and topology at the first year graduate level of an american university is assumed. Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps. Your print orders will be fulfilled, even in these challenging times. This thesis is concerned with the study of the largescale geometry of asymptotically hyperbolic manifolds. More carefully, mostow showed that if there was an isomorphism between the fundamental groups of two closed hyperbolic 3manifolds, then there was an isometry taking one to the other. We give a brief introduction to hyperbolic geometry, including its.
A hyperbolic manifold is a complete riemannian manifold with constant sectional curvature 1. This third edition greatly expands upon the second with an abundance of additional content, including a section dedicated to arithmetic hyperbolic groups. In the closed case, one uses thurstons dehn surgery theorem to relate volumes of closed hyperbolic 3manaifolds with volumes of cusped hyperbolic 3 manifolds, as for example in this 1985 paper of neumann and zagier, or a later paper of neumann, combinatorics of triangulations and the chernsimons invariant for hyperbolic 3 manifolds. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. The volume spectrum of hyperbolic 4manifolds ratcliffe, john g. Consider the operator a on even forms on m, f2 0 2, defined on f22v p 0, 1 by a1p,dd. Introduction to hyperbolic geometry arizona state university. Kirby diagrams and the ratcliffetschantz hyperbolic 4manifolds. By the mostowprasad rigidity theorem,14, if a knot complement admits a hyperbolic structure, then that structure is unique.
This family contains all the manifolds obtained by surgery along the hyperbolic 2bridge knots. Well aim for this result1 starting from the much simpler goingson of 2. Arithmetic of hyperbolic manifolds 283 moreover, if. The chapter ends with a detailed study of hyperbolic dehn surgery on the figureeight knot complement. In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. Foundations of hyperbolic manifolds graduate texts in mathematics 9780387331973. Hyperbolic nmanifolds exist for every n 2, but the small dimensions n 2 and n 3 have been the object of a much wider and more intense study than others. We construct complete, open, hyperbolic 4manifolds of smallest volume by gluing together the sides of a regular ideal 24cell in hyperbolic 4space.
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