Hyperbolic Structures on 3-Manifolds Via Volume Maximization
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The Arithmetic Hyperbolic 3-Manifold of Smallest Volume
THE ARITHMETIC HYPERBOLIC 3-MANIFOLD OF SMALLEST VOLUME Ted Chinburg * Eduardo Friedman ** Kerry N. Jones y Alan W. Reid yy Abstract. We show that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold. The next smallest one is the Meyerhoff manifold. 0. Introduction x A hyperbolic 3-manifold is a 3-manifold admitting a complete Riemannian metric all of whose sectional curvatures are 1. The universal cover of such a manifold can therefore be identified with the hyperbolic− 3-space, that is, the unique connected and simply connected hyperbolic 3-manifold. We denote by H3 = (z; t) C R t > 0 ; f 2 ⊕ j g the upper half space model of hyperbolic 3-space. With this convention, the full group of orientation-preserving isometries of H3 is simply PGL(2; C). A Kleinian group Γ is a discrete subgroup of PGL(2; C). Hence an orientable hyperbolic 3- manifold is the quotient of H3 by a torsion-free Kleinian group, since this acts properly discontinuously and freely on H3. If we relax the condition that Γ act freely, allowing Γ to contain torsion elements, we obtain a complete orientable hyperbolic 3-orbifold (cf. [Th] for further details). We will only be concerned with the case that M = H3=Γ has a hyperbolic structure of finite volume (we say Γ has finite covolume). Therefore in what follows, by a hyperbolic 3-manifold or 3-orbifold we shall always mean a complete orientable hyperbolic 3-manifold or 3-orbifold of finite volume. By the Mostow and Prasad Rigidity Theorems [Mo] [Pr], hyperbolic volume is a topological invariant of a hyperbolic 3-orbifold and is therefore a natural object to study. -
On the Number of Hyperbolic 3-Manifolds of a Given Volume 3
On the number of hyperbolic 3-manifolds of a given volume Craig Hodgson and Hidetoshi Masai Abstract. The work of Jørgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. We show that there is an infinite sequence of closed orientable hyperbolic 3- manifolds, obtained by Dehn filling on the figure eight knot complement, that are uniquely determined by their volumes. This gives a sequence of distinct volumes xi converging to the volume of the figure eight knot complement with N(xi) = 1 for each i. We also give an infinite sequence of 1-cusped hyper- bolic 3-manifolds, obtained by Dehn filling one cusp of the (−2, 3, 8)-pretzel link complement, that are uniquely determined by their volumes amongst ori- entable cusped hyperbolic 3-manifolds. Finally, we describe examples showing that the number of hyperbolic link complements with a given volume v can grow at least exponentially fast with v. 1. Introduction Thurston and Jørgensen (see [43]) showed that the set of volumes of finite volume orientable complete hyperbolic 3-manifolds is a closed, non-discrete, well ω ordered subset of R>0 of order type ω . Further, the number N(v) of complete orientable hyperbolic 3-manifolds of volume v is finite for all v R>0. In this paper we study how N(v) varies with v. ∈ There has been much work on determining the lowest volumes in various classes of hyperbolic 3-manifolds. The work of Gabai-Meyerhoff-Milley ([21, 22, 35]) shows that the Weeks manifold with volume v1 =0.9427 .. -
LOWER BOUNDS on VOLUMES of HYPERBOLIC HAKEN 3-MANIFOLDS 1. Introduction in This Paper We Prove a Volume Inequality for 3-Manifol
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 20, Number 4, October 2007, Pages 1053–1077 S 0894-0347(07)00564-4 Article electronically published on May 31, 2007 LOWER BOUNDS ON VOLUMES OF HYPERBOLIC HAKEN 3-MANIFOLDS IAN AGOL, PETER A. STORM, AND WILLIAM P. THURSTON, WITH AN APPENDIX BY NATHAN DUNFIELD 1. Introduction In this paper we prove a volume inequality for 3-manifolds having C0 metrics “bent” along a surface satisfying certain curvature conditions. The method is a direct generalization of work of Bray [7] and Miao [28] on the Riemannian Penrose conjecture. We also make use of Perelman’s results on geometrization [34, 33]. Perelman’s montonicity formula for the Ricci flow with surgery (discovered by Hamilton in the case of the standard Ricci flow [18, Thm. 2.1]) implies that if (M,g) is a hyperbolic 3-manifold and (M,h) is a Riemannian metric such that the scalar curvature R(h) ≥−6=R(g), then Vol(M,h) ≥ Vol(M,g). More generally, if M is not hyperbolic, then Vol(M,h) ≥ V3M,whereV3M denotes the sim- 3 plicial volume of M (V3 is the volume of a regular ideal tetrahedron in H ,and M is the Gromov norm of the fundamental class of the 3-manifold M). This sort of curvature condition is much weaker than the conditions on Ricci curvature used in the work of Besson, Courtois, and Gallot. Their work implies a similar volume estimate if Ric(h) ≥−2h [6]. Theorem 7.2 is the main result of this paper, which states that if (M,g)isa compact hyperbolic 3-manifold with minimal surface boundary, then Vol(M,g) ≥ 1 2 V3 DM ,whereDM denotes the double of M along its boundary. -
Embedding Closed Hyperbolic 3-Manifolds in Small Volume Hyperbolic 4-Manifolds
EMBEDDING CLOSED HYPERBOLIC 3-MANIFOLDS IN SMALL VOLUME HYPERBOLIC 4-MANIFOLDS M. CHU AND A. W. REID Abstract. In this paper we study existence and lack thereof of closed embedded ori- entable co-dimension one totally geodesic submanifolds of minimal volume cusped ori- entable hyperbolic manifolds. 1. Introduction Let W n denote a minimal volume, orientable, cusped hyperbolic n-manifold. In this paper we will be concerned with the existence of closed embedded orientable totally geodesic hyperbolic submanifolds M n−1 ,! W n. When n = 2, W 2 is either the thrice-punctured sphere or a once-punctured torus, and in the former case there are no such submanifolds, whilst in the latter there are many. In dimension 3, by work of Cao and Meyerhoff, [CM], the manifolds in question are the com- plement of the figure-eight knot and its sister. These do not contain any closed embedded orientable totally geodesic surfaces (although they do contain infinitely many immersed closed totally geodesic surfaces [MR, Chapter 9]). Indeed, they do not contain any closed embedded essential surfaces (see [Th], [CJR] and [FH]). These 3-manifolds are arithmetic hyperbolic 3-manifolds, as is the case of the thrice-punctured sphere. The once-punctured torus has a positive dimensional Teichm¨ullerspace, but there are a finite number of exam- ples which are arithmetic. This paper is concerned with the following conjecture. As we discuss in Section 2.2, Conjecture 1.1 is easily reduced to the only non-trivial case, that of dimension 4. Conjecture 1.1. Let W n denote a minimal volume, orientable, cusped arithmetic hyper- bolic n-manifold. -
Heegaard Diagrams of Certain 3-Manifolds
HEEGAARD DIAGRAMS OF CERTAIN 3-MANIFOLDS Katarzyna Krawiec [email protected] November 18, 2020 ABSTRACT This paper is an exercise in topology of 3-manifolds. We present three examples of genus 2 Heegaard splittings of manifolds with different geometries diagramatically. 1 Introduction 1.1 Brief history of classification of 3-manifolds The history of manifolds began in 1851, when Bernhard Riemann used the word for the first time. It was the beginning of low-dimensional topology as we know it today. Mathematicians soon started working on the classification of manifolds1. Trivial classification of 0-dimensional and 1-dimensional manifolds, and then the uniformization theorem for 2-dimensional manifolds, painted a promising picture for classifying manifolds of higher dimensions. However, the simplest question in classification of 3-manifolds, the Poincaré conjecture, was only settled after almost eighty years of work. Of course there was much progress in general theory of 3-manifolds, to just mention the equivalence of smooth, piecewise linear and topological categories (which we will encounter below via the Moise theorem). This allowed for the development of new methods and new paths of investigation. 1.1.1 Prime decomposition, JSJ decomposition and Geometrization Conjecture One of the first approaches to the classification of 3-manifolds was to cut them into smaller and hopefully simpler pieces. Around 1930 Hellmuth Kneser proved the existence of the prime decomposition (Theorem 1.5 in [10]), which was based on cutting 3-manifolds along the spheres. He showed that every compact orientable 3-manifold factors as a connected sum of prime manifolds, i.e. -
William P. Thurston Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry Bull
William P. Thurston Three-dimensional manifolds, Kleinian groups and Hyperbolic geometry Bull. Am. Math. Soc., New Ser. 6, 357{379 (1982) Jean-Pierre Otal This paper is the written version of a talk that Thurston gave at the AMS Sym- posium on the Mathematical Heritage of Henri Poincar´eheld at Bloomington in April 1980. It is a Research announcement.1 In his \Analysis Situs" paper, Poincar´eintroduces simplicial homology and the fundamental group, with the aim of finding invariants which could distinguish 3- manifolds. He concludes the Vth complement [88] discussing the question: Is any simply connected closed 3-manifold homeomorphic to the 3-sphere? This ques- tion, often understood as a conjecture, motivated much of the development of 3- dimensional topology. At the beginning of his paper Thurston proposes, \fairly con- fidently", his Geometrization Conjecture: The interior of any compact 3-manifold has a canonical decomposition into pieces that carry a geometric structure. By Kneser and Milnor any orientable compact 3-manifold is homeomorphic to the connected sum of manifolds which are either irreducible or homeomorphic to a sphere bundle over S1. By Jaco-Shalen and Johannson any orientable irreducible 3-manifold can be decomposed along a disjoint union of embedded tori into pieces that are atoroidal or Seifert-fibered (see in [15] the corresponding statement for non-orientable manifolds). Saying that a manifold carries a geometric structure means that it has a complete Riemannian metric which is locally modelled on one of the eight geometries (which are introduced in Section 4 of the paper): the con- stant curvature ones H3, S3, R3, the Seifert-fibered ones PSL(2^;R), H2 × R, S2 × R, Nil and Sol. -
Embedding Closed Hyperbolic 3-Manifolds in Small Volume Hyperbolic 4-Manifolds
EMBEDDING CLOSED HYPERBOLIC 3-MANIFOLDS IN SMALL VOLUME HYPERBOLIC 4-MANIFOLDS M. CHU AND A. W. REID Abstract. In this paper we study existence and lack thereof of closed embedded ori- entable co-dimension one totally geodesic submanifolds of minimal volume cusped ori- entable hyperbolic manifolds. 1. Introduction Let W n denote a minimal volume, orientable, cusped hyperbolic n-manifold. In this paper we will be concerned with the existence of closed embedded orientable totally geodesic hyperbolic submanifolds M n−1 ,! W n. When n = 2, W 2 is either the thrice-punctured sphere or a once-punctured torus, and in the former case there are no such submanifolds, whilst in the latter there are many. In dimension 3, by work of Cao and Meyerhoff, [CM], the manifolds in question are the com- plement of the figure-eight knot and its sister. These do not contain any closed embedded orientable totally geodesic surfaces (although they do contain infinitely many immersed closed totally geodesic surfaces [MR, Chapter 9]). Indeed, they do not contain any closed embedded essential surfaces (see [Th], [CJR] and [FH]). These 3-manifolds are arithmetic hyperbolic 3-manifolds, as is the case of the thrice-punctured sphere. The once-punctured torus has a positive dimensional Teichm¨ullerspace, but there are a finite number of exam- ples which are arithmetic. This paper is concerned with the following conjecture. As we discuss in Section 2.2, Conjecture 1.1 is easily reduced to the only non-trivial case, that of dimension 4. Conjecture 1.1. Let W n denote a minimal volume, orientable, cusped arithmetic hyper- bolic n-manifold. -
ARITHMETICITY, DISCRETENESS and VOLUME 1. Introduction This
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 3611{3643 S 0002-9947(97)01989-2 ARITHMETICITY, DISCRETENESS AND VOLUME F. W. GEHRING, C. MACLACHLAN, G. J. MARTIN, AND A. W. REID Abstract. We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of PSL(2; c). We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple. We then list all \small" discrete groups generated by elliptics of order 2 and n, n =3;4;5;6;7. 1. Introduction This paper is concerned with the geometry of Kleinian groups and hence the geometry and topology of hyperbolic 3–manifolds and orbifolds; see [4, 32, 33, 41] as basic references. In [18, 21] we initiated a program to describe the analytic space of discrete two-generator subgroups of PSL(2, C) and developed a disk covering procedure to identify all two-generator Kleinian groups with elliptic generators. This method allowed us to identify many groups in [16] through [23] as extremals for various geometric problems or as finite index subgroups of extremal groups. A principal obstruction encountered in this program, and hence to solving related geometric problems, was in establishing the discreteness of candidates identified by this procedure.