W.P. Thurston. The Geometry and Topology of Three-Manifolds. Princeton University, Princeton, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be the closure of R S in R . For simplicity, a measured lamination coming from a pair (s; will be denoted s . In the presence of a hyperbolic structure on S, it is typical to de ne measured lami 5 nations in terms of geodesic laminations equipped with a measure on transverse arcs (see [Th82] or [Bo88] for more details) We can also use a hyperbolic structure on S to de ne a notion of the length L( of a measured lamination : one de nes L(s ) to be the product of s and the hyperbolic length of on S, and then extends L : R S R to all of ML by continuity (see e.g. Ke85] ....

W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton Univ., Princeton, 1982.

.... are used to describe the structure of the orbit spaces of discrete groups acting on manifolds not necessarily freely. Fixed points of such a group action lead to singularities in the orbit space, thus the stabilizer group of such a point is memorized as part of the orbifold structure (c.f. Thu80] Sco83] 2 OLAF DELGADO FRIEDRICHS one of which is an as of yet, hopelessly impractical algorithm which solves the problem for so called Haken manifolds. It proceeds by splitting a manifold along certain embedded surfaces. In the general case, a vast number of possible splitting surfaces ....

....contains exactly one or two one dimensional subsets which are non trivially branched, and if two such subsets exist, then their branching numbers are di erent (c.f. Bal90] An equivalent to Theorem 2.4. 5 in the terminology of orbifolds can be found, for example, in [Con90] or, of course, in [Thu80] 24 OLAF DELGADO FRIEDRICHS 2.5. Symmetry breaking in dimension three. Now we present an important partial result on the existence of geometric realizations in dimension three. Obviously, if a branched set (D; L) has a geometric realization in some space X, then any cover of (D; L) has a ....

[Article contains additional citation context not shown here]

W.P. Thurston. The Geometry and Topology of Three-Manifolds. Princeton University, Princeton, 1980. Chapter 13.

....the topological type of M . In fact, even in dimension 3, where Einstein metrics are hyperbolic, i.e. of constant curvature, there are numerous examples of non uniqueness. These examples come from the Thurston theory of Dehn surgery, or the well known process of opening cusps , c.f. 18] [35]. This construction gives in nitely many isometrically distinct hyperbolic 3 manifolds, all with a given conformal in nity. AH EINSTEIN METRICS ON 4 MANIFOLDS 3 Of course such Thurston Dehn surgery is a special feature of hyperbolic 3 manifolds, and does not generalize to hyperbolic ....

....but xed (T 2 ; g o ) As L = L( 1; the length of the core geodesic tends to 0. Any sequence of such metrics thus converges to the complete (rank 2) hyperbolic cusp g C = dr 2 e 2r g o ; 4.10) on R T 2 : This process is the formation of a cusp, or opening a cusp , c.f. 18] [35]. Remark 4.3. The process described above of opening a cusp may also be reversed. Thus, given a complete hyperbolic cusp as in (4.10) the Dehn surgery process above closes this cusp by lling in with a hyperbolic solid torus, keeping the conformal structure at in nity xed. As discussed above, ....

W. Thurston, The Geometry and Topology of Three-Manifolds, (preprint), Princeton Univ.,(1978)

....N 2 H(S) correspond to simple loops on S; and ffl If a geodesic fl ae N represents a simple loop on S, then fl has bounded diameter modulo the thin part of N . The key to proving these results is that one can construct a pleated surface through fl whenever fl represents a simple loop on S; see [Th, x8, x9], Bon] Using train tracks, one can even show that fl varies continuously on the space of laminations on S. The factorization of elements of 1 (S) into simple curves can also be carried out on surfaces of higher genus. Indeed, if one picks a complex structure on S together with a holomorphic ....

W. P. Thurston. Geometry and Topology of Three-Manifolds. Lecture

....Proposition 4.1. If G is a finite subgroup of the mapping class group of M 3 , then there is a quasiisometry f mapping M 3 to another complete hyperbolic manifold N 3 , on which the group G is realized as isometries. A proof can be given along the lines of Theorem 2. 1, using e.g. Thurston [Th, Chapter 11] or Douady Earle [DE] to obtain the quasi isometry statement. It is interesting to speculate on the connection between our finiteness result for Mod(J,f) when J is connected and known results in the topology of three manifolds. By Dehn s lemma and the loop theorem, the limit set L is connected if ....

W. Thurston. The Geometry and Topology of Three Manifolds. Lecture notes, Princeton University (1979).

....2 C and [z] g: When M is orientable, de ne the simplicial volume of M , denoted kMk, to be the simplicial norm of the fundamental class. The simplicial volume is also called Gromov s invariant, since it was rst introduced by Gromov [6] The following lower bound on kMk is due to Thurston [28]. Theorem 2.6. 28, Theorem 6.5.5] Suppose that M is a closed orientable 3 manifold and that H M is a 3 dimensional submanifold whose interior admits a complete hyperbolic structure of nite volume. Suppose further that H is embedded in M and that H is incompressible in M . Then, kMk ....

W. P. Thurston, `The geometry and topology of three-manifolds', electronic version 1.0-October 1997.

....hyperbolic n manifolds of finite volume is called the volume spectrum of hyperbolic n manifolds. It has been known for over a hundred years that the volume spectrum of hyperbolic 2 manifolds is the set of all positive integral multiples of 2. In contrast to dimension two, Jrgensen and Thurston [10] have shown that the volume spectrum of hyperbolic 3 manifolds is a closed, non discrete, well ordered subset of the positive real numbers of order type . In particular, there is a smallest volume of a complete hyperbolic 3 manifold. The smallest volume of a complete hyperbolic 3 manifold ....

....M 3 2 has an inside out symmetry that interchanges the two octahedrons and has no fixed points. The quotient space under the action of this inside out symmetry of M 3 2 is the Whitehead link complement obtained by gluing together the sides of a regular octahedron as in Thurston s lecture notes [10]. From the classification of all the index 24 torsion free subgroups of the Picard group [1] we deduce that M 3 2 is homeomorphic to the link complement 8 3 9 . One can also derive a presentation for the fundamental group of M 3 2 from the side pairing for M 3 2 in Table 1 and transform ....

[Article contains additional citation context not shown here]

Thurston, W. P., The Geometry and Topology of Three-Manifolds, lecture notes, Princeton Univ., 1979.

....Can the paths in P be ( C) coloured so that no intersecting pair of paths receive the same colour Note that if P consists of the set of single edge paths and C = 1, then this is equivalent to finding a ( Delta 1) edge colouring of G. We remark on another special case. It was shown (see [8]) that any planar graph G is the intersection graph of a family of non crossing circles in the plane, i.e. the vertices of G correspond to circles, and two vertices are adjacent if and only if their corresponding circles meet at a point. Thus in this special case, the contours can be 4 coloured. ....

Thurston, "The geometry and topology of three-manifolds" unpublished.

....whereas in the second case, the corresponding group would have precisely two fixed points on the sphere, but of different orders, again impossible. The other two bad cases are obtained by adding a reflection in the two above situations. For details of this theorem, see [Vin93] page 185] [Thu80][Chapter 13] or [Mon87] page 91] What do 2 dimensional orbifolds look like The well known classification of compact 2 dimensional surfaces (see e.g. ST80] tells us that all compact, orientable 2 manifolds can be obtained from the 2 sphere by drilling q holes in it, and attaching h handles to ....

W.P. Thurston. The Geometry and Topology of ThreeManifolds. Princeton University, Princeton, 1980.

....of R Theta S in R S . For simplicity, a measured lamination coming from a 4 pair (s; fl) will be denoted sfl. In the presence of a hyperbolic structure on S, it is typical to define measured laminations in terms of geodesic laminations equipped with a measure on transverse arcs (see [Th82] or [Bo88] for more details) We can also use a hyperbolic structure on S to define a notion of the length L( of a measured lamination : one defines L(sfl) to be the product of s and the hyperbolic length of fl on S, and then extends L : R Theta S R to all of ML by continuity (see e.g. ....

W. P. Thurston, The Geometry and Topology of Three-Manifolds, Princeton Univ., Princeton, 1982.

.... ( Gamma p 1 p 2 ) is the geodesic orthogonal to both (p 1 ) p 2 ) Geodesics (p) and l are orthogonal ( p 2 l. Furthermore Gamma p 1 p 2 is tangent to Q ( p 1 ) and (p 2 ) are parallel. For more information on this model for hyperbolic geometry, see Coxeter [C1] or Thurston [T,x2]. This model for non Euclidean geometry seems to have first been discovered by Cayley in 1858. 3.3 We shall discuss the metric on hyperbolic space, however, in the more general setting of the Hilbert Carath eodory Kobayashi metric on a convex domain P = P n . Let V = R n 1 be the ....

Thurston, W., "The geometry and topology of three--manifolds," notes available from Princeton University mathematics department (1978)

....N 2 H(S) correspond to simple loops on S; and ffl If a geodesic fl ae N represents a simple loop on S, then fl has bounded diameter modulo the thin part of N . The key to proving these results is that one can construct a pleated surface through fl whenever fl represents a simple loop on S; see [Th, xx8, 9], Bon] Using train tracks, one even show that fl varies continuously on the space of laminations on S. The factorization of elements of 1 (S) into simple curves can also be carried out on surfaces of higher genus. Indeed, if one picks a complex structure on S together with a holomorphic ....

W. P. Thurston. Geometry and Topology of Three-Manifolds. Lecture

....that the threedimensional spaces of constant curvature are just a very special case of more general homogeneous spaces. As we shall see in more details in section 5, there are eight types of homogeneous three dimensional geometries , five of them not admitting a metric of constant curvature [150, 151]. Let us give mathematical substance to these notions. Quite generally, to any manifold (M;g) is associated a group G of isometries, i.e. transformations which leave the meric invariant. The manifold M is said homogeneous if G is non trivial 7 . The group G is said to act transitively on M ....

....idea of the extreme complication of such multi connected topologies. 66 8 Three dimensional hyperbolic space forms 8.1 The geometry of IH 3 Locally hyperbolic manifolds are by far less well understood than the other homogeneous spaces. However, according to the pionneering work of Thurston [150], almost all 3 manifolds can be endowed with a hyperbolic structure. Here we present some elements of the theory; for a recent report, see [9] It is not easy to have an intuitive representation of IH 3 because it cannot be imbedded in IR 4 . Instead, it can be seen as an hypersurface of ....

[Article contains additional citation context not shown here]

Thurston, W.P. 1979, The geometry and topology of three manifolds, Princeton Lecture Notes

....are the only quadriculated surfaces with no boundary. Quadriculated cylinders and quadriculated Mobius bands are even easier to construct: start with any simply connected region in the plane and glue along congruent boundaries. As for Euclidean manifolds, it is easy to define a developing map ([10]) from the universal cover of a quadriculated surface to the plane. Similarly, define the holonomy of a quadriculated surface: it is a homomorphism from the fundamental group of the surface to the group of isometries of Z 2 . If the surface is a topological disk, it has trivial holonomy and may ....

....c i 1 is to the left of c i . The contradiction arises from proving that the existence of a closed sequence of cycles as above implies that A is a torus or a Klein bottle and that the height sections 1 and 2 never coincide. By going to the universal cover and using the developing map as in [10], each cycle c i becomes a periodic line c i in Z 2 , the period being an orientation preserving isometry of R 2 preserving Z 2 , thus either a translation or a rotation of period 2 or 4. If the period of c 0 is not a translation, the curve c 0 surrounds a certain signed area, which ....

[Article contains additional citation context not shown here]

W. P. Thurston, Geometry and Topology of Three-manifolds, Department of Mathematics, Princeton University, 1979.

....its Corollary highlight a difference between these two types of conformal dynamical systems. The questions addressed here emerged from joint work with Mike Freedman [FM] See [BV] for more on John domains and Julia sets. Basic facts about hyperbolic manifolds used in the sequel can be found in [Th], BP] and [Rat] 2 2 John domains Let H n denote hyperbolic n space and S n Gamma1 1 its sphere at infinity. A region U ae S n Gamma1 1 is a John domain if there is an a 2 U and an ffl 0 such that for any b 2 U , there is a path p : 0; 1] U with p(0) a, p(1) b and d(p(t) U) ....

....( 2 1 1 1 ) An extended discussion of such groups can be found in [Mc2, x3] and [Mc3, x7] The domain of discontinuity U = S 2 1 Gamma is uniformly connected, but not a John domain, as is evident from the narrow fjords reaching towards the center of the picture. In fact U has measure zero [Th] but Hausdorff dimension two [Sul1] BJ] The center of symmetry c of the picture is a cut point of the limit set; Gamma fcg has six components. However, under blowups about c, the limit set converges to the plane and the region U converges to the empty set [Mc2, p.68] in contrast to the ....

W. P. Thurston. Geometry and Topology of Three-Manifolds. Lecture Notes, Princeton University, 1979.

....The grafting construction appears in works of Goldman, Hejhal and Maskit on exotic projective structures [Gol] Hej] Mas] These works concern grafting a 2 annulus along a simple closed curve. The bending lamination for the convex hull of hyperbolic 3 manifold was introduced by Thurston [Th1]. A detailed development is given by Epstein and Marden in [EpM] Thurston also showed the bending lamination can be defined for an arbitrary projective structure, and thus every projective structure is given by a canonical grafting along a lamination. This result, Theorem 2.4 below, is presented ....

....simple geodesics. On any surface X 2 T (S) any 2 ML(S) can be represented by a compact, transversally measured geodesic lamination. The space of such compactly supported laminations is also sometimes denoted ML 0 (S) An extensive discussion of measured laminations can be found, for example, in [Th1], EpM] Bon2] and [Ot] To allow complex multiples of laminations, we let ML C (S) f( z) 2 ML(S) Theta C g=h( tz) t; z) t 2 R i: This tensor product can be formed with other sets of scalars such as R and H , and we have ML(S) ae MLR (S) ae ML H (S) ae ML C (S) For 2 ML C ....

[Article contains additional citation context not shown here]

W. P. Thurston. Geometry and Topology of Three-Manifolds. Lecture Notes, Princeton University, 1979.

....topology of Kleinian groups, Marden showed that geometrically finite manifolds are tame [Mar] and raised the question of tameness in general. Through work of Thurston and Bonahon, the Marden and Ahlfors conjectures were both established for 3 manifolds with incompressible ends [Bon] [Th]. These are manifolds admitting a Scott core with incompressible boundary) The proofs use pleated surfaces sweeping out the geometrically infinite ends. Canary showed, using branched coverings, that tame manifolds with compressible ends can also be equipped with sufficiently many pleated ....

W. P. Thurston. Geometry and Topology of Three-Manifolds. Lecture Notes, Princeton University, 1979.

No context found.

W.P. Thurston. The Geometry and Topology of Three-Manifolds. Princeton University, Princeton, 1980.

....of 1 injective embedded surfaces. This result is one of the major results in 3 dimensional topology and gives to hyperbolic geometry and Kleinian group theory a fundamental role in 3 dimensional topology. It has been announced by Thurston in the mid 70 s and provided with evidence in his Notes [Thu3] and a series of papers. Only in recent years, a number of complete accounts became available. We will focus in this workshop on one partial but substantial aspect of the Hyperbolization Theorem, namely on the case of 3 manifolds fibering over the circle. It already contains a wealth of ....

....group actions on trees and Skoras Theorem for the proof of the Double Limit Theorem which is the heart of hyperbolization. The main references for this workshop will be: On Hyperbolization: The books by Jean Pierre Otal [O] and Misha Kapovich [Ka] Sullivans Bourbaki talk [Su3] Thurston s Notes [Thu3] and the notes on The Notes [Ep] On diffeomorphisms of surfaces: the book [CB] and the Orsay seminar notes [FLP] For background on the Geometrisation Programm: Thurstons book [Thu4] and the article by Peter Scott [Sc] Lecture 1. Geometrisation of mapping tori. The purpose of this lecture is ....

[Article contains additional citation context not shown here]

W. Thurston, The Geometry and Topology of Three-Manifolds, Princeton Notes 1979, http://www.msri.org/publications/books/gt3m/

No context found.

THURSTON, W. P. 1980. The Geometry and Topology of Three-Manifolds . Available at http://www.msri.org/publications/books/gt3m/.

No context found.

W. P. Thurston, The Geometry and Topology of Three-Manifolds, 2002.

No context found.

W. P. Thurston, The Geometry and Topology of Three-Manifolds, 2002.

No context found.

William P. Thurston, The Geometry and Topology of Three Manifolds, Princeton University lecture notes. 8

No context found.

W. P. Thurston. Geometry and Topology of Three-Manifolds. Lecture Notes, Princeton University, 1979.

No context found.

W. P. Thurston. Geometry and Topology of Three-Manifolds. Lecture Notes, Princeton University, 1979.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC