P.Scott, The geometries of 3-manifolds, Bull.London Math. Soc., 15(1983), 401-487.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 structure. There are three exceptional cases: D 2 S 1 , T 2 (0, 1) and a manifold 2 fold covered by the latter (interval bundle over Klein bottle) each have infinite volume complete E 3 structures but no finite volume geometric structure. For more details on the above see [30] or [41]. Case 2 of the easy cases is manifolds with Sol structures. Only closed manifolds occur for this geometry. This leaves only H 3 to discuss. 10. The hyperbolic geometrization conjecture There are several models for H 3 . One of the often used ones is the upper half space model : z, r) ....

G.P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401--478.

....on a 3 manifold is a complete, locally homogeneous Riemannian metric. There are exactly eight such structures, namely the three constant curvature geometries, together with two further product geometries, H 2 R, S 2 R and three twisted product geometries, SL(2; R) Nil and Sol, c.f. 37] [33]. The decomposition of M is along certain essential 2 spheres and essential tori embedded in M . Recall that a 3 manifold M is irreducible if every embedded 2 sphere S 2 in M bounds a 3 ball B 3 M: The sphere decomposition [22] 24] states that M may be decomposed as a union of ....

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc., 15, (1983), 401-487.

....to the case of solvmanifolds. These are torus bundles over the circle in which the monodromy L 2 SL(2; Z) is Anosov (that is, if it is neither periodic nor does it fix a circle) Equivalent formulations are that jtrace(L)j 2 or that L has two irrational eigenvalues or that L is hyperbolic. See [Sco]. Proposition 4.1. If M L is a solvmanifold, then the only irreducible and weakly reducible Heegaard splitting is the standard genus three splitting. Proof: Suppose M L = A [ S B is a weakly reducible splitting. The main theorem of [CG] shows that, a maximal family of disjoint compressions of S ....

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401--487.

....over S 1 or over an interval or nite quotients of such bundles. The topology of graph manifolds is completely classi ed, c.f. 41] The Seifert bered spaces above each admit a geometric structure in the sense of Thurston, i.e. a complete locally homogeneous Riemannian metric, c.f. 39] [38]. Thus if has a strong geometrization as above, then admits a further decomposition by incompressible tori into domains, each of which has a complete geometric structure. Such a structure is called the geometrization of in the sense of Thurston. Not all 3 manifolds have such a geometric ....

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. vol. 15, (1983), 401-487.

....equipped with the given features. All of the cone points indicated have order 2. The quotient of the braid group by the Artin group is given in the third column, and in each case the group extension splits. sets in Euclidean space taken modulo nite groups. See chapter 13 of [18] section 2 of [17] or appendix A of [14] for more information. The braid groups of certain two orbifolds turn out to contain the Artin groups A(D n ) and A( D n ) as subgroups of very small index. The relevant orbifolds are among the simplest possible ones: one is the plane with a single cone point of order 2 ....

P. Scott. The Geometries of 3-Manifolds. Bull. Lond. Math. Soc., 15:401-487, 1983.

.... wide open and are central conjectures in 3 dimensional topology: The Generalized Poincar e Conjecture that 3 manifolds with finite fundamental group are spherical, and the Hyperbolization Conjecture that compact irreducible atoroidal 3 manifolds with incompressible boundary are hyperbolic, cf. [Sc]. Thurston s Hyperbolisation Theorem asserts that the Hyperbolization Conjecture holds for Haken manifolds, i.e. in the presence 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 ....

.... 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 to explain the geometrisation program in the special case of mapping tori of surface diffeomorphisms. In this case only six of the eight geometries are needed for geometrisation The case of sphere bundles is trivial. ....

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P. Scott, The geometries of 3-manifolds, London Math. Soc. 15 (1983), 401487.

....maps into a closed subset of the Teichmuller component. For the purpose of the section, we prove that if two 2 orbifolds are homotopy equivalent, then they are homeomorphic using harmonic di#eomorphisms. A good reference on orbifolds is the Chapter 5 of Thurston s note [19] or Scott s survey paper [18]. Also, Ratcli#e [17] devotes a chapter to orbifolds with geometric structures but for one with invariant Riemannian metrics. We would like to thank Yves Benoist, Karston Grove, Silvio Levi, Misha Kapovich, Hyuk Kim, Inkang Kim, John Millson, and Shmuel Weinberger for their helpful comments and ....

P. Scott. The geometries of 3-manifolds. Bull. London Math. Soc., 15:401--487, 1983.

....locally conformally flat manifolds. In this case G = SO(n 1; 1) H = P; X = S n where P is the maximal parabolic subgroup and S n is equipped with the standard conformal structure. Other examples which have been studied in great detail are the geometric structures on 3 manifolds (see [13] [12]) 1 INTRODUCTION 3 A (G; X) structure on a manifold M determines a developing map dev : M X where M M is the universal cover. This map is unique up to translation by an element of G. Moreover there is the holonomy map dev ] 1 (M) G unique up to conjugation such that dev ] ....

P. Scott. The geometries of 3-manifolds. Bull.London math.Soc., 15(5):401--487, 1983.

.... is a codimension 1 foliation on M which is transversely oriented and possessing the property that through every leaf there passes a transverse closed curve, then each leaf of F is quasi area minimizing [Ha] It follows that in the fibration M # S 1 , where M is a 3 dimensional Sol manifold [Sc], the fibre is a minimal surface in M for a suitable Riemannian metric on M . The algebraic counterpart to this fact is that in the split extension G = Z 2 o # Z, where # is a hyperbolic matrix in Gl 2 (Z) so Trace(#) 2 2) the combinatorial area of loops in a finite presentation for ....

.... class of pairs (G, H) up to quasiisometry adapted to pairs, where G = Z 2 oA Z, H is the normal Z 2 subgroup, and A is a hyperbolic matrix in Sl 2 (Z) To see this, note that G is a lattice in the 3 dimensional Lie group Sol; here Sol = R 2 o # R, where #(t) x, y) e t x, e t y) [Sc]. Since the the subgroup H G is commensurable with a lattice in the nilradical R 2 of Sol, it follows that (G, H) # (Sol, R 2 ) An open question in this connection is whether the pair (G, H) is rigid , i.e. whether every quasi isometry of G is adapted to the pair (G, H) Example 7.9. ....

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P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 405--487.

....knot complements among the 5 tetrahedra census manifolds as arising from either Berge knots or knots with less than or equal to 10 crossings. A Seifert bered space is small if the base space is S 2 and there are 3 or fewer critical bers. For more information about Seifert bered spaces see [Sco83] A construction of knots with small Seifert bered Dehn surgeries which generalizes Berge s construction was developed in [Dea96] Many of the knots that had not been identi ed in Hodgson s survey were knots arising from this construction. In fact, many of the knots were observed to be twisted ....

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401-487.

....(C 2 ) To get the cover, consider the cone type orbifold O obtained by attaching a disk D 1 to S along C 1 , and a disk D 2 with a cone point of order 2 around C 2 . Then O is an orbifold of hyperbolic type and hence admits a finite orbifold covering p 0 : O O where O is a manifold (see [Sc]) Now remove p Gamma1 0 (Interior(D 1 ) Interior(D 2 ) from the surface O, and call the resulting surface F . Then p : p 0 j F : F S is the covering with the required properties. Let m denote the number of boundary components of F which cover C 2 . Now define a complex L by ....

Scott, P., The geometries of 3-manifolds, Bull. London Math. Soc., vol. 15 (1983) p. 401--487.

....or the two eigenvalues are distinct. If it is minus the identity then S has a two fold cover which is a torus and for our purposes is essentially the same as when N is the identity. When the eigenvalues are distinct the manifold S admits a geometric structure of type Sol in the sense of Thurston [18]. The metrics obtained in that case include ones which are of 4 Bianchi type VI 0 . There is a polarized case, where reflections in the eigendirections of N are supposed to be isometries of the metric g fffi on the universal covering space. If N has a non standard Jordan form then, by passing ....

....N are supposed to be isometries of the metric g fffi on the universal covering space. If N has a non standard Jordan form then, by passing to a two fold cover if necessary, we can assume that these eigenvalues are equal to unity. The resulting manifold S admits a geometric structure of type Nil [18]. The metrics obtained in that case include those of Bianchi type II. Lemma 2.1 Let (M; g) be a non flat spacetime with local U(1) Theta U(1) symmetry having a symmetric constant mean curvature Cauchy hypersurface and satisfying the dominant and strong energy conditions. Then given any point p ....

Scott, P.: The geometries of 3-manifolds. Bull. London Math. Soc. 15, 401-487 (1983).

....There are three exceptional cases: D 2 Theta S 1 , T 2 Theta (0; 1) and a manifold 2 fold covered by the latter (interval bundle over Klein bottle) each have infinite volume complete E 3 structures but no finite volume geometric structure. For more details on the above see [30] or [41]. Case 2 of the easy cases is manifolds with Sol structures. Only closed manifolds occur for this geometry. This leaves only H 3 to discuss. 10. The hyperbolic geometrization conjecture There are several models for H 3 . One of the often used ones is the upper half space model : f(z; r) ....

G.P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401--478.

....non primitive otherwise. Non primitive elliptic element of order p is conjugate to a rotation F (z) exp(2 iq=p)z, where p, q 2 N are relatively prime, q 1 and q=p 1=2. Further information on Kleinian groups and hyperbolic geometry can be found in [A] Br1] F] GeM] KAG] M1] Mi] [S], Th] V] ZVC] A class for which the problem could be solved We return to our problem: when Gamma = hA; Bi PSL(2; C ) is discrete Consider the following class: G = Phi Gamma = hA; Bi j A; B and [A; B] ABA Gamma1 B Gamma1 are not strictly loxodromic Psi : We have the ....

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401--487.

....in X is (the image of) an isometric, bilipschitz, respectively quasi isometric embedding of the euclidean 2 plane into X. Convention 2.1 All flats, bilipschitz flats and quasi flats considered in the present paper are 2 dimensional. 2. 2 3 manifolds and their canonical decomposition We refer to [S] for information about the geometrization of 3 manifolds and a description of the eight 3 dimensional homogeneous geometries. Here we only recall a few facts which are important for this paper. A compact smooth 3 manifold P is called geometric if its interior admits a geometric structure, i.e. a ....

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-487.

....[20] Thus M k n is a K ( Gamma k n ; 1) space with infinite torsion free fundamental group, whence M k n is large [14, p. 91] Therefore large Seifert manifolds Sigma k n and M k n are homeomorphic. 2 Corollary 3.1. The groups Gamma k n are automatic for (n; k) 6= 4; 1) Proof: Recall [15], that the geometry on a Seifert fibred manifold is completely determined by the Euler characteristic of the base orbifold and the Euler number of the fibration. Calculating these values for manifolds Sigma k n , one can see that Sigma 1 3 is a spherical manifold, Sigma 2 3 is an euclidean ....

P.Scott, The geometries of 3-manifolds. Bull. London Math. Soc. 15 (1983) 401--487.

....Thurston s geometries are studied. It is shown that for a wide class of Coxeter groups there are finite index subgroups which uniformize hyperelliptic 3 manifolds. Let X 3 be one of three dimensional geometries IH 3 , IE 3 , S 3 , IH 2 Theta IE 1 or S 2 Theta IE 1 [6]. In the paper we consider 3 manifolds M 3 which are quotientspaces M 3 = X 3 = Gamma, where Gamma is a discrete group of isometries acting on X 3 without fixed points. A 3 manifold M 3 is said to be hyperelliptic, if there exists an isometric involution such that the quotient space ....

Scott P. The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1986) P. 401--487.

....OF REPRESENTATIONS OF DISCRETE SUBGROUPS OF SO(3,1) Michael Kapovich October 1, 1993 1. Introduction Let M be a closed hyperbolic 3 dimensional orbifold (see [T] [Sc] for definitions) ae 0 : 1 (M) Isom(H 3 ) be its holonomy representation. Denote the conjugacy class of ae 0 by [ae 0 ] In this paper we discuss whether for n = 4 the point [ae 0 ] is isolated in the space R( 1 (M) n) Hom( 1 (M) Isom(H n ) Isom (H n ) If [ae 0 ] is isolated, ....

Scott, P.: The geometries of 3- manifolds. Bull. London Math. Soc., 15 , 401-- 478 (1983)

....on R compatible with the original structure. Similarly the Riemann surface lamination can be viewed as a topological lamination with transversally continuous conformal structure on the leaves. 2.1. Orbifold laminations. In analogy with Thurston s notion of orbifolds (see Thurston [51] Scott [42] and also Satake s similar notion of V manifolds, 41] we may de ne an orbifold lamination to be a space for which every point has a neighborhood that is either homeomorphic to a standard product box neighborhood in a lamination, or to a quotient of such a box by a nite leaf preserving group ....

....is isolated means that all backward orbits z with z 0 2 X eventually escaping X hit a critical point. In other words, L s C is double branched at all points of L s 1 X , except the singular periodic point. Thus this map is an orbifold cover. See e.g. Thurston [51] or Scott [42] for a discussion of orbifolds and orbifold covers) Let q : e L s L s be the double covering associated to the orbifold structure of L s , e L s C. It follows that q : e L s ( C; X) is an orbifold universal cover. The group of deck translations for such a cover is ....

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-487.

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P.Scott, The geometries of 3-manifolds, Bull.London Math. Soc., 15(1983), 401-487.

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Scott, G.P., The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (1983), 401--487 66

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P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401--487.

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P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (1983) 401-487.

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P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc.. 15(1983), 404-487.

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P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983) 401-487.

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