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Quasi–Geodesic Flows Quasi{geodesic flows Maciej Czarnecki UniwersytetL´odzki,Katedra Geometrii ul. Banacha 22, 90-238L´od´z,Poland e-mail: [email protected] 1 Contents 1 Hyperbolic manifolds and hyperbolic groups 4 1.1 Hyperbolic space . 4 1.2 M¨obiustransformations . 5 1.3 Isometries of hyperbolic spaces . 5 1.4 Gromov hyperbolicity . 7 1.5 Hyperbolic surfaces and hyperbolic manifolds . 8 2 Foliations and flows 10 2.1 Foliations . 10 2.2 Flows . 10 2.3 Anosov and pseudo{Anosov flows . 11 2.4 Geodesic and quasi{geodesic flows . 12 3 Compactification of decomposed plane 13 3.1 Circular order . 13 3.2 Construction of universal circle . 13 3.3 Decompositions of the plane and ordering ends . 14 3.4 End compactification . 15 4 Quasi{geodesic flow and its end extension 16 4.1 Product covering . 16 4.2 Compactification of the flow space . 16 4.3 Properties of extension and the Calegari conjecture . 17 5 Non{compact case 18 5.1 Spaces of spheres . 18 5.2 Constant curvature flows on H2 . 18 5.3 Remarks on geodesic flows in H3 . 19 2 Introduction These are cosy notes of Erasmus+ lectures at Universidad de Granada, Spain on April 16{18, 2018 during the author's stay at IEMath{GR. After Danny Calegari and Steven Frankel we describe a structure of quasi{ geodesic flows on 3{dimensional hyperbolic manifolds. A flow in an action of the addirtive group R o on given manifold. We concentrate on closed 3{dimensional hyperbolic manifolds i.e. having locally isometric covering by the hyperbolic space H3. Such flow is quasi{geodesic if every flowline of the lifted flow is a quasi{geodesic in H3. Quasi{geodesic flows are probably the only reasonable metric objects which are foliations of hyperbolic 3{manifolds which do not carry neither geodesic foliation in any dimension (Zeghib) nor quasi{geodesic foliations of dimension 2 (Fenley). We start with foundations of hyperbolic manifolds, hyperbolic groups and their asympotic properties. Then we describe shortly notions for folia- tions and flows mentioning their type like (quasi){isometric, (quasi){geodesic, (pseudo){Anosov etc. We take care of topology of the plane focusing on decompositions into continua. For such decompositions we construct a circular order in the set of their topological ends. Since after Calegari any quasi{geodesic flow on a hyperbolic 3 manifold has the Hausdorff flowspace (i.e. the plane) we are able to apply decompo- sitions for a compactification the flowspace by ends of flowlines making it a closed disc. Finally, we study new results of Frankel on extension properties of quasi{ geodesic flow. In particular we take a look for its proof of Calegari conjecture stating that such flows need to have closed orbits. At the end, we add some remarks about (quasi){geodesic flows in non- compact case. I would like to thank Prof. Antonio Mart´ınezL´opez and Prof. Joaqu´ın P´erezMu~nozfor organizing my visit to IEMath{GR and their overall hospi- tality. 3 1 Hyperbolic manifolds and hyperbolic groups 1.1 Hyperbolic space [BP] n+1 Definition 1.1. In the Minkowski space R1 with the Lorentz form h:j:i given by hxjyi = −x0y0 + x1y1 + : : : xnyn we define the n{dimensional hyperbolic space as n n+1 H = x 2 R1 j hxjxi = −1; x0 > 0 n ? On every tangent space TxH = x (? means Lorentz orthogonality) the Lorentz form in an inner product making Hn a Riemannian manifold with constant sectional curvature equal −1. The most useful models of Hn are (B) the ball model in the unit ball Bn ⊂ Rn together with standard Riemannian metric multiplied at x by 4 , (1−kxk2)2 (Π) the half{space model in the open upper half{space Πn;+ ⊂ Rn 1 together with standard Riemannian metric multiplied at x by 2 . xn In the ball and the half{space models complete geodesci are circle arcs or rays perpendicular (in the Euclidean sense) to their topological boundaries Sn−1 and Rn−1 × f0g [ f1g, respectively. Definition 1.2. We say that unit speed geodesic rays in Hn are asymptotic iff the distance between them is bounded. We associate to Hn its ideal boundary Hn(1) consisting of asymptoticity class of geodesics. The cone topology in Hn [ Hn(1) is such that ideal points are close to each other if representing geodesic make small angle at their common origin, and close to ordinary points lying on such geodesics far awy from the origin. The Hadamard{Cartan theorem states that every n{dimensional Hada- mard manifold i.e. connected, simply connected Riemannian n{manifold which is complete and nonpositively curved is diffeomorphic to Rn and its ideal boundary is defined analogously. 4 1.2 M¨obiustransformations [R] Definition 1.3. We call an inversion in a sphere S(c; r) ⊂ Rn a transfor- mation given by x − c ι (x) = r2 + c: c;r kx − ck2 After obvious extension we could treat inversions as transformations of Sn including reflections in hyperplanes. Inversions imitate properties of standard reflection but they are only con- formal i.e. preserve angles instead of distances. Definition 1.4. The M¨obiusgroup M¨obn is a group generated by all the inversions in (n − 1){dimensional subspheres of Sn. 1.5. Orientation preserving M¨obiustransformations of the upper half plane Π2;+ are simply homographies az + b z 7! cz + d of the complex plane C with real coefficients and determinant equal 1. 1.3 Isometries of hyperbolic spaces [BP], [R] 1.6. Isometries of Hn in the hyperboloid model are easily observed as n + T Isom(H ) = O (1; n) = A 2 Mn+1;n+1 j AJA = J where J is diagonal matrix with entries −1; 1;:::; 1. Thus matrices from O+(1; n) preserve the Lorentz form and upper sheet of hyperboloid hxjxi = −1. In the ball model isometries are compositions of an orthogonal trans- formation of Rn with inversion in sphere orthogonal to Sn−1 or orthogonal transformations. Similarly, in the half{space model we compose an inversion or identity with an orthogonal transformation pereserving the last coordi- nate. Thus isometries of Hn have canonical extension to the ideal boundary as M¨obiustransformations. 5 + 1.7. In the case n = 2, the most popular description of Isom (H2) comes from the half{plane model where isometries are described as homographies (cf. 1.5). Thus we obtain the full group of orientation preserving isometries as projectivization P SL(2; R) which is doubly covered by SL(2; R) (opposite matrices identify). Definition 1.8. An isometry f of Hn is elliptic if it has a fixed point in Hn, parabolic if it has no fixed points in Hn and fixes a unique fixed point of Sn−1 = Hn(1) hyperbolic if it has no fixed points in Hn and fixes two points of Sn−1 = Hn(1). 1.9. Isom(Hn) is a Lie group. For n = 2 its Lie algebra serving as tangent space to P SL(2; R) is sl(2; R) consisting of matrices of zero trace. 1.10. The group P SL(2; R) acts transitively on the unit tangent bundle T 1H2 of H2 and stabilizer of any vector is trivial then P SL(2; R) could be identified (but not canonically) with T 1H2. 1.11. Since isometries of Hn act on its ideal boundary Sn−1, the group + P SL(2; R) could treated as a subgroup of Homeo (S1) of orientation pre- serving homeomorpshisms of the circle. A subgroup of Homeo(S1) is a M¨obius{likegroup if every its element is (topologically) conjugate to a M¨obiustransformation. 1.12. Finally, observe that all M¨obiustransformations could be described in terms of isometries of Hn. + M¨obn ' O (1; n + 1) In fact, if ι is an inversion in an (n − 1){sphere S ⊂ Sn then there is a unique n{sphere S~ ⊂ Rn+1 which contains S and is orthogonal to Sn and then the inversion ~ι in S~ is extension of ι. Think of Sn as ideal boundary of hyperbolic space Hn+1 in the ball model. Thus ~ι is an hyperbolic reflection and could be treated as element of O+(1; n+1). The converse implication fol- lows that every isometry of Hn+1 is a composition of at most n+2 hyperbolic reflections. 6 1.4 Gromov hyperbolicity [BH] Definition 1.13. A geodesic in a metric space (X; d) is an isometric embed- ding of an interval I ⊂ R in to X i.e. such a map c : I ! X that d(c(t); c(t0)) = jt − t0j for any t; t0 2 I: In a geodesic metric space every two points could joined by a geodesic. Definition 1.14. We say that a geodesic metric space X is δ{hyperbolic (in the sense of Gromov) if in any triangle made of geodesics between points x; y; z 2 X the geodesic segment [y; z] in the δ{neighbourhood of [x; y][[x; z]. The above definition does not need assumption on geodesicity. Gromov hyperbolic condition could formulated using only distances between quadru- ples of points. p Example 1.15. Hn is (ln(1 + 2)){hyperbolic. 1.16. The definition of the ideal boundary @X of a Gromov hyperbolic space X is based on asymptotic geodesics as in case of Hadamard manifolds but the topology in X [ @X comes from the uniform convergence of rays on compact sets. A special role in this theory plays the notion "quasi{geodesic" which more flexible and more invariant. Even in case of Riemannian manifolds we know that "(totally) geodesic" is fragile and could be lost even with small perturbations.
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