Publication: Geodesic flows on hyperbolic orbifolds, and universal orbifolds
Full text at PDC
Lozano Imízcoz, María Teresa
Advisors (or tutors)
Pacific Journal of Mathematics
The authors discuss a class of flows on 3-manifolds closely related to Anosov flows, which they call singular Anosov flows. These are flows which are Anosov outside of a finite number of periodic "singular orbits'', such that each singular orbit has a Poincaré section on which the first return map has an "n-pronged singularity'' for some n≥1, n≠2. If only 1-pronged singularities occur the flow is called V-Anosov; the authors observe, for example, that the geodesic flow of a compact, hyperbolic 2-orbifold is V-Anosov. The main theorem is that every closed 3-manifold has a singular Anosov flow. The theorem is proved by constructing a certain link L in the 3-sphere such that L is a universal branching link, so every closed 3-manifold M is a branched cover of the 3-sphere branched over L, and L is the set of singular orbits of some V-Anosov flow on S3, so the lifted flow is a singular Anosov flow on M. In the literature, a singular Anosov flow whose n-pronged singularities always satisfy n≥3 is called pseudo-Anosov. The main theorem should be contrasted with the fact that an Anosov or pseudo-Anosov flow can only occur on an aspherical 3-manifold—an irreducible 3-manifold with infinite fundamental group. The literature contains many constructions of Anosov and pseudo-Anosov flows, but it remains unknown exactly which aspherical 3-manifolds support such flows
D.V. Anosov, Geodesic Flows on Closed Riemannian Manifolds with Negative Cur- vature, Proceeding of the Steklov Institute of Math., 90 (1967), English translations Amer. Math. Soc., (1969). J. Birman, Braids, links, and mapping class groups, Study 82, Princeton Univ. Press, 1976. F. Bonahon and L. Siebenmann, The classification of Seifert fibred 3-orbifolds, in Low Dimensional Topology. LMS. Lecture Note Series 95 (1985), 19-85. G. Burde and H. Zieschang, Knots, de Gruiter Studies in Mathematics, 5. Walter de Gruyter, Berlin, New York, 1985. J.P. Christy, Anosov Flows on three-Manifolds, Ph.D. Thesis. University of Cali- fornia, Berkeley, 1984. P. Eberlein, Geodesic flows on negatively curved manifolds I, Ann. of Math. (2), 95 (1972), 58-82. A. Fathi, F. Laudenbach and V. Po'enaru, Travaux de Thurston sur les surfaces, Asterisque 66-67. Soci'et'e Math'ematique de France (1979). D. Fried, Transitive Anosov flows and pseudo-Anosov maps, Topology, 22 (1983), 299-303. E. Ghys, Flots d'Anosov dont les feuilletages stables sont differentiables, Ann. Sci. Ec. Norm. Super., IV. Ser., 20 (1987), 251-270. S. Goodman, Dehn surgery on Anosov flows, in Geometric Dynamics, SLN 1007, Springer, New York, (1983). MR1691596 J. Hempel, Construction of Orientable 3-manifold, in Topology of 3-manifold and related topics, Prentice-Hall (1962), 207-212. H. Hilden, M.T. Lozano and J.M. Montesinos, The Whitehead link, the Borromean rings and the knot 946 are universal, Collect. Math., 34 (1983), 19-28. H. Hilden, M.T. Lozano and J.M. Montesinos, On knots that are universal, Topology, 24 (1985), 499-504. H. Hilden, M.T. Lozano, J.M. Montesinos and W. Whitten, On universal groups and three-manifold, Invent. Math., 87 (1987), 441-456. M. Handel and W. Thurston, Anosov flows on New three Manifolds, Invent. Math., 59 (1980), 95-103. G.A. Margulis, Y-Flows on three-dimensional manifolds, Appendix to Anosov-Sinai: "Some smooth ergodic systems''. Uspekhi Math. Nauk, 22 (1967), 107-172; Russian Math. Surveys, 22 (1967), 103-168. J.M. Montesinos, Variedades de Seifert que son recubridores cíclicos ramificados de dos hojas, Bol. Soc. Mat. Mexicana, 18 (1973), 1-32. J.M. Montesinos, Revêtement ramifies de noeuds, espaces fibrés de Seifert et scindements de Heegard, Publ. Orsay, (1980). J.M. Montesinos, Classical Tessellations and Three-Manifolds, Springer-Verlag (1987). J.M. Montesinos, Una nota a un teorema de Alexander, Revista Mat. Hisp.-Amer. (4), 32(1972), 167-187. L. Mosher, The classification of Pseudo-Anosovs, Low dimensional Topology and Klenian Groups. LMS LNS112. Cambridge University Press, 1986. J. Palis and W. de Melo, Geometric theory of Dinamical Systems, Springer-Verlag, New York (1982). D. Rolfsen, Knots and Links, Publish or Perish, Berkeley (1976). W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. AMS, 19 (1988), 417-431. W. Thurston, Universal links, preprint, 1982. cf. R.F. Williams, The "DA'' maps of Smale and structural stability, Proc. on Sym. in Pure Math., XIV (1970), 329-334.