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Geodesic flows on hyperbolic orbifolds, and universal orbifolds

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Lozano Imízcoz, María Teresa and Montesinos Amilibia, José María (1997) Geodesic flows on hyperbolic orbifolds, and universal orbifolds. Pacific Journal of Mathematics, 177 (1). pp. 109-147. ISSN 0030-8730

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Official URL: http://msp.org/pjm/1997/177-1/pjm-v177-n1-p08-p.pdf



Abstract

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


Item Type:Article
Uncontrolled Keywords:hyperbolic 2-orbifolds; branched coverings; singular Anosov flows
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:22212
Deposited On:05 Jul 2013 15:20
Last Modified:12 Dec 2018 15:13

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