Cusp geometry of fibered 3-manifolds

Details Cusp geometry of fibered 3-manifolds Cusp geometry of fibered 3-manifolds

2011-08-29

arxiv.org/abs/1108.5748v1

Mathematics

Geometric Topology

34 pages, 9 figures

Scientific paper

Let F be a surface and suppose that \phi: F -> F is a pseudo-Anosov homeomorphism fixing a puncture p of F. The mapping torus M = M_\phi\ is hyperbolic and contains a maximal cusp C about the puncture p. We show that the area and height of the cusp torus bounding C are equal, up to explicit multiplicative error, to the stable translation distance of \phi\ acting on the arc complex A(F,p). Our proofs rely on elementary facts about the hyperbolic geometry of pleated surfaces. In particular, we do not use any deep results in Teichmueller theory, Kleinian group theory, or the coarse geometry of A(F,p). A similar result holds for quasi-Fuchsian manifolds N = (F x R). In that setting, we prove a combinatorial estimate on the area and height of the cusp annulus in the convex core of N and give explicit multiplicative and additive errors.

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