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Iterated grafting and holonomy lifts of Teichmüller space

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Abstract

Let X be a closed hyperbolic surface and λ, η be weighted geodesic multicurves which are short on X. We show that the iterated grafting along λ and η is close in the Teichmüller metric to grafting along a single multicurve which can be given explicitly in terms of λ and η. Using this result, we study the holonomy lifts gr λ ρ X,λ of Teichmüller geodesics ρ X,λ for integral laminations λ and show that all of them have bounded Teichmüller distance to the geodesic ρ X,λ. We obtain analogous results for grafting rays. Finally we consider the asymptotic behaviour of iterated grafting sequences gr nλ X and show that they converge geometrically to a punctured surface.

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  1. Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Sebastian W. Hensel

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  1. Sebastian W. Hensel
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Correspondence to Sebastian W. Hensel.

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Hensel, S.W. Iterated grafting and holonomy lifts of Teichmüller space. Geom Dedicata 155, 31–67 (2011). https://doi.org/10.1007/s10711-011-9577-0

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  • DOI: https://doi.org/10.1007/s10711-011-9577-0

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