Iterated grafting and holonomy lifts of Teichmüller space

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 Teichmül-ler 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.

Tame and relatively elliptic $\mathbb{CP}^1$-structures on the thrice-punctured sphere

“…Pick a simple closed geodesic and let σ n be the structure obtained by grafting along it n times. For n → ∞ we obtain a punctured surface Σ with two punctures (possibly disconnected if the geodesic is separating) which is endowed with a complex projective structure in P • (S) (see [Hen11]). However it is not tame, and peripherals have hyperbolic holonomy, so it is not in P (Σ).…”

“…For a more extreme behavior, take a closed hyperbolic surface, and graft it along a geodesic pants decomposition infinitely many times. The underlying complex structure is being pinched along each pants curve, and in the limit the structure decomposes into a collection of thrice-punctured spheres (see [Hen11,§6]). There, punctures do not give rise to well-defined ideal points; indeed, the structure has hyperbolic peripheral holonomy, hence it is not tame (by Lemma 3.1.3).…”