Teichmueller空间的迭代嫁接与Holonomy提升
Iterated Grafting and Holonomy Lifts of Teichmueller space
摘要：
设$X$是闭双曲曲面，$λ, η$是在X上短的加权测地多重曲线.我们证明了在Teichmueller度量下沿$λ$和$η$的迭代嫁接与沿一条多重曲线的嫁接是接近的，这条多重曲线可以用$λ$和$η$明确给出.利用这一结果，我们研究了积分叠层$λ$的Teichmueller测地线$ρ_{X,λ}$的完整提升$gr_λρ_{X,λ}$，并证明了它们到测地线$ρ_{X,λ}$的Teichmueller距离都有界.我们得到类似的结果嫁接射线。最后，我们考虑了迭代嫁接序列$\gr_{nλ}X$的渐近行为，并证明了它们几何收敛到穿孔曲面。
英文摘要：
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 Teichmueller 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 $gr_λρ_{X,λ}$ of Teichmueller geodesics $ρ_{X,λ}$ for integral laminations $λ$ and show that all of them have bounded Teichmueller 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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