Asymptotic behavior of grafting rays

Abstract

In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmüller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmüller geodesics and lines of minima. We also show that the rays grafted along a weighted system of simple closed curves are at bounded distance from Teichmüller geodesics.

Iterated grafting and holonomy lifts of Teichmüller space

“…where θ = min i θ i , which coincides with our claim for n = 1 and yields the upper bound on l gr λ X (γ i ) for all n. For convenience (and to emphasize why their length estimate also works for δ i ) we will shortly summarize the proof given in [3]. To see the upper bound one constructs an embedded holomorphic disc in the universal cover of gr λ X , such that the imaginary axis is sent to a lift of the curve δ i .…”

“…(i) for n = 1 This is a length estimate obtained by Díaz and Kim (Proposition 3.4 in [3]). They show that…”

“…The most important property for us is that grafting rays have the same asymptotic behavior as Teichmüller geodesics. Namely, Díaz and Kim [3] have shown that for any X ∈ T (S) and weighted multicurve λ, the grafting ray gr tλ X is contained in a L-tube around the Teichmüller geodesic ray from X in direction λ, where L depends on X . This result then allows us to acquire Corollary 1.3 from the variant for grafting rays.…”

“…where θ = θ(l) is the angle corresponding to the standard hyperbolic collar on X (see the proof of Proposition 4.1 and [3] or [9] for more details). This cylinder carries a natural flat metric realizing it as C = γ × [−θ, t + θ ] such that γ × [0, t] ⊂ C is exactly the natural flat metric on the grafting cylinder.…”

Asymptoticity of grafting and Teichmüller rays

“…In [26] Masur proved that if is uniquely ergodic, then for any two initial surfaces X; Y 2 T g the Teichmüller rays determined by .X; / and .Y; / are asymptotic. The comparison of grafting rays and Teichmüller rays has been less explored, however a recent result along these lines (see also Díaz and Kim [8]) is the following "fellow-traveling" result in Choi, Dumas and Rafi [7]:…”