Limit points of lines of minima in Thurston’s boundary of Teichmüller space

Abstract

Given two measured laminations µ and ν in a hyperbolic surface which fill up the surface, Kerckhoff [8] defines an associated line of minima along which convex combinations of the length functions of µ and ν are minimised. This is a line in Teichmüller space which can be thought as analogous to the geodesic in hyperbolic space determined by two points at infinity. We show that when µ is uniquely ergodic, this line converges to the projective lamination [µ], but that when µ is rational, the line converges not to [µ], but rather to the barycentre of the support of µ. Similar results on the behaviour of Teichmüller geodesics have been proved by Masur [9].

Asymptotic behavior of grafting rays

“…In this paper, we study a third kind of curves, grafting rays, and their convergence properties, and prove analogous results as in [15] for Teichmüller geodesics and in [4] for lines of minima.…”

“…The proofs follow the similar lines as in [3,4,15]. To prove the convergence properties, we need to estimate the length of simple closed curves along the ray.…”

“…Under this condition, the length of any other curve α can be approximated up to a bounded additive error by the length of a polygonal curve. We describe here this estimate, which is mainly obtained in [4]. Let (S, σ ) be a hyperbolic surface and {γ 1 , .…”

“…The convergence properties of these lines towards the Thurston boundary has been studied by Díaz-Series [4], and their comparison with Teichmüller geodesics by Choi-Rafi-Series [3].…”

Limits of quasi-Fuchsian groups with small bending

“…Thus the length bound in Proposition 5.9 fails, in other words, l α 1 → ∞ and the groups G n diverge. (For the limiting behaviour along lines of minima, see [5]. )…”