The asymptoticity of pairs of Teichm\"uller rays

Abstract

In this paper, we study the limit of Teichm\"uller distance between two points along a pair of Teichm\"uller rays. We obtain an explicit formula for the limiting Teichm\"uller distance when the vertical measured foliations of the quadratic differentials are finite sums of weighted simple closed curves and uniquely ergodic measures. The limit is expressed in terms of ratios of the corresponding moduli and the Teichm\"uller distance between the limit surfaces when the vertical measured foliations are absolutely continuous. Consequently, two Teichm\"uller rays are asymptotic if and only if their vertical measured foliations are modularly equivalent and their limit surfaces coincide, which implies a main result of Masur on the asymptoticity of Teichm\"uller rays determined by uniquely ergodic quadratic differentials. Furthermore, we prove that the infimum of the limiting Teichm\"uller distances can be represented in terms of the distance between the limit surfaces of the Teichm\"uller rays and the detour metric of their endpoints on the Gardiner-Masur boundary, when the initial points of the rays vary along the Teichm\"uller geodesics.