The asymptoticity of pairs of Teichmüller rays

TL;DR

The paper analyzes the asymptotic behavior of Teichmüller distance between two points on a pair of Teichmüller rays, proving an explicit limiting formula when the vertical foliations decompose into weighted simple closed curves and uniquely ergodic measures. The main result expresses the limit as the maximum of a moduli-based term $\frac{1}{2}\log\max_j\{m'_j/m_j, m_j/m'_j\}$ and the Teichmüller distance between the corresponding limit surfaces $d_{ar{\mathcal{T}}}(X_\infty,Y_\infty)$, provided the vertical foliations are absolutely continuous; otherwise the limit is infinite. The authors construct the limit surfaces as unions of half-plane pieces via a rectangular decomposition and a corresponding half-plane gluing, establishing convergence in the pointed Gromov–Hausdorff sense. They also connect the infimum of the limiting distance to the detour metric on the Gardiner–Masur boundary, yielding a sharp link between asymptotic geometry in Teichmüller space and boundary horofunction structures. Together, these results generalize prior work on Jenkins–Strebel rays and recover Masur’s asymptotic theorem for uniquely ergodic foliations, with a robust framework for measuring asymptotic distances through moduli, limit surfaces, and boundary metrics.

Abstract

In this paper, we study the limit of Teichmüller distance between two points along a pair of Teichmüller rays. We obtain an explicit formula for the limiting Teichmüller 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 Teichmüller distance between the limit surfaces when the vertical measured foliations are absolutely continuous. Consequently, two Teichmüller 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 Teichmüller rays determined by uniquely ergodic quadratic differentials. Furthermore, we prove that the infimum of the limiting Teichmüller distances can be represented in terms of the distance between the limit surfaces of the Teichmüller rays and the detour metric of their endpoints on the Gardiner-Masur boundary, when the initial points of the rays vary along the Teichmüller geodesics.

Figures (5)

• Figure 1: The rectangular decomposition of a Riemann surface of genus $2$ by the first return map on a horizontal segment $\tau$. The rectangles $R_2$ and $R_3$ are formed by splitting a rectangle containing singularities on both vertical edges. The rectangles $R_7$ and $R_8$ are the same case.
• Figure 2: Glue the rectangles along their edges containing singularities. The Riemann surface is divided into two subsurfaces which depend on the connected subgraph of the finite critical graph.
• Figure 3: The half-plane surface is formed by gluing two half planes and two semi-infinite cylinders along the edges of the admissible metric graph.
• Figure 4: The two half-plane surfaces with the same admissible metric graph are obtained by gluing one half plane and two semi-infinite cylinders.
• Figure 5: Pick a point on each half-infinite critical trajectory such that all singularities are between the picked points for each $R_i$ and $R_i^\prime$.

Theorems & Definitions (32)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3: Masur Mas1980
• Proposition 1.4
• Example 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Remark
• Proposition 3.4