On behavior of pairs of Teichm\"{u}ller geodesic rays

TL;DR

This paper explicitly calculates the limit of the Teichmüller distance between two Jenkins-Strebel geodesic rays sharing an endpoint and identifies conditions for their asymptotic behavior, extending Farb and Masur's results.

Contribution

It provides explicit limit values and asymptotic conditions for pairs of Teichmüller geodesic rays determined by Jenkins-Strebel differentials.

Findings

Explicit limit value of Teichmüller distance between rays

Condition for asymptotic behavior of rays

Extension of Farb and Masur's results

Abstract

In this paper, we obtain the explicit limit value of the Teichm\"{u}ller distance between two Teichm\"{u}ller geodesic rays which are determined by Jenkins-Strebel differentials having a common end point on the augmented Teichm\"{u}ller space. Furthermore, we also obtain a condition under which these two rays are asymptotic. This is similar to a result of Farb and Masur.