Length orders of curves on hyperbolic surfaces

YMSC Topology Seminar

Organizers：

陈伟彦、高鸿灏、黄意、林剑锋、孙巍峰

Speaker：

Binbin XU 徐彬斌 (Nankai)

Time：

Mon., 10:00-11:00 am, Mar.17, 2025

Venue：

B725, Shuangqing Complex Building A

Online：

Zoom Meeting ID: 405 416 0815

Password: 111111

Title:

Length orders of curves on hyperbolic surfaces

Abstract：

Let S be an oriented topological surface of finite type. Given a hyperbolic metric on S, there is an order among all homotopy classes of curves on S induced by comparing the lengths of their geodesic representatives. We call it the length order of curves induced by the given hyperbolic metric. In a collaboration with Hugo Parlier and Hanh Vo, we show that given any pair of distinct points in the Teichmüller space T(S) of S, there exist two homotopy classes of curves on S, such that the two given points of T(S) induce different length order on them. Hence the length orders of curves on S can determine points in T(S). This result is a generalization of a result of Greg McShane and Hugo Parlier. We also study the homotopy classes of curves whose length order never changes as the hyperbolic metric varies, and introduce a way to construct such examples.