Annales Academić Scientiarum Fennicć
Mathematica
Volumen 33, 2008, 159-170

TOPOLOGICAL EQUIVALENCE OF METRICS IN TEICHMÜLLER SPACE

Lixin Liu, Zongliang Sun and Hanbai Wei

Zhongshan (Sun Yat-sen) University, Department of Mathematics
Guangzhou 510275, P.R. China; mcsllx 'at' mail.sysu.edu.cn

Zhongshan (Sun Yat-sen) University, Department of Mathematics
Guangzhou 510275, P.R. China

Jiujiang Vocational & Technical College
1188 Shili Road, Jiujiang, Jiangxi, P.R. China

Abstract. For d_T, d_L and d_{p_i}, i =1,2, the Teichmüller metric, the length spectrum metric and the Thurston's pseudo-metrics on Teichmüller space T(X), we first give some estimations of the above (pseudo)metrics on the thick part of T(X). Then we show that there exist two sequences {\tau_n}_n=1^\infty and {~\tau_n}_n=1^\infty in T(X), such that as n -> \infty, d_L (\tau_n,~\tau_n) -> 0, d_{P_1} (\tau_n,~\tau_n) -> 0, d_{P_2} (\tau_n, ~\tau_n) -> 0, while d_T (\tau_n,~\tau_n) -> \infty. As an application, we give a proof that for certain topologically infinite type Riemann surface X, d_L, d_{P_1} and d_{P_2} are not topologically equivalent to d_T on T(X), a result originally proved by Shiga [18]. From this we obtain a necessary condition for the topological equivalence of d_T to any one of d_L, d_{P_1} and d_{P_2} on T(X).

2000 Mathematics Subject Classification: Primary 32G15, 30F60, 32H15.

Key words: Length spectrum, Teichmüller metric, Thurston's pseudo-metrics.

Reference to this article: L. Liu, Z. Sun and H. Wei: Topological equivalence of metrics in Teichmüller space. Ann. Acad. Sci. Fenn. Math. 33 (2008), 159-170.

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