Annales Academiæ Scientiarum Fennicæ

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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