Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 221-234

COMPARISON BETWEEN TEICHMÜLLER METRIC AND LENGTH SPECTRUM METRIC UNDER PARTIAL TWISTS

Jun Hu and Francisco G. Jimenez-Lopez

Brooklyn College of CUNY, Department of Mathematics, Brooklyn, NY 11210, U.S.A.
and Graduate Center of CUNY, Ph.D. Program in Mathematics
365 Fifth Avenue, New York, NY 10016, U.S.A.; junhu 'at' brooklyn.cuny.edu, JHu1 'at' gc.cuny.edu

Cinvestav-IPN, Department of Mathematics
3 Libramiento Norponiente 2000, Real De Juriquilla, 76230 Santiago de Queretaro, Queretaro, Mexico; jimenez 'at' math.cinvestav.edu.mx

Abstract. Let d_L and d_T denote, respectively, the length spectrum metric and the Teichmüller metric on the Teichmüller space T(S₀) of a Riemann surface S₀. Wolpert showed that d_L(τ,τ^*) ≤ d_T(τ,τ^*) for any two points τ and τ^* in T(S₀). If S₀ is a hyperbolic Riemann surface with non-elementary Fuchsian group, then there are two sequences {τ_n} and {τ_n^*} of points in T(S₀) such that d_L(τ_n,τ_n^*) → 0 as n &rarr ∞, but d_T(τ_n,τ_n^*) ≥ b for some positive constant b and any n. This property was proved in [11] for any hyperbolic compact Riemann surface and in [13] for any hyperbolic one with non-elementary Fuchsian group. It is further shown in [13] that the two sequences can be modified to keep d_L(τ_n,τ_n^*) → 0 but have d_T(τ_n,τ_n^*) → ∞ as n → ∞. For all these results, each τ_n^* is constructed from a Riemann surface τ_n by taking a number of Dehn twists (full twists) along a closed curve on τ_n. When τ_n^* is constructed from τ_n through a Dehn twist, one can use the maximal dilatation of a quasiconformal self map of τ_n to control and compare d_L(τ_n,τ_n^*) and d_T(τ_n,τ_n^*). But when τ_n^* is constructed from τ_n in a similar pattern but with partial twist, the method of using a self map of τ_n to control and compare d_L(τ_n,τ_n^*) and d_T(τ_n,τ_n^*) fails. In this paper, we show how to control and compare d_L(τ,τ^*) and d_T(τ,τ^*) under such partial twists, which enables us to obtain continuous versions of the results of [11] and [13] by using partial twists to connect the points τ_n^*.

2010 Mathematics Subject Classification: Primary 30F60.

Key words: Teichmüller metric, length spectrum metric, partial twist, earthquake map.

Reference to this article: J. Hu and F.G. Jimenez-Lopez: Comparison between Teichmüller metric and length spectrum metric under partial twists. Ann. Acad. Sci. Fenn. Math. 41 (2016), 221-234.

Full document as PDF file

doi:10.5186/aasfm.2016.4121