Length Spectrums of Riemann Surfaces and the Teichmüller Metric

Suppose that T ( S 0 ) is the Teichmüller space of a compact Riemann surface S 0 of genus g > 1. Let d T (·, ·) be the Teichmüller metric of T ( S 0 ) and let d S (·, ·) be a metric of T ( S 0 ) defined by the length spectrums of Riemann surfaces. The author showed in a previous paper that d T and d S are topologically equivalent andd S ( τ 1 , τ 2 ) ⩽ d T ( τ 1 , τ 2 ) ⩽ 2 d S ( τ 1 , τ 2 ) + C ( τ 1 ) , where C (τ 1 ) is a constant depending on τ 1 . In this paper, it is shown that d T and d S are not metrically equivalent; that is, there is no constant C > 0 such thatd S ( τ 1 , τ 2 ) ⩽ d T ( τ 1 , τ 2 ) ⩽ C d S ( τ 1 , τ 2 )for all τ 1 and τ 2 in T ( S 0 ). 2000 Mathematics Subject Classification 32G15, 30C62, 30C75.