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- W2054210908 abstract "We consider the action of the Teichmuller modular group on the Teichmuller space of a topologically infinite Riemann surface. For a Riemann surface S, the Teichmuller space TðSÞ is the set of all equivalence classes of the pair ð f ; sÞ, where f : S ! Ss is a quasiconformal homeomorphism of S onto another Riemann surface Ss of a complex structure s. Two pairs ð f1; s1Þ and ð f2; s2Þ are considered to be equivalent if s1 1⁄4 s2 and f2 f 1 1 is isotopic to a conformal map. Here the isotopy is considered to be relative to the boundary at infinity. A distance between p1 1⁄4 1⁄2 f1; s1 and p2 1⁄4 1⁄2 f2; s2 in TðSÞ is defined by dðp1; p2Þ 1⁄4 log KðhÞ for an extremal quasiconformal homeomorphism h whose maximal dilatation KðhÞ is minimal in the isotopy class of f2 f 1 1 . Then d becomes a complete metric on TðSÞ, which is called the Teichmuller distance. The Teichmuller modular group ModðSÞ of S is a group of the isotopy classes of quasiconformal automorphisms of S. An element g of ModðSÞ acts on TðSÞ in such a way that 1⁄2 f ; s 7! 1⁄2 f g ; s , where g also denotes a representative of the isotopy class. It is evident from definition that ModðSÞ acts on TðSÞ isometrically with respect to the Teichmuller distance. In the case that TðSÞ is finite dimensional (equivalently S is of analytically finite type), ModðSÞ acts on TðSÞ properly discontinuously and the orbit of any point p A TðSÞ is discrete. However, when TðSÞ is infinite dimensional, these are not always true. See recent works [4], [5] and a monograph [6, Chap. 10]. For finite dimensional Teichmuller spaces, Bers [2] classified the elements of ModðSÞ by certain analytic criteria in comparison with Thurston’s topological classification. This can be extended to infinite dimensional Teichmuller spaces in the same way. For example, an element g A ModðSÞ is elliptic if g has a fixed point in TðSÞ, and parabolic if inf dðp; gðpÞÞ 1⁄4 0 where the infimum is taken over all points p in TðSÞ. An elliptic element is realized as a conformal automorphism of the Riemann surface corresponding to the fixed point of g. A" @default.
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- W2054210908 date "2003-10-01" @default.
- W2054210908 modified "2023-09-27" @default.
- W2054210908 title "The infinite direct product of Dehn twists acting on infinite dimensional Teichmüller spaces" @default.
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- W2054210908 doi "https://doi.org/10.2996/kmj/1073670609" @default.
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