Abstract
Let Tg be the Teichmüller space of compact Riemann surfaces of genus g. Then Tg is the space of conformal structures on a fixed surface W modulo equivalence under conformal maps homotopic to the identity. The Teichmüller modular group [unk] is the group of all orientation preserving homeomorphisms of W onto itself modulo those which are homotopic to the identity. Each element of [unk] induces a biholomorphic map of Tg onto itself, and the present note outlines a proof of the converse statement: Every biholomorphic map of Tg onto itself is induced by an element of [unk]. It is first shown that every isometry of Tg with the Teichmüller metric arises from an element of [unk]. The Teichmüller metric is then shown to be the Kobayashi metric for Tg and hence invariant under biholomorphic maps.
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