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A Primer on Mapping Class Groups (PMS-49) by Dan Margalit, Benson Farb

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Chapter Eleven

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Teichmüller Geometry

Teichmüller space Teich(S) was defined in Chapter 10 as the space of hyperbolic structures on the surface S modulo isotopy. But Teich(S) parameterizes other important structures as well, for example, complex structures on S modulo isotopy and conformal classes of metrics on S up to isotopy.

We would like to have a way to compare different complex or conformal structures on S to each other. A natural way to do this is to search for a quasiconformal homeomorphism f : SS that is homotopic to the identity map and that has the smallest possible quasiconformal dilatation with respect to the two structures. Informally, ...

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