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

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Chapter One

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Curves, Surfaces, and Hyperbolic Geometry

A linear transformation of a vector space is determined by, and is best understood by, its action on vectors. In analogy with this, we shall see that an element of the mapping class group of a surface S is determined by, and is best understood by, its action on homotopy classes of simple closed curves in S. We therefore begin our study of the mapping class group by obtaining a good understanding of simple closed curves on surfaces.

Simple closed curves can most easily be studied via their geodesic representatives, and so we begin with the fact that every surface may be endowed with a constant-curvature ...

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