By Benson Farb

The learn of the mapping classification team Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and workforce conception. This e-book explains as many vital theorems, examples, and strategies as attainable, speedy and at once, whereas even as giving complete information and protecting the textual content approximately self-contained. The publication is acceptable for graduate students.A Primer on Mapping type teams starts off via explaining the most group-theoretical houses of Mod(S), from finite new release by way of Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, relevant items and instruments are brought, similar to the Birman precise series, the complicated of curves, the braid workforce, the symplectic illustration, and the Torelli workforce. The ebook then introduces Teichmller area and its geometry, and makes use of the motion of Mod(S) on it to turn out the Nielsen-Thurston class of floor homeomorphisms. issues contain the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov idea, and Thurston's method of the class.

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**Sample text**

If S has nonempty boundary and has a hyperbolic metric, then S is isometric to a totally geodesic subspace of H2 . Similarly, if S has a Euclidean metric, then S is isometric to a totally geodesic subspace of the Euclidean plane E2 . 2 Let S be any surface (perhaps with punctures or boundary). If χ(S) < 0, then S admits a hyperbolic metric. If χ(S) = 0, then S admits a Euclidean metric. A surface endowed with a ﬁxed hyperbolic metric will be called a hyperbolic surface. A surface with a Euclidean metric will be called a Euclidean surface or ﬂat surface.

13 gives us a way to replace homeomorphisms with diffeomorphisms. We can also replace isotopies with smooth isotopies. In other words, if two diffeomorphisms are isotopic, then they are smoothly isotopic; see, for example, [30]. In this book, we will switch between the topological setting and the smooth setting as is convenient. For example, when deﬁning a map of a surface to itself (either by equations or by pictures), it is often easier to write down a homeomorphism than a smooth map. On the other hand, when we need to appeal to transversality, extension of isotopy, and so on, we will need to assume we have a diffeomorphism.

There is then a homotopy from γ to the standard straight-line representative of (p, q) ∈ π1 (T 2 ); indeed, the straight-line homotopy from the lift of γ to the straight line through (0, 0) and (p, q) is equivariant with respect to the group of deck transformations and thus descends to the desired homotopy. Now, if a closed curve in T 2 is simple, then its straight-line representative is simple. Thus we have the following fact. 5 The nontrivial homotopy classes of oriented simple closed curves in T 2 are in bijective correspondence with the set of primitive elements of π1 (T 2 ) ≈ Z2 .