By Benson Farb

ISBN-10: 0691147949

ISBN-13: 9780691147949

The research of the mapping category workforce Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and workforce idea. This publication explains as many vital theorems, examples, and methods as attainable, speedy and at once, whereas while giving complete info and preserving the textual content approximately self-contained. The e-book is acceptable for graduate students.The e-book starts by way of explaining the most group-theoretical homes of Mod(S), from finite iteration through Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, primary gadgets and instruments are brought, corresponding to the Birman specified series, the complicated of curves, the braid team, the symplectic illustration, and the Torelli team. The booklet then introduces Teichmüller area and its geometry, and makes use of the motion of Mod(S) on it to end up the Nielsen-Thurston class of floor homeomorphisms. subject matters contain the topology of the moduli house of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov concept, and Thurston's method of the category.

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**Extra info for A Primer on Mapping Class Groups (Princeton Mathematical)**

**Example text**

8, α intersects β in exactly one point x. It cannot be that the axes of the hyperbolic isometries corresponding to α and β share exactly one endpoint at ∂H2 , because this would violate discreteness of the action of π1 (S) on H2 ; indeed, in this case the commutator of these isometries is parabolic and the conjugates of this parabolic isometry by either of the original hyperbolic isometries have arbitrarily small translation length. Further, these axes cannot share two endpoints on ∂H2 , for then the corresponding hyperbolic isometries would have the same axis, and so they would have to have a common power φ (otherwise the action of π1 (S) on this axis would be nondiscrete).

Since ∂D embeds under the covering map, φ(∂D) ∩ ∂D is either empty or all of ∂D (in the case that φ is the identity). By the Jordan curve theorem, we then see that either φ(D) or φ−1 (D) must be contained in D. Now, by the Brouwer fixed point theorem, φ has a fixed point, which is a contradiction, unless φ is the identity. ✷ We give two proofs of the bigon criterion. One proof uses hyperbolic geometry and one proof uses only topology. We give both proofs since each of the techniques will be important later in this book.

In particular it makes sense to write ˆi(a, b) for a and b the free homotopy classes (or homology classes) of closed curves α and β. The most naive way to count intersections between homotopy classes of closed curves is to simply count the minimal number of unsigned intersections. This idea is encoded in the concept of geometric intersection number. The geometric intersection number between free homotopy classes a and b of simple closed curves in a surface S is defined to be the minimal number 30 CHAPTER 1 of intersection points between a representative curve in the class a and a representative curve in the class b: i(a, b) = min{|α ∩ β| : α ∈ a, β ∈ b}.

### A Primer on Mapping Class Groups (Princeton Mathematical) by Benson Farb

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