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

By Benson Farb

The learn of the mapping category crew Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and crew idea. This e-book explains as many vital theorems, examples, and strategies as attainable, fast and without delay, whereas even as giving complete info and retaining the textual content approximately self-contained. The e-book is acceptable for graduate students.The ebook starts by way of explaining the most group-theoretical homes of Mod(S), from finite iteration by means of Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, primary gadgets and instruments are brought, reminiscent of the Birman specified series, the complicated of curves, the braid team, the symplectic illustration, and the Torelli crew. The e-book then introduces Teichmüller 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. themes comprise the topology of the moduli house of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov thought, and Thurston's method of the type.

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We claim that ψ(α) = α. We know that ψ(α) is some lift of α. Since α is simple, all of its lifts are disjoint and no two lifts of α have the same endpoints in ∂H2 . Thus, ψ(α) and α are disjoint and have distinct endpoints. Now, we know that ψ n−1 (ψ(α)) = φ(α) = α. Since the fixed points in ∂H2 of ψ n−1 are the same as the endpoints of α, the only way ψ n−1 (ψ(α)) can have the same endpoints at infinity as α is if ψ(α) does. This is to say that ψ(α) = α, and the claim is proven. Thus, the restriction of ψ to α is a translation.

By a slight abuse of notation we will denote this element of π1 (S) by α as well. There is a bijective correspondence:     Nontrivial Nontrivial free     conjugacy classes homotopy classes of oriented ←→     in π1 (S) closed curves in S An element g of a group G is primitive if there does not exist any h ∈ G so 24 CHAPTER 1 that g = hk where |k| > 1. The property of being primitive is a conjugacy class invariant. In particular, it makes sense to say that a closed curve in a surface is primitive.

The property of being primitive is a conjugacy class invariant. In particular, it makes sense to say that a closed curve in a surface is primitive. A closed curve in S is a multiple if it is a map S 1 → S that factors through ×n the map S 1 −→ S 1 for n > 1. In other words, a curve is a multiple if it “runs around” another curve multiple times. If a closed curve in S is a multiple then no element of the corresponding conjugacy class in π1 (S) is primitive. Let p : S → S be any covering space. By a lift of a closed curve α to S we will always mean the image of a lift R → S of the map α ◦ π, where π : R → S 1 is the usual covering map.

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