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

By Benson Farb

The research of the mapping classification workforce Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and staff idea. This publication explains as many very important theorems, examples, and methods as attainable, fast and without delay, whereas even as giving complete information and holding the textual content approximately self-contained. The publication is acceptable for graduate students.A Primer on Mapping classification teams starts via explaining the most group-theoretical houses of Mod(S), from finite new release via Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the way in which, significant gadgets and instruments are brought, corresponding to the Birman unique series, the advanced of curves, the braid crew, the symplectic illustration, and the Torelli team. The booklet then introduces Teichmller house and its geometry, and makes use of the motion of Mod(S) on it to turn out the Nielsen-Thurston type of floor homeomorphisms. issues contain the topology of the moduli house of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov conception, and Thurston's method of the category.

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This completes the proof of the existence of γ. Note that we do not need to worry that the resulting parameterization of γ is geodesic since any two parameterizations of the same closed curve are homotopic as parameterized maps. 1 A lift α e of a closed curve α and the axis A for the corresponding isometry φ. from α to a multiple γ of some simple closed geodesic γ0 . By compactness of S 1 × I, there exists a constant C ≥ 0 such that no point of α is moved a distance greater than C by the homotopy.

4. Pairs of simple closed curves {α, β} with i(α, β) = |α ∩ β| = 2 and ˆi(α, β) = 0 and whose union does not separate. 5. Nonseparating simple proper arcs in a surface S that meet the same number of components of ∂S. 6. Chains of simple closed curves. A chain of simple closed curves in a surface S is a sequence α1 , . . , αk with the properties that i(αi , αi+1 ) = 1 for each i and i(αi , αj ) = 0 whenever |i − j| > 1. A chain is nonseparating if the union of the curves does not separate the surface.

However, since the covering map R → R is single-sheeted on the boundary, it follows that R ≈ R . The annulus R between α and β gives the desired isotopy. 6 E XTENSION OF I SOTOPIES An isotopy of a surface S is a homotopy H : S × I → S so that, for each t ∈ [0, 1], the map H(S, t) : S × {t} → S is a homeomorphism. Given an isotopy between two simple closed curves in S, it will often be useful to promote this to an isotopy of S, which we call an ambient isotopy of S. 11 Let S be any surface. If F : S 1 × I → S is a smooth isotopy of simple closed curves, then there is an isotopy H : S × I → S so that H|S×0 is the identity and H|F (S 1 ×0)×I = F .

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