By Benson Farb

The research of the mapping classification staff Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and workforce idea. This ebook explains as many very important theorems, examples, and methods as attainable, quick and at once, whereas while giving complete information and retaining the textual content approximately self-contained. The e-book is acceptable for graduate students.The ebook starts off via explaining the most group-theoretical homes of Mod(S), from finite new release by means of Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, vital gadgets and instruments are brought, comparable to the Birman specific series, the complicated of curves, the braid crew, the symplectic illustration, and the Torelli team. The publication then introduces Teichmüller area and its geometry, and makes use of the motion of Mod(S) on it to end up the Nielsen-Thurston type of floor homeomorphisms. themes comprise the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov thought, and Thurston's method of the class.

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**A Primer on Mapping Class Groups (Princeton Mathematical)**

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Geodesics are in minimal position. Note that if two geodesic segments on a hyperbolic surface S together bounded a bigon then, since the bigon is simply connected, one could lift this bigon to the universal cover H2 of S. But this would contradict the fact that the geodesic between any two points of H2 is unique. 7 we have the following. 9 Distinct simple closed geodesics in a hyperbolic surface are in minimal position. The bigon criterion gives an algorithmic answer to the question of how to find representatives in minimal position: given any pair of transverse simple closed curves, we can remove bigons one by one, until none remain, and 35 CURVES AND SURFACES the resulting curves are in minimal position.

Since filling is a topological property, it follows that φ(γ) is the curve we are looking for, since it together with α = φ(β) fills S2 . 7 Two simple closed curves that fill a genus 2 surface. We think of φ as “changing coordinates” so that the complicated curve α becomes the easy-to-see curve β. The second question can be answered similarly. 3 E XAMPLES OF THE CHANGE OF COORDINATES PRINCIPLE The change of coordinates principle applies to more general situations. We give several examples here.

It follows that the endpoints of α in ∂H2 are the same as those of γ0′ . Since a geodesic in H2 is uniquely determined by its endpoints in ∂H2 , this proves that the geodesic closed curve γ0′ is the same as γ0 up to sign. The closed curve γ ′ is then specified by which multiple of γ0 it is. But different multiples of γ0 correspond to conjugacy classes in Isom+ (H2 ) that have different translation lengths and/or translation directions. Conjugacy classes with differing translation lengths are distinct, and so distinct multiples of γ0 do not lie in the same free homotopy class.