By Benson Farb
The research 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 crew thought. This booklet explains as many very important theorems, examples, and methods as attainable, quick and without delay, whereas even as giving complete info and conserving the textual content approximately self-contained. The e-book is acceptable for graduate students.The booklet 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, primary gadgets and instruments are brought, resembling the Birman targeted series, the advanced of curves, the braid crew, the symplectic illustration, and the Torelli staff. The booklet then introduces Teichmüller house 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 category.
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Additional info for A Primer on Mapping Class Groups (Princeton Mathematical)
It follows that f extends uniquely to a map f : H2 → H2 . As any pair of distinct points in ∂H2 are the endpoints of a unique geodesic in H2 , it follows that f maps distinct points to distinct points. It is easy to check that in fact f is a homeomorphism. Classification of isometries of H2 . We can use the above setup to classify nontrivial elements of Isom+ (H2 ). Suppose we are given an arbitrary nontrivial element f ∈ Isom+ (H2 ). Since f is a self-homeomorphism of a closed disk, the Brouwer fixed point theorem gives that f has a fixed point in H2 .
This makes sense because for any parabolic isometry of H2 , there is no positive lower bound to the distance between a point in H2 and its image. All other nontrivial elements of π1 (S) correspond to hyperbolic isometries of H2 , and hence have associated axes in H2 . We have the following fact, which will be used several times throughout this book: CURVES AND SURFACES 23 If S admits a hyperbolic metric then the centralizer of any nontrivial element of π1 (S) is cyclic. In particular, π1 (S) has trivial center.
We first need to introduce an essential concept. Given a simple closed curve α in a surface S, the surface obtained by cutting S along α is a compact surface Sα equipped with a homeomorphism h between two of its boundary components so that: 1. The quotient Sα /(x ∼ h(x)) is homeomorphic to S, and 2. the image of these distinguished boundary components under this quotient map is α. It also makes sense to cut a surface with boundary or marked points along a simple proper arc; the definition is analogous.
A Primer on Mapping Class Groups (Princeton Mathematical) by Benson Farb