By Benson Farb
The learn 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 concept. This e-book explains as many very important theorems, examples, and strategies as attainable, quick and without delay, whereas even as giving complete info and holding the textual content approximately self-contained. The publication is acceptable for graduate students.A Primer on Mapping category teams starts through explaining the most group-theoretical houses of Mod(S), from finite iteration through Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, crucial gadgets and instruments are brought, resembling the Birman targeted series, the complicated of curves, the braid workforce, the symplectic illustration, and the Torelli team. The e-book then introduces Teichmller area and its geometry, and makes use of the motion of Mod(S) on it to end up the Nielsen-Thurston category of floor homeomorphisms. issues comprise the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov idea, and Thurston's method of the category.
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Extra resources for A Primer on Mapping Class Groups (Princeton Mathematical Series)
The arc α is simple if it is an embedding on its interior. The homotopy class of a proper arc is taken to be the homotopy class within the class of proper arcs. Thus points on ∂S cannot move off the boundary during the homotopy; all arcs would be homotopic to a point otherwise. But there is still a choice to be made: a homotopy (or isotopy) of an arc is said to be relative to the boundary if its endpoints stay ﬁxed throughout the homotopy. An arc in a surface S is essential if it is neither homotopic into a boundary component of S nor a marked point of S.
Also, by a free homotopy of loops in S we simply mean an 22 CHAPTER 1 unbased homotopy. If a nontrivial element of π1 (S) is represented by a loop that can be freely homotoped into the neighborhood of a puncture, then it follows that the loop can be made arbitrarily short; otherwise, we would ﬁnd an embedded annulus whose length is inﬁnite (by completeness) and where the length of each circular cross section is bounded from below, giving inﬁnite area. The deck transformation corresponding to such an element of π1 (S) is a parabolic isometry of the universal cover H2 .
This closed curve is null homotopic—indeed, H(∆) is the null homotopy. It follows that H(δ ∪ δ ) lifts to a closed curve in the universal cover S; what is more, this lift has one arc in a lift of α and one arc in a lift of β. 8 implies that α and β form a bigon. ✷ 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.
A Primer on Mapping Class Groups (Princeton Mathematical Series) by Benson Farb