By Farb B., Margalit D.
The research of the mapping type team Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and team idea. This publication explains as many vital theorems, examples, and methods as attainable, fast and without delay, whereas even as giving complete information and retaining the textual content approximately self-contained. The booklet is acceptable for graduate students.A Primer on Mapping category teams starts by means of explaining the most group-theoretical houses of Mod(S), from finite new release through Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the way in which, imperative items and instruments are brought, equivalent to the Birman particular series, the complicated of curves, the braid staff, 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 end up the Nielsen-Thurston type of floor homeomorphisms. issues comprise the topology of the moduli house of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov conception, and Thurston's method of the type.
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Additional info for A primer on mapping class groups
Proof. We give the proof for the case when S is hyperbolic. Fix a covering map H2 → S and let φ ∈ Isom+ (H2 ) be the hyperbolic isometry corresponding to some element of the conjugacy class of α. The primitivity of the elements of the conjugacy class of α is equivalent to the primitivity of φ in the deck transformation group. Assume that φ = ψ n , where ψ is another element of the deck transformation group and n ∈ Z. In any group, powers of the same element commute, and so φ commutes with ψ. Thus, φ and ψ have the same set of fixed points in ∂H2 .
Given two simple closed curves α and β, how do we find homotopic simple closed curves that are in minimal position? While the first question is a priori a minimization problem over an infinite dimensional space, we will see that the question can be reduced to a finite check—the bigon criterion given below. For the second question, we will see that geodesic representatives of simple closed curves are always in minimal position. 2. 2 A bigon. The following proposition gives a simple, combinatorial condition for deciding whether or not two simple closed curves are in minimal position.
2. Is there a simple closed curve δ in S with i(α, δ) = 0? i(α, δ) = 1? i(α, δ) = k? 6 A simple closed curve on a genus 2 surface. 6. 7 gives a proof that the answer to the first question is “yes” in this case, as we now show. ). By the classification of simple closed curves in a surface, there is a homeomorphism φ : S2 → S2 with φ(β) = α. 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.
A primer on mapping class groups by Farb B., Margalit D.