By Farb B., Margalit D.

ISBN-10: 0691147949

ISBN-13: 9780691147949

The research of the mapping category staff Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and workforce conception. This booklet explains as many vital theorems, examples, and strategies as attainable, quick and without delay, whereas even as giving complete info and retaining the textual content approximately self-contained. The ebook is appropriate for graduate students.A Primer on Mapping type teams starts by way of explaining the most group-theoretical homes of Mod(S), from finite new release via Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the best way, imperative gadgets and instruments are brought, resembling the Birman unique series, the advanced of curves, the braid team, the symplectic illustration, and the Torelli team. The ebook 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 comprise the topology of the moduli area of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov thought, and Thurston's method of the type.

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Let α be a closed curve in S, not homotopic into a neighborhood of a puncture. 3. If α is simple then γ is simple. CURVES AND SURFACES 29 Proof. We begin by applying the following fact. A closed curve β in a hyperbolic surface S is simple if and only if the following properties hold: 1. Each lift of β to H2 is simple. 2. No two lifts of β intersect. 3. β is not a nontrivial multiple of another closed curve. Thus if α is simple, then no two of its lifts to H2 intersect. It follows that for any two such lifts, their endpoints are not linked in ∂H2 .

The genus of a bounding pair in a closed surface is defined similarly to the genus of a separating simple closed curve. 8 A genus one bounding pair. 3. Pairs (or k–tuples) of disjoint simple closed curves whose union does not separate. 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 , .

If a point of α1 and a point of β1 have the same image in S, then both points would be an intersection of p−1 (α) with p−1 (β), violating the assumption that D is innermost. If two points of α1 (or two points of β1 ) map to the same point in S, then there is a lift of p(v1 ) between these two points, also contradicting the assumption that D is innermost. CURVES AND SURFACES 33 We can now argue that D projects to an embedded disk in S. Indeed, if x and y in D project to the same point in S, then x = φ(y) for some deck transformation φ.

### A primer on mapping class groups by Farb B., Margalit D.

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