A computation of the action of the mapping class group on isotopy classes of curves and arcs in surfaces (original) (raw)
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Let S be a closed orientable surface of genus g ≥ 1. The mapping class group of S, denoted by Mod(Sg), is the group of isotopy classes of orientation-preserving self diffeomorphisms of Sg which are identity on the boundary and preserve the set of punctures. We start by introducing some basic properties of Mod(S) followed by some explicit computation of the group for some surfaces such as a closed disk, the sphere, etc.. Then we discuss some fundamental examples of infinite-order elements in Mod(Sg), known as Dehn twists. Further introducing the representation Mod(Sg) −→ Sp(2g,Z) afforded by the natural action of Mod(Sg) on H1(Sg,Z), we conclude the project by showing that the kernel of this representation namely the Torelli group, is torsion-free.
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The action of the mapping class group of a surface on the collection of homotopy classes of disjointly embedded curves or arcs in the surface is discussed here as a tool for understanding Riemann's moduli space and its topological and geometric invariants. Furthermore, appropriate completions, elaborations, or quotients of the set of all such homotopy classes of curves or arcs give Thurston's boundary for Teichmüller space or a combinatorial description of moduli space in terms of fatgraphs. Related open problems and questions are discussed.
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This paper has three parts. The first part is a general introduction to rigidity and to rigid actions of mapping class group actions on various spaces. In the second part, we describe in detail four rigidity results that concern actions of mapping class groups on spaces of foliations and of laminations, namely, Thurston’s sphere of projective foliations equipped with its projective piecewise-linear structure, the space of unmeasured foliations equipped with the quotient topology, the reduced Bers boundary, and the space of geodesic laminations equipped with the Thurston topology. In the third part, we present some perspectives and open problems on other actions of mapping class groups. 2010 Mathematics Subject Classification: 57M50, 30F40, 20F65, 57R30, 57M60.
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Let S g,b,p denote a surface which is connected, orientable, has genus g, has b boundary components, and has p punctures. Let Σ g,b,p denote the fundamental group of S g,b,p . Let Out g,b,p denote the algebraic mapping-class group of S g,b,p .
Proceedings of the American Mathematical Society, 2008
Let S be a surface of finite type which is not a sphere with at most four punctures, a torus with at most two punctures, or a closed surface of genus two. Let MF be the space of equivalence classes of measured foliations of compact support on S and let U MF be the quotient space of MF obtained by identifying two equivalence classes whenever they can be represented by topologically equivalent foliations, that is, forgetting the transverse measure. The extended mapping class group Γ * of S acts as by homeomorphisms of U MF. We show that the restriction of the action of the whole homeomorphism group of U MF on some dense subset of U MF coincides with the action of Γ * on that subset. More precisely, let D be the natural image in U MF of the set of homotopy classes of not necessarily connected essential disjoint and pairwise nonhomotopic simple closed curves on S. The set D is dense in U MF, it is invariant by the action of Γ * on U MF and the restriction of the action of Γ * on D is faithful. We prove that the restriction of the action on D of the group Homeo(U MF) coincides with the action of Γ * (S) on that subspace.
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We describe an algorithm to compute ®nite presentations for the mapping class group of a connected, compact, orientable surface, possibly with boundary and punctures. By an inductive process, such an algorithm, starting from a presentation well known for the mapping class group of the sphere and the torus with``few'' boundary components and/or punctures, produces a presentation for the mapping class group of any other surface.
Rigid actions of mapping class groups
I study various actions of the extended mapping class group of a surface. This includes actions on simplicial complexes and on CW complexes, like the Hatcher-Thurston complex, the Harvey curve complex and various generalizations of these complexes; actions on spaces of measured foliations and of unmeasured foliations; actions on Teichmueller space equipped with various structures; and algebraic actions. The paper grew out of a set of lectures that I gave at the Center for the Topology and Quantization of Muduli spacse (University of Aarhus). A large part of the material presented is expository, but there are also new results. These include results on the action of the extended mapping class group on spaces of foliations, and an outline of recent joint work with John McCarthy on the action of the complex of domains of a surface.
Rank one phenomena for mapping class groups
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Let Σg be a closed, orientable, connected surface of genus g ≥ 1. The mapping class group Mod(Σg) is the group Homeo+(Σg)/Homeo0(Σg) of isotopy classes of orientation-preserving homeomorphisms of Σg. It has been a recurring theme to compare the group Mod(Σg) and its action on the ...
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