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We introduce an operation which measures the self intersections of paths on an oriented surface.

Problems on Mapping Class Groups and Related Topics

As applications, we give a criterion of the realizability of a generalized Dehn twist, and derive a geometric constraint on the image of the Johnson homomorphisms. Dedicata , Tome , pp.

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Invariant functions on Lie groups and Hamiltonian flows of surface group representations , Invent. Tome 20 , pp. France , Tome no.

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Fourier Grenoble , Tome 63 no. IMRN no. Japan Acad. A Math. Groups, Geomet. Number of pseudo-- Anosov elements in the mapping class group of a four--holed sphere J. Turkish Journal of Mathematic 34 , no 4, Lefschetz fibrations on 4-manifolds J. Lefschetz fibrations and an invariant of finitely presented groups. Notices , No. Problems on homomorphisms of mapping class groups. Farb Ed. Pure Math. Click here for the whole book.

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On sections of elliptic fibrations J. Michigan Math. Automorphisms of the Hatcher-Thurston complex J. More precisely, the braid group on n strands is naturally isomorphic to the mapping class group of a disc with n punctures. The Dehn—Nielsen—Baer theorem states that it is in addition surjective. The image of the mapping class group is an index 2 subgroup of the outer automorphism group, which can be characterised by its action on homology.

In this case the fundamental group is a free group and the outer automorphism group Out Fn is strictly larger than the image of the mapping class group via the morphism defined in the previous paragraph. The image is exactly those outer automorphisms which preserve each conjugacy class in the fundamental group corresponding to a boundary component.

This is an exact sequence relating the mapping class group of surfaces with the same genus and boundary but a different number of punctures. It is a fundamental tool which allows to use recursive arguments in the study of mapping class groups. It was proven by Joan Birman in For example, the matrix.

There is a classification of the mapping classes on a surface, originally due to Nielsen and rediscovered by Thurston, which can be stated as follows. The main content of the theorem is that a mapping class which is neither of finite order nor reducible must be pseudo-Anosov, which can be defined explicitly by dynamical properties. The study of pseudo-Anosov diffeomorphisms of a surface is fundamental. They are the most interesting diffeomorphisms, since finite-order mapping classes are isotopic to isometries and thus well understood, and the study of reducible classes indeed essentially reduces to the study of mapping classes on smaller surfaces which may themselves be either finite order or pseudo-Anosov.

Pseudo-Anosov mapping classes are "generic" in the mapping class group in various ways. For example, a random walk on the mapping class group will end on a pseudo-Anosov element with a probability tending to 1 as the number of steps grows.

[math/] Some problems on mapping class groups and moduli space

This action has many interesting properties; for example it is properly discontinuous though not free. Namely: [10].

The action is not properly discontinuous the stabiliser of a simple closed curve is an infinite group. This action, together with combinatorial and geometric properties of the curve complex, can be used to prove various properties of the mapping class group.

watch The stabilisers of the mapping class group's action on the curve and pants complexes are quite large. It is in opposition to the curve or pants complex a locally finite complex which is quasi-isometric to the mapping class group. Two distinct markings are joined by an edge if they differ by an "elementary move", and the full complex is obtained by adding all possible higher-dimensional simplices.