Castel F. Geometric Representations of the Braid Groups

Société Mathématique de France, 2016. — 184 p. — (Astérisque 378). — ISBN: 2856298354.Since the braid group and the mapping class group were defined in the first half of the last century, mathematicians have atempted, in vain, to compute the endomorphisms for both. In addition, each partial result in this direction has seemed to confirm a tight connection between the braid group and the mapping class group, without revealing the nature of this connection. In this paper, the author changes the point of view: determining all the homomorphisms from the braid group to the mapping class group via Thurston's theory. He explains their geometric nature and shows that they are almost all embeddings. Thanks to these new results, the author has found answers to these questions in a unified way.Corollaries of Theorems 1, 2 and 3
Conventions and definitions
On monodromy homomorphisms
About the number of boundary components
Irreducible geometric representations of Bn
An nth standard generator for Bn
The special curves σs(G0)
The special curves are not separating
Description of σ(X) in Σ
Expression of the mapping classes of G0
Appendix. Miscellaneous results on the mapping class group