Geometric Representations of the Braid Groups

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.