Geometric Representations of the Braid Groups

We define a geometric representation to be any representation of a group in the mapping class group of a surface. Let _g,b be the orientable connected compact surface of genus g with b boundary components, and (_g,b) the associated mapping class group globally preserving each boundary component. The aim of this paper consists in describing the set of the geometric representations of the braid group _n with n6 strands in (_g,b) subject to the only condition that gn/2. We prove that under this condition, such representations are either cyclic, that is, their images are cyclic groups, or are transvections of monodromy homomorphisms, that is, up to multiplication by an element in the centralizer of the image, the image of a standard generator of _n is a Dehn twist, and the images of two consecutive standard generators are two Dehn twists along two curves intersecting in one point. This leads to different results. They will be proved in later papers, but we explain how they are deduced from our main theorem. These corollaries concern four families of groups: the braid groups _n for all n6, the Artin groups of type D_n for all n6, the mapping class groups (_g,b) (preserving each boundary component) and the mapping class groups (_g,b,_g,b) (preserving the boundary pointwise), for g2 and b0. We describe the remarkable structure of the sets of the endomorphisms of these groups, their automorphisms and their outer automorphism groups. We also describe the set of the homomorphisms between braid groups _n_m with mn+1 and the set of the homomorphisms between mapping class groups of surfaces (possibly with boundary) whose genera (greater than or equal to 2) differ by at most one. Finally, we describe the set of the geometric representations of the Artin groups of type E_n (n6,7,8).