Hurwitz Spaces and Braid Group Representations Partially Supported by the National Science Foundation

In this paper we investigate certain moduli spaces (\Hurwitz spaces") of branched covers of the Riemann sphere S 2 , and representations of nite index subgroups of the spherical braid group which arise from these Hurwitz spaces. (By spherical braid group, we mean the group of braids in the 2-sphere; we will refer to the more classical group of braids in the plane as the planar braid group.) Hurwitz spaces play an important role in realizing groups as Galois groups, a role which has been explored primarily by Fried and Volklein in F1], F2], FV], V1], V2], etc. In these works, they have given a couple of constructions of Hurwitz spaces, examined their algebraic structure, and explored their applications to the inverse Galois problem. In section 1 of this paper we give an alternative construction of the Hurwitz spaces, exhibiting them as homogeneous spaces of Aut(S 2), the group of orientation preserving homeomorphisms of S 2. This point of view enables us to prove that the universal cover of a Hurwitz space is homotopy equivalent to S 3 (see the discussion just after Proposition 4), which is equivalent to showing that the Teichmuller space of a sphere with 3 or more punctures is contractible. Of course this is not a new result, but we believe our rather elementary topological proof is interesting enough to include. In A], Arnol'd described representations of the planar braid groups on the homology of hyperelliptic curves. Later, Magnus and Peluso MP] analyzed these representations and related ones obtained from \generalized" hyperelliptic curves in more detail and expressed them in terms of Burau representations. (Note that the Burau representation doesn't satisfy the extra relation it would need to provide a representation of the spherical braid group.) In F2], M. Fried used Hurwitz spaces to describe an action of certain nite index subgroups of the spherical braid group on the homology of Riemann surfaces given as branched covers of S 2 without automorphisms. (Since these subgroups have nite index, one may obtain representations of the whole spherical braid group by inducing.) In Section 1 of this paper, we describe these representations using the topological point of view developed in our construction of the Hurwitz spaces. We also show that in the case where the original branched cover has nontrivial automorphisms, instead of obtaining a representation of a subgroup of the spherical braid group, one obtains a representation …