We show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces that is, lift to embeddings in the product of the surface with R². This result is nontrivial already for unramified coverings, in which case the lifting problem is well-known to reduce to the purely algebraic problem of factoring the monodromy map to the symmetric group S_n through the braid group B_n. In our approach, this factorization is often achieved as a consequence of a stronger property: a factorization through a free group. In the reverse direction we show that any non-abelian surface group has infinitely many finite simple non-abelian groups quotients with characteristic kernels which do not contain any simple loop and hence the quotient maps do not factor through free groups. By a pullback construction, finite dimensional Hermitian representations of braid groups provide invariants for the braided surfaces. We show that the strong equivalence classes of braided surfaces are separated by such invariants if and only if they are profinitely separated.

The authors are grateful to P. Bellingeri, G. Kuperberg, L. Liechti, M. Lönne, J. Marché, J.B. Meilhan, S. Melikhov, E. Samperton and E. Wagner for useful conversations and to the referees for pointing out several errors and incomplete arguments in the previous versions of this paper and improving the exposition.