Monodromy and the Burau-Gassner representations

Deligne Mostow considered certain families of cyclic coverings of the projective line and showed
that under some restrictions, the monodromy of the family can be non-arithmetic groups. We show
that when the number of branch points of the cover is sufficiently large relative to the degree of
the cover, then the monodromy group is an arithmetic group. The proof exploits the inductive
properties of the Burau-Gassner representations.