  We consider subgroups of the braid groups which are generated by $k$-th
powers of the standard generators and prove that any infinite intersection
(with even $k$) is trivial. This is motivated by some conjectures of Squier
concerning the kernels of Burau's representations of the braid groups at roots
of unity. Furthermore, we show that the image of the braid group on 3 strands
by these representations is either a finite group, for a few roots of unity, or
a finite extension of a triangle group, by using geometric methods.
