  Let X be a normal complex algebraic variety, and p a prime. We show that
there exists an integer N=N(X, p) such that: any non-trivial, irreducible
representation of the fundamental group of X, which arises from geometry, must
be non-trivial mod p^N.
  The proof involves an analysis of the action of the Galois group of a
finitely generated field on the etale fundamental group of X. We also prove
many arithmetic statements about fundamental groups which are of independent
interest, and give several applications.
