  The fundamental group of a smooth projective variety is fibered if it maps
onto the fundamental group of smooth curve of genus 2 or more. The goal of this
paper is to establish some strong restrictions on these groups, and in
particular on the fundamental groups of Kodaira surfaces. In the specific case
of a Kodaira surface, these results are in the form of restrictions on the
monodromy representation into the mapping class group. When the monodromy is
composed with certain standard representations, the images are Zariski dense in
a semisimple group of Hermitian type.
