  Let $X$ be the surface $\T\times\T$ where $\T$ is the complex torus. This
paper is the third in a series, studying the fundamental group of the Galois
cover of $X$ \wrt a generic projection onto $\C\P^2$.
  Van Kampen Theorem gives a presentation of the fundamental group of the
complement of the branch curve, with 54 generators and more than 2000
relations. Here we introduce a certain natural quotient (obtained by
identifying pairs of generators), prove it is a quotient of a Coxeter group
related to the degeneration of $X$, and show that this quotient is virtually
nilpotent.
