A NORM ON THE FUNDAMENTAL GROUP OF NON-HAKEN 3-MANIFOLDS

A canonical (presentation-independent) conjugacy-invariant norm is con structed on the fundamental group of any 3-manifold which is orientabl e, irreducible, has infinite fundamental group, and contains no incomp ressible surface. More generally, this norm exists on any torsion-free group whose commutator quotient is finite. This norm is then computed explicitly in an example which shows that the induced metric on the g roup is not quasi-isometric to any word metric.