Commentarii Mathematici Helvetici

If a finitely presented groupG is negatively curved, automatic or asynchronously automatic thenG has an asynchronously bounded “almost prefix closed” combing. Results in [Br1] and [E] imply that the fundamental group of any closed 3-manifold satisfying Thurston's geometrization conjecture has an asynchronously bounded, almost prefix closed combing. MAIN THEOREM. IfM is a compactP 2-irreducible 3-manifold,π 1 (M) has an asynchronously bounded, almost prefix closed combing, andH, a subgroup ofπ 1 (M), is quasiconvex with respect to this combing, then the cover ofM corresponding toH is a missing boundary manifold.

We give a generalization of N. A'Campo's formula for the zeta-function of the monodromy of a singularity and show some examples of its applications.