FORMAL LANGUAGE THEORY AND THE GEOMETRY OF 3-MANIFOLDS

Automatic groups were introduced in connection with geometric problems , in particular with the study of fundamental groups of 3-manifolds. I n this article the class of automatic groups is extended to include th e fundamental group of every compact 3-manifold which satisfies Thurst on's geometrization conjecture. Toward this end, the class C-A of asyn chronously A-combable groups is introduced and studied, where A is an arbitrary full abstract family of languages. For example A may be the family of regular languages Reg, context-free languages CF,or indexed languages Ind. The class C-Reg consists of precisely those groups whic h are asynchronously automatic. It is proved that C-Ind contains all o f the above fundamental groups, but that C-CF does not. Indeed a virtu ally nilpotent group belongs to C-CF if and only if it is virtually ab elian.