ALMOST CONVEX GROUPS AND THE 8 GEOMETRIES

If M is a closed Nil geometry 3-manifold then pi(1)(M) is almost convex with respect to a fairly simple ''geometric'' generating set. If G is a central extension or a Z extension of a word hyperbolic group, then G is also almost convex with respect to some generating set. Combining these with previously known results shows that if M is a closed 3-manifold with one of Thurston's eight geometries, pi(1)(M) is almost convex with respect to some generating set if and only if the geometry in question is not Sol.