Almost Convex Groups and the Eight Geometries

Abstract

If $M$ is a closed Nil geometry 3-manifold then $π_1(M)$ is almost convex with respect to a fairly simple ``geometric'' generating set. If $G$ is a central extension or a ${\Bbb Z}$-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, $π_1(M)$ is almost convex with respect to some generating set if and only if the geometry in question is not Sol.