Frattini subgroups of 3-manifold groups

Abstract

In this paper it is shown that if the Frattini subgroup of the fundamental group of a compact, orientable, irreducible, sufficiently large 3-manifold is nontrivial then the 3-manifold is a Seifert fibered space. We show further that the Frattini subgroup of the group of a Seifert fibered space is trivial or cyclic. As a corollary to our work we prove that every knot group has trivial Frattini subgroup.