Frattini subgroups of 3-manifold groups

Abstract

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.

Engulfing and subgroup separability for hyperbolic groups

“…Our next result covers some of the cases not covered in [1]. Notice that it is not necessary to assume engulfing here.…”

“…□ We conclude this section with a slight digression. To place this in context, recall that if G is a group, then the Frattini subgroup of G, usually written $(67), is the intersection of all the maximal subgroups of G. There has been some interest (see [1]) in the Frattini subgroups of knot complements and 3-manifolds.…”

Frattini and related subgroups of mapping class groups

“…Following [1], let C denote the centralizer of N 1 in G 1 , and as in [1], we can deduce various properties about the groups N 1 and C. Namely: (i) since N 1 is a finite group, its centralizer C in G 1 is of finite index in G 1 .…”

Conjugacy separability of certain Seifert 3-manifold groups

“…Its fundamental group π 1 (M) has one of the following three possible presentations, where α i , β i are coprime integers and γ is an integer (see [2] or [14]). …”

“…(1) Free products of conjugacy separable groups are conjugacy separable [17]; (2) The group obtained by adjoining a root to a free group is conjugacy separable [18];…”