Practically perfect three-manifold groups

Abstract

Let M 3 {M^3} be a 3 3 -manifold containing no 2 2 -sided projective plane. Let G ≠ { 1 } G \ne \{ 1\} be a finitely-generated subgroup of π 1 ( M 3 ) {\pi _1}({M^3}) such that G G is indecomposable relative to free product, and such that G G abelianized is finite. ( G G is “practically perfect".) Then, it is shown that there is a compact 3 3 -submanifold Z 3 ⊂ M 3 {Z^3} \subset {M^3} such that π 1 ( Z 3 ) {\pi _1}({Z^3}) contains a subgroup of finite index conjugate to G G , and Z 3 {Z^3} is bounded by a 2 2 -sphere. Some related extensions of this result are given, plus an application to compact absolute neighborhood retracts in 3 3 -manifolds.