Compactifying coverings of 3-manifolds
Let \(M\) denote a compact, \(P^2\)-irreducible 3-manifold. \textit{J. Simon} [Mich. Math. J. 23(1976), 245-256 (1977; Zbl 0331.57002)] conjectured that any cover \(M_H\) of \(M\) corresponding to a finitely generated subgroup \(H\) of \(\pi_1(M)\) is a missing boundary manifold, meaning that \(M_H\) is homeomorphic to \(\overline M\smallsetminus K\), where \(\overline M\) is compact and \(K\subset\partial\overline M\). A key involves a geometric group theoretic concept, newly introduced here, of 1-tameness for pairs of groups. A revealing characterization establishes that, when \(H\) denotes a finitely generated subgroup of \(\pi_1(M)\), the pair \((\pi_1(M),H)\) is 1-tame if and only if, for each finite subcomplex \(C\) of \(M_H\) (the cover corresponding to \(H\)), \(\pi_1(M_H\smallsetminus C)\) is finitely generated. More importantly, in the same context \((\pi_1(M),H)\) is 1-tame if and only if \(M_H\) is a missing boundary manifold. The main result shows that if \(\pi_1(M)\) has an asynchronously bounded, almost prefix closed combing, and if \(H\) is a finitely generated subgroup of \(\pi_1(M)\) which is quasi-convex with respect to the combing, then \((\pi_1(M),H)\) is 1-tame, so \(M_H\) is a missing boundary manifold. It is known that any finitely presented group that is either word hyperbolic, automatic, or asynchronously automatic has an asynchronously bounded, almost prefix closed combing, as does the fundamental group of any compact 3-manifold satisfying the hypotheses of Thurston's geometrization conjecture.
scientific article; zbMATH DE number 949849