On Asymptotic Cones and Quasi-isometry Classes of Fundamental Groups of 3-manifolds.
Let be a compact, orientable, irreducible 3-manifold with a torus. We show that there can be infinitely many slopes on realized by the boundary curves of immersed, incompressible, - incompressible surfaces in which are embedded in a neighborhood of .
Let be a non-trivial knot in the -sphere, its exterior, its group, and its peripheral subgroup. We show that is malnormal in , namely that for any with , unless is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in attached to which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups in the fundamental group of a...
This is a survey of results and open problems on compact 3-manifolds which admit spines corresponding to cyclic presentations of groups. We also discuss questions concerning spines of knot manifolds and regular neighborhoods of homotopically PL embedded compacta in 3-manifolds.