Coherence, planar boundaries, and the geometry of subgroups

Abstract: This will be a broad talk about coherence of groups, and how it relates to conjectures about hyperbolic groups with planar boundaries. A group is coherent if every finitely generated subgroup is finitely presented.  This is a property enjoyed by the fundamental groups of 3-manifolds. A natural stepping stone to proving that certain hyperbolic groups are 3-manifold groups is to prove that they are coherent.  A stronger property that some hyperbolic groups possess is  "local quasi-convexity", in that every finitely generated subgroup is quasi-convex.   It is conjectured that a hyperbolic group whose Gromov boundary is a Sierpinski carpet is locally quasi-convex. I'll also discuss several examples of coherent and incoherent groups that have 3-manifold like properties.