Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams

Let Gamma be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin-Razborov diagrams for Gamma. We also prove that every system of equations over Gamma is equivalent to a finite subsystem, and a number of structural results about Gamma-limit groups.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper54.abs.htm

Residually free 3-manifolds

We classify those compact 3-manifolds with incompressible toral boundary whose fundamental groups are residually free. For example, if such a manifold $M$ is prime and orientable and the fundamental group of $M$ is non-trivial then $M \cong \Sigma\times S^1$, where $\Sigma$ is a surface.Comment: 19 pages, referee's comments incorporated, to appear in Algebraic & Geometric Topolog