Solutions to Bestvina & Feighn's Exercises on Limit Groups

This article gives solutions to the exercises in Bestvina and Feighn's paper on Sela's work on limit groups. We prove that all constructible limit groups are limit groups and give an account of the shortening argument of Rips and Sela.Comment: 31 pages, 2 figures. To appear in `Geometry and Cohomology in Group Theory (Durham, 2003)

On surface subgroups of doubles of free groups

We give several sufficient conditions for a double of a free group along a cyclic subgroup to contain a surface subgroup.Comment: 21 pages, 1 figur

The membership problem for 3-manifold groups is solvable

We show that the Membership Problem for finitely generated subgroups of 3-manifold groups is solvable.Comment: 19 page

Conjugacy classes of solutions to equations and inequations over hyperbolic groups

We study conjugacy classes of solutions to systems of equations and inequations over torsion-free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many conjugacy classes of solutions to such a system. The class of immutable subgroups of hyperbolic groups is introduced, which is fundamental to the study of equations in this context. We apply our results to enumerate the immutable subgroups of a torsion-free hyperbolic group.Comment: 28 pages; referee's comments incorporated; to appear in the Journal of Topolog