The problems in this program are in the area of overlap between low-dimensional topology and combinatorial group theory. W. Thurston's work indicates that the most interesting 3-manifolds are hyperbolic, i.e. ones that admit a Riemannian metric of constant negative curvature. By letting a sequence of such structures degenerate, a tree is obtained. The first problem is to understand the space of actions of a group on trees. The motivating question here is whether or not the space of simplicial actions is dense in the space of all actions. The second problem is to bound the complexity of 3-orbifolds in terms of their fundamental groups. A main question here is whether there is a bound (in terms of the number of generators needed for the group) for the number of conjugacy classes of maximal finite subgroups. There is a generalization, due to M. Gromov, to arbitrary groups of the notion of hyperbolicity of the fundamental group of a closed manifold. The final problem is to show that the fundamental group of a mapping cylinder of an irreducible automorphism of a free group is hyperbolic in this sense. All the problems have in common that they heavily mix topology and some other branch of mathematics, geometry or algebra, or both. This is a perennial phenomenom in mathematics, but it waxes and wanes. At the moment projects which form bridges between neighboring fields are proliferating. The trend to blur boundaries, as in this project, is particularly pronounced.
