Group rings and hyperbolic geometry

In the 60's Cohn showed that all ideals in the group algebra of a free group are free. Bass and Wall used this result to show that all two-dimensional complexes with free fundamental groups are standard: they are all homotopy equivalent to wedges of circles and 2-spheres. The goal of this talk is to describe recent results of this type for groups acting on hyperbolic spaces. I will discuss an algorithm showing that in the group algebra of a group acting on a hyperbolic space, ideals generated by ``few'' elements are free (where "few'' is a function of the minimal displacement of the action) and an application to complexity of cell decompositions of hyperbolic manifolds. Joint work with Thomas Delzant.