Topological geodesics and virtual rigidity

We introduce the notion of a topological geodesic in a 3-manifold. Under suitable hypotheses on the fundamental group, for instance word-hyperbolicity, topological geodesics are shown to have the useful properties of, and play the same role in several applications as, geodesics in negatively curved spaces. This permits us to obtain virtual rigidity results for 3-manifolds.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-18.abs.htm