Symmetric patterns of geodesics and automorphisms of surface groups (Q1359216)

The author proves a non-equivariant version of Mostow rigidity for symmetric patterns of geodesics in hyperbolic space \(H^n\), \(n\geq 3\). Such a pattern is defined by three axioms; it is also the lift of a finite union of closed geodesics in a compact hyperbolic orbifold. Let \(\Gamma_1\), \(\Gamma_2\) be two such patterns and \(G_1 \subset\Gamma_1\), \(G_2 \subset \Gamma_2\) be of bounded geometry. Then any uniformly proper bijection from \(G_1\) to \(G_2\) is induced by a hyperbolic isometry. On the way to and around this main theorem there are results and techniques which are interesting on their own: extension lemma, eccentricity lemma, scattering lemma, flexible differentiability principle, finiteness theorem for closed hyperbolic surface groups and co-compact hyperbolic lattices, cone lemma, etc. The main result allows a classification of pseudo-Anosov automorphisms, based on the large scale geometric structure of their orbits.