Group actions on quasi-trees

Abstract

This talk will be about negative curvature in spaces and groups. To be precise, the difference between negative curvature and non-positive curvature is important. I start with the celebrated Rank rigidity theorem proved by Ballmann in 80’s after a sequence of works by many people. In this note, I usually state theorems for closed manifolds, but many of the conclusions hold for manifolds of finite volume. Let X be a simply connected, complete Riemannian manifold of non-positive sectional curvature, of dimension at least two. (From now on, I only say curvature instead of sectional curvature.) A geodesic γ is rank-1 unless it is the boundary of a flat Euclidean half plane in X. An (hyperbolic) isometry of X is rank-1 if it leaves some rank-1 geodesic invariant and acts on it by a non-trivial translation. A manifold of non-positive curvature M is rank-1 if π1(M) contains a rank-1 element when it acts on M by a Deck transformation.