### Negative Curvature Phenomena in Outer Space

#### Yael Algom Kfir, University of Utah

Abstract: In a negatively curved space, closest point projections to geodesics are "strongly contracting" in the following sense. A ball disjoint from a geodesic projects to a bounded diameter set on the geodesic and the bound on the diameter of the image is independent of the diameter of the ball. We will discuss Culler and Vogtmann's Outer Space endowed with the Lipschitz metric, a space which admits an isometric action of Out(F_n). We will then prove that axes of fully irreducible automorphisms have strongly contracting projections. This can be applied to show that axes of fully irreducible automorphisms in the Cayley graph of Out(F_n) are Morse (stable).