Negative curvature phenomena in Outer Space

In a negatively curved space, closest-point projections to geodesics are “strongly contracting” in the following sense. Every ball disjoint from the geodesic projects to a set whose diameter is uniformly bounded (independent of the radius of the ball). While Teichmueller space is known not to be negatively curved, Minsky showed that geodesics in the thick part of Teichmueller space have uniformly strongly contracting projections. I will discuss the analogy between Teichmueller space and “Outer Space” which models the group Out(F_n), $the outer automorphism group of the free group. I will then sketch a proof that an axis of a fully irreducible element in Out(F_n) $has a strongly contracting projection.$