Lipschitz metric on outer space

In 1986 Culler and Vogtmann invented a topological space, which later became known as Outer Space, to serve as a topological model for the group Out(Fn), the outer automorphism group of the free group of rank n. Recently, attempts have been made to endow Outer Space with a metric so as to make it a geodesic space on which Out(Fn) acts by isometries. One immediately encounters some obstacles because of certain nonsymmetric phenomena which occur in Out(Fn). This work is concerned with investigating the properties of the Lipschitz metric, an asymmetric metric on Outer Space. We prove two main theorems: - the Asymmetry Theorem, which investigates how d(x; y) and d(y; x) might di er. This is joint work with Mladen Bestvina. - the Contracting Geodesics Theorem, which shows that Outer Space is negatively curved in certain directions. We remark that Outer Space has been shown by Bridson and Vogtmann not to admit a ?-hyperbolic or a CAT(0) metric. In addition to proving these theorems we describe their signi cance and prove some applications.