Asymmetry of Outer Space

We study the asymmetry of the Lipschitz metric d on Outer space. We introduce an (asymmetric) Finsler norm that induces d. There is an Out(F_n)-invariant potential \Psi on Outer space such that when the Lipschitz norm is corrected by the derivative of \Psi, the resulting norm is quasisymmetric. As an application, we give new proofs of two theorems of Handel-Mosher, that the Lipschitz metric is quasi-symmetric when restricted to a thick part of Outer space, and that there is a uniform bound, depending only on the rank, on the ratio of logs of growth rates of any irreducible outer automorphism f in Out(F_n) and its inverse.Comment: 15 pages, accepted to Geometriae Dedicata, omitted the comment about the potential function in rank 2 being equal to injrad (because it was false

Lone Axes in Outer Space

Handel and Mosher define the axis bundle for a fully irreducible outer automorphism in "Axes in Outer Space." In this paper we give a necessary and sufficient condition for the axis bundle to consist of a unique periodic fold line. As a consequence, we give a setting, and means for identifying in this setting, when two elements of an outer automorphism group $Out(F_r)$ have conjugate powers.Comment: Significant revisions for the sake of readability, several arguments filled i

Relative twisting in Outer space

Subsurface projection has become indispensable in studying the geometry of the mapping class group and the curve complex of a surface. When the subsurface is an annulus, this projection is sometimes called relative twisting. We give two alternate versions of relative twisting for the outer automorphism group of a free group. We use this to describe sufficient conditions for when a folding path enters the thin part of Culler-Vogtmann's Outer space. As an application of our condition, we produce a sequence of fully irreducible outer automorphisms whose axes in Outer space travel through graphs with arbitrarily short cycles; we also describe the asymptotic behavior of their translation lengths.Comment: updated version, incorporates referee comment

On the bordification of outer space

We give a simple construction of an equivariant deformation retract of Outer space which is homeomorphic to the Bestvina-Feighn bordification. This results in a much easier proof that the bordification is (2n-5)-connected at infinity, and hence that $Out(F_n)$ is a virtual duality group.Comment: Accepted version, to appear in the Journal of the London MS. Section 7, giving the homeomorphism to the Bestvina-Feighn bordification, has been substantially revise

From Outer Space by Parachute

Capability of parachutists and aircraft pilots to perform astronaut function

Joining inner space to outer space

The purpose of this paper is to demonstrate that it is possible, in principle, to obtain knowledge of the entire universe at the present time, even if the radius of the universe is much larger than the radius of the observable universe