Geometric and Analytic Aspects of Free Group Automorphisms

We will pioneer the study of analytic properties of the group of outer automorphisms of a finite rank free group. We will determine the limits and explore the extent of the geometric and analytic similarities between the outer automorphism group of a free group and the mapping class group of a hyperbolic surface. Objective A is to prove that the outer automorphism group is coarsely amenable. Coarse amenability is equivalent to the functional analytic property that the outer automorphism group is `exact`. This would be one of the first significant results about functional analytic properties of the outer automorphism group. This would also give the first proof that the outer automorphism group satisfies the famous Novikov Conjecture. In analogy with surface mapping class groups, it has been shown that the outer automorphism group acts on a hyperbolic `curve graph`. We will extend this analogy by proving, as in the mapping class group case, that this curve graph is in fact coarsely amenable. We will then use properties of the action of the outer automorphism group on this graph to prove that the coarse amenability extends to the outer automorphism group. Objective B is to prove that there are infinitely many quasi-isometry types of mapping torus groups of free group automorphisms that are irreducible with irreducible powers. The surface analogues of such automorphisms are the pseudo-Anosov homeomorphisms. Thurston proved that the mapping torus of a pseudo-Anosov homeomorphism is a compact hyperbolic three dimensional manifold, so all such mapping torus groups are quasi-isometric to three dimensional hyperbolic space. Exhibiting infinitely many quasi- isometry types of mapping torus groups of free group automorphisms that are irreducible with irreducible powers will be a dramatic failure of the usually strong analogy between the outer automorphism group of a free group and a surface mapping class group. We will accomplish this objective by relating invariants of a free group automorphism to the conformal dimension of the boundary of the mapping torus group of the automorphism. We will construct a sequence of automorphisms such that the conformal dimensions of the boundaries of their mapping tori are unbounded. The conformal dimension of the boundary is a quasi- isometry invariant of the group, so this implies there are infinitely many distinct quasi-isometry types in the sequence. Objective C is to write a software module for computations in the outer automorphism group. Exponential growth in the free group means that only the simplest examples can be computed by hand, so computerizing these computations will allow us to test conjectures and verify interesting examples of higher complexity. Achieving these objectives will yield some of the first results on analytic properties of the outer automorphism group of a free group.

We introduce and develop the notion of a contracting geodesic.' This is a precise, quantitative way to describe a path that is much shorter than any other path with the same endpoints. Such paths give the most efficient ways to travel through a geometric space, and help us to understand the large-scale structure of the space. From the large-scale geometric structure we derive algebraic and analytic conclusions.