06 Sep 2019
Today I posted my first paper, Train Tracks, Orbigraphs and CAT(0) Free-by-cyclic Groups to the ArXiv! In it I give an extension to automorphisms of free products of the algorithmic construction of relative train track maps for outer automorphisms of free groups. I use it to prove that mapping tori of polynomially-growing automorphisms of certain kinds are CAT(0) groups. I want to talk about the main ideas in the proof. I haven’t addressed the question of “why” study these groups in this post, but I will another time.
The length of the paper comes from constructing relative train track maps for outer automorphisms of free products. Relative train track maps always seem to scare people who hear about them, so let me attempt an explanation of what’s going on. The name and most motivations in the literature derive from the Nielsen–Thurston normal forms for homotopy classes of surface homeomorphisms. My application has a different feel, so let me attempt to sketch another connection.
One of the best ways to study a linear transformation is by considering its eigenvalues, and thus the dynamics of the linear transformation’s action on the associated vector space. The Jordan normal form is designed to present the dynamical properties of a linear transformation in a clean, easily understood way.
Given an automorphism of a free group , its action on the homology of the free group gives one a matrix in . However, since free groups are nonabelian, this matrix tends to lose a lot of information about the automorphism except in particular cases. A little better is the transition matrix of a self-map of a graph with fundamental group . The th entry of this matrix records how many times the -image of the th edge crosses the th edge in either direction. Here some information is lost again in general. The point of a relative train track map is to find such a map whose transition matrix retains as much information as possible.
Like the Jordan normal form, the transition matrix for a relative train track map is block upper-triangular, corresponding to an increasing sequence of -invariant subgraphs. Each square submatrix corresponding to a block is irreducible. Polynomially-growing automorphisms have (after taking powers) genuinely upper-triangular transition matrices with 1s on the diagonal.
Write , let be an automorphism and suppose that , where . In the free-by-cyclic group , this gives us the relation , which we can rewrite as , which, if you squint, starts to look like a relation in an HNN extension with stable letter . In fact, by our assumption on and , we actually do get a description of as an iterated HNN extension.
This situation can be good and bad. Gersten gave an example of such a description of a free-by-cyclic group which is a “poison subgroup” for nonpositive curvature. There, and are such that and have no business being conjugate in a nice action on a CAT(0) space. On the other hand, in the situation where they should be conjugate, Bridson–Haefliger give a construction that allows you to bootstrap a geometric action on a CAT(0) space up the HNN extension.
What I realized was that in the class of automorphisms I was studying, if it were the case that the automorphism was already upper-triangular with respect to the basis we were handed, then and were already conjugate, so Bridson–Haefliger says we win. So, my goal in constructing relative train track maps was to find a basis that has this nice property for a (power of) an arbitrary polynomially-growing outer automorphism. In the paper, the application is in Section 7, while the heavy lifting is to find a way to adapt the relative train track technology appropriately.
Since relative train track maps with nice properties have become a foundational tool in the study of , my hope is that this construction will help in the study of outer automorphism groups of free products.