Train Track Map Research Articles

Train track maps on graphs of groups

In this paper we develop the theory of train track maps on graphs of groups. Expanding a definition of Bass, we define a notion of a map of a graph of groups, and of a homotopy equivalence. We prove that under one of two technical hypotheses, any homotopy equivalence of a graph of groups may be represented by a relative train track map. The first applies in particular to graphs of groups with finite edge groups, while the second applies in particular to certain generalized Baumslag–Solitar groups.

Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification – the free splitting complex – with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of A u t ( F n ) Aut(F_n) is a well-ordered subset of R \mathbb {R} , this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call partial train tracks (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement – minpoints – coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.

Graph towers, laminations and their invariant measures

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow \Gamma$ and vector towers $\overleftarrow v$ on $\overleftarrow \Gamma$, which allows us to efficiently describe all invariant measures $\mu = \mu^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$, and in particular the set of projectively fixed currents under the action of a (possibly reducible) endomorphism $\varphi: F_N \to F_N$ is determined, when $\varphi$ is represented by a train track map.

Algorithmic constructions of relative train track maps and CTs

Building on [BH92, BFH00], we proved in [FH11] that every element \psi of the outer automorphism group of a finite rank free group is represented by a particularly useful relative train track map. In the case that \psi is rotationless (every outer automorphism has a rotationless power), we showed that there is a type of relative train track map, called a CT, satisfying additional properties. The main result of this paper is that the constructions of these relative train tracks can be made algorithmic. A key step in our argument is proving that it is algorithmic to check if an inclusion \mathcal F \sqsubset \mathcal F' of \phi -invariant free factor systems is reduced. We also give applications of the main result.

Endomorphisms, train track maps, and fully irreducible monodromies

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphismof the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application,we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group depends only on the component of the BNS-invariant containing the associated homomorphism to the integers.

McMullen polynomials and Lipschitz flows for free-by-cyclic groups

Consider a group G and an epimorphism u_0\colon G\to \mathbb Z inducing a splitting of G as a semidirect product ker (u_0)\rtimes_\varphi \mathbb Z with ker (u_0) a finitely generated free group and \varphi\in Out (ker (u_0) ) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which we realized G as \pi_1(X) for an Eilenberg–Maclane 2-complex X equipped with a semiflow \psi , and inspired by McMullen's Teichmüller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant \mathfrak m \in \mathbb Z[H_1(G;\mathbb Z)/ torsion] for (X,\psi) and investigate its properties. Specifically, \mathfrak m determines a convex polyhedral cone \mathcal C_X\subset H^1(G;\mathbb R) , a convex, real-analytic function \mathfrak H\colon \mathcal C_X\to \mathbb R , and specializes to give an integral Laurent polynomial \mathfrak m_u(\zeta) for each integral u\in \mathcal C_X . We show that \mathcal C_X is equal to the "cone of sections" of (X,\psi) (the convex hull of all cohomology classes dual to sections of of \psi ), and that for each (compatible) cross section \Theta_u\subset X with first return map f_u\colon \Theta_u\to \Theta_u , the specialization \mathfrak m_u(\zeta) encodes the characteristic polynomial of the transition matrix of f_u . More generally, for every class u\in \mathcal C_X there exists a geodesic metric d_u and a codimension-1 foliation \Omega_u of X defined by a "closed 1-form" representing u transverse to \psi so that after reparametrizing the flow \psi^u_{s} maps leaves of \Omega_u to leaves via a local e^{s\mathfrak H(u)} -homothety. Among other things, we additionally prove that \mathcal C_X is equal to (the cone over) the component of the BNS-invariant \Sigma(G) containing u_0 and, consequently, that each primitive integral u\in \mathcal C_X induces a splitting of G as an ascending HNN-extension G = Q_u\ast_{\phi_u} with Q_u a finite-rank free group and \phi_u\colon Q_u\to Q_u injective. For any such splitting, we show that the stretch factor of \phi_u is exactly given by e^{\mathfrak H(u)} . In particular, we see that \mathcal C_X and \mathfrak H depend only on the group G and epimorphism u_0 .

Detecting Fully Irreducible Automorphisms: A Polynomial Time Algorithm

ABSTRACTIn [Kapovich 14] we produced an algorithm for deciding whether or not an element is an iwip (“fully irreducible”) automorphism. At several points that algorithm was rather inefficient as it involved some general enumeration procedures as well as running several abstract processes in parallel. In this article we refine the algorithm from [Kapovich 14] by eliminating these inefficient features, and also by eliminating any use of mapping class groups algorithms. Our main result is to produce, for any fixed N ⩾ 3, an algorithm which, given a topological representative f of an element ϕ of , decides in polynomial time in terms of the “size” of f, whether or not ϕ is fully irreducible. In addition, we provide a train-track criterion of being fully irreducible which covers all fully irreducible elements of , including both atoroidal and non-atoroidal ones. We also give an algorithm, alternative to that of Turner, for finding all the indivisible Nielsen paths in an expanding train-track map, and estimate the complexity of this algorithm. An Appendix by Mark Bell provides a polynomial upper bound, in terms of the size of the topological representative, on the complexity of the Bestvina–Handel algorithm[Bestvina and Handel 92] for finding either an irreducible train-track representative or a topological reduction.

Cubulating hyperbolic free-by-cyclic groups: The irreducible case

Let V be a finite graph, and let ϕ:V→V be an irreducible train track map whose mapping torus has word-hyperbolic fundamental group G. Then G acts freely and cocompactly on a CAT(0) cube complex. Hence, if F is a finite-rank free group and if Φ:F→F is an irreducible monomorphism so that G=F∗Φ is word-hyperbolic, then G acts freely and cocompactly on a CAT(0) cube complex. This holds, in particular, if Φ is an irreducible automorphism with G=F⋊ΦZ word-hyperbolic.

Dynamics on free-by-cyclic groups

Given a free-by-cyclic group $G = F_N \rtimes_\varphi \mathbb{Z}$ determined by any outer automorphism $\varphi \in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $\mathcal{A} \subset H^1(X;\mathbb{R}) = \mathrm{Hom}(G;\mathbb{R})$ containing the homomorphism $u_0 \colon G \to \mathbb{Z}$ having $\mathrm{ker}(u_0) = F_N$, a homology class $\epsilon \in H_1(X;\mathbb{R})$, and a continuous, convex, homogeneous of degree $-1$ function $\mathfrak H\colon\mathcal{A} \to \mathbb{R}$ with the following properties. Given any primitive integral class $u \in \mathcal{A}$ there is a graph $\Theta_u \subset X$ such that: (1) the inclusion $\Theta_u \to X$ is $\pi_1$-injective and $\pi_1(\Theta_u) = \mathrm{ker}(u)$, (2) $u(\epsilon) = \chi(\Theta_u)$, (3) $\Theta_u \subset X$ is a section of the semiflow and the first return map to $\Theta_u$ is an expanding irreducible train track map representing $\varphi_u \in \mathrm{Out}(\mathrm{ker}(u))$ such that $G = \mathrm{ker}(u) \rtimes_{\varphi_u} \mathbb{Z}$, (4) the logarithm of the stretch factor of $\varphi_u$ is precisely $\mathfrak H(u)$, (5) if $\varphi$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u\in \mathcal{A}$ the automorphism $\varphi_u$ of $\mathrm{ker}(u)$ is also hyperbolic and fully irreducible.

Digraphs and cycle polynomials for free-by-cyclic groups

Let 2 Out.Fn/ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism determines a freeby-cyclic group AD Fn A Z and a homomorphism 2 H 1 .AIZ/. By work of Neumann, Bieri, Neumann and Strebel, and Dowdall, Kapovich and Leininger, has an open cone neighborhood A in H 1 .AIR/ whose integral points correspond to other fibrations of A whose associated outer automorphisms are themselves representable by expanding irreducible train-track maps. In this paper, we define an analog of McMullen’s Teichmuller polynomial that computes the dilatations of all outer automorphisms in A. 57M20

Index realization for automorphisms of free groups

The goal of this paper is to introduce a new tool, called long turns, which is a useful addition to the train track technology for auto-morphisms of free groups, in that it allows one to control periodic INPs in a train track map and hence the index of the induced automorphism.

Long turns, INP’s and indices for free group automorphisms

The goal of this paper is to introduce a new tool, called {\em long turns}, which is a useful addition to the train track technology for automorphisms of free groups, in that it allows one to control periodic INPs in a train track map and hence the index of the induced automorphism.

Mapping tori of small dilatation expanding train-track maps

An expanding train-track map on a graph of rank n is P-small if its dilatation is bounded above by Pn. We prove that for every P there is a finite list of mapping tori X1,…,XA, with A depending only on P and not n, so that the mapping torus associated with every P-small expanding train-track map can be obtained by surgery on some Xi. We also show that, given an integer P>0, there is a bound M depending only on P and not n, so that the fundamental group of the mapping torus of any P-small expanding train-track map has a presentation with less than M generators and M relations. We also provide some bounds for the smallest possible dilatation.

The Recognition Theorem for $\mathrm{Out}(F_n)$

Our goal is to find dynamic invariants that completely determine elements of the outer automorphism group \mathrm{Out}(F_n) of the free group F_n of rank n . To avoid finite order phenomena, we do this for forward rotationless elements. This is not a serious restriction. For example, there is K_n>0 depending only on n such that, for all \phi\in\mathrm{Out}(F_n) , \phi^{K_n} is forward rotationless. An important part of our analysis is to show that rotationless elements are represented by particularly nice relative train track maps.

ENTROPIES OF BRAIDS

Pseudo-Anosov homeomorphisms are classified by their invariant train tracks. The decomposition of any train track map in elementary folding maps gives a normal form for each train track class. In the case of 4-braids there are three train tracks classes and we give an explicit automaton that generates a normal form for each class. This enables us, for instance, to exhibit the pseudo-Anosov 4-braid with the minimal growth rate. We also show that the growth rate of a pseudo-Anosov braid appears as a root of the Alexander polynomial of a link that shares a common sub-link with the closure of the braid. We finally give a criterion for the faithfulness of the Burau representation for 4-braids.