Train track maps on graphs of groups

Train track maps for graphs of groups

We define train track maps for graphs-of-groups $\cal G$ and exhibit the precise conditions under which the fundamental finiteness properties known for classical train track maps extend to this generalization. These finiteness properties are the crucial tool to control the decrease of illegal turns under iteration of the train track map, and they are a principal ingredient in the answer to basic algorithmic questions about automorphisms induced by such train track maps on $π_1 \cal G$.

Algorithmic constructions of relative train track maps and CTs

Every rotationless outer automorphism of a finite rank free group is represented by a particularly useful relative train track map called a CT. The main result of this paper is that the constructions of CTs can be made algorithmic. A key step in our argument is proving that it is algorithmic to check if an inclusion of one invariant free factor system in another is reduced. Several applications are included, as well as algorithmic constructions for relative train track maps in the general case.

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 endomorphism of 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-inv…

Stretching factors, metrics and train tracks for free products

In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of $G$-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices. In particular, we describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths. We s…

A Bers-like proof of the existence of train tracks for free group automorphisms

Using Lipschitz distance on Outer space we give another proof of the train track theorem.

Out($F_r$)的火车轨道自动机I：真全折叠分解

Out($F_r$) train track automata I: Proper full fold decompositions

该研究刻画了Out($F_r$)中大类完全不可约元素对应的火车轨道自动机，以及这些自动机在Culler-Vogtmann外空间中的关联测地线。