Free-group automorphisms, train tracks and the beaded decomposition
Martin R. Bridson, Daniel Groves
备注：
Version 2 is 35 pages. The section on growth contained an error pointed out by Gilbert Levitt, and has been removed. A corrected verion will appear in a subsequent paper. Otherwise, changes are mostly cosmetic
We study the automorphisms ϕof a finitely generated free group F. Building on the train-track technology of Bestvina, Feighn and Handel, we provide a topological representative f:G\to G of a power of ϕthat behaves very much like the realization on the rose of a positive automorphism. This resemblance is encapsulated in the Beaded Decomposition Theorem which describes the structure of paths in G obtained by repeatedly passing to f-images of an edge and taking subpaths. This decomposition is the key to adapting our proof of the quadratic isoperimetric inequality for $F\rtimes_ϕ\mathbb Z$, with ϕpositive, to the general case. To illustrate the wider utility of our topological normal form, we provide a short proof that for every w in F, the function $n\mapsto |ϕ^n(w)|$ grows either polynomially or exponentially.
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…
We define fully irreducible automorphisms of generalized Baumslag-Solitar groups in analogy with fully irreducible automorphisms of free groups. We first obtain a characterization of fully irreducible automorphisms analogous to a condition given by Kapovich. Next we discuss the existence of pseudo-periodic conjugacy classes based on the study of Nielsen paths in a train track representative. Finally we give an algorithm which finds out whether an automorphism with a train track representative is pseudo-atoroidal and fully irreducible.
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.
We introduce a new class of automorphisms $φ$ of the non-abelian free group $F_N$ of finite rank $N \geq 2$ which contains all iwips (= fully irreducible automorphisms), but also any automorphism induced by a pseudo-Anosov homeomorphism of a surface with arbitrary many boundary components. More generally, there may be subgroups of $F_N$ of rank $\geq 2$ on which $φ$ restricts to the identity. We prove some basic facts about such {\em tree-irreducible} automorphisms, and show that, together with Dehn twist automorphisms, they are the natural basic building blocks from which any automorphism of …
Similarly to the action of $Out(F_N)$ on Outer Space, the outer automorphism group of a Generalized Baumslag Solitar group acts on a deformation space endowed with the Lipschitz metric and the action of any fully irreducible automorphism with a train track representative is hyperbolic. Inspired by previous work by Algom-Kfir, we prove that the axis of such an automorphism is strongly contracting.
We prove that if F is a finitely generated free group and f:F -> F is an automorphism with polynomial growth of degree d, then there exists a characteristic subgroup S < F of finite index such that the induced automorphism of the abelianisation of S also grows polynomially of degree d. The proof is geometric in nature and makes use of improved relative train track representatives.
We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordifica…
Among the simplest invariants of the sporadic finite simple groups are their outer automorphism groups. For 12 of the 26 possible isomorphism types of a sporadic simple group G, the outer automorphism group Out(G) has order 2, and in the remaining 14 cases, Out(G) is trivial. Historically the suspicion of the existence of a sporadic group was followed in fairly short order by the calculation of a good upper bound on the size of its outer automorphism group. In a few cases establishing the existence of certain outer automorphisms, like the existence of the groups themselves, presented difficult…
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.
The main theorem of this document emulates, in the context of Out(F_r) theory, a mapping class group theorem (by H. Masur and J. Smillie) that determines precisely which index lists arise from pseudo-Anosov mapping classes. Since the ideal Whitehead graph gives a finer invariant in the analogous setting of a fully irreducible outer automorphism, we instead focus on determining which of the 21 connected, loop-free, 5-vertex graphs are ideal Whitehead graphs of ageometric, fully irreducible outer automorphisms of the free group of rank 3. Our main theorem accomplishes this by showing that there …
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 $π_1 \cal G$.
This is the second of two papers in which we investigate the properties of displacement functions of automorphisms of free groups (more generally, free products) on the Culler-Vogtmann Outer space $CV_n$ and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of $CV_n$ we have to deal with general deformation spaces, for this reason we developed the theory in such generality. In first paper~\cite{FMpartI} we studied general properties of the displacement functions, such as well-orderability of the spectrum and the t…
We present an effective algorithm for detecting automorphic orbits in free groups, as well as a number of algorithmic improvements of train tracks for free group automorphisms.
In \cite{Ka14} we produced an algorithm for deciding whether or not an element $ϕ\in Out(F_N)$ 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 paper we refine the algorithm from \cite{Ka14} 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 $N\ge 3$, an algorithm which, given a topological representative $f$ of an elemen…
We prove that abelian subgroups of the outer automorphism group of a free group are quasi-isometrically embedded. Our proof uses recent developments in the theory of train track maps by Feighn-Handel. As an application, we prove the rank conjecture for $\mathrm{Out}(F_n)$.
The fundamental group of a finite graph of groups with trivial edge groups is a free product. We are interested in those outer automorphisms of such a free product that permute the conjugacy classes of the vertex groups. We show that in particular cases of interest, such as where vertex groups are themselves finite free products of finite and cyclic groups, given such an outer automorphism, after passing to a positive power, the outer automorphism is represented by a particularly nice kind of relative train track map called a CT. CTs were first introduced by Feighn and Handel for outer automor…
We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group $\Z$) and the one-sided case (spatially acting monoid $\N$, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sof…
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…
Given a free-by-cyclic group $G = F_N \rtimes_φ\mathbb{Z}$ determined by any outer automorphism $φ\in \mathrm{Out}(F_N)$ 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…
