Axis Bundles in Free-by-Cyclic Groups

Let $F_{N}$ be the free group of rank $N$. An automorphism $\varphi$ of $F_{N}$ is fully irreducible if no nontrivial conjugacy class of a proper free factor is periodic under $\varphi$. It is atoroidal if there is no $n\geq 1$ and $1\neq w\in F_{N}$ such that $\varphi^{n}(w)$ is conjugate to $w$. Some automorphisms are induced by homeomorphisms of punctured surfaces; these are the geometric automorphisms. Automorphisms which are not geometric are called nongeometric. It is a result of Bestvina–Handel [BH92] that a fully irreducible automorphism is atoroidal if and only if it is not induced by a pseudo-Anosov homeomorphism on a surface with one boundary component.

Associated to a fully irreducible automorphism $\varphi$ is the attracting tree $T_{+}$ and the repelling tree $T_{-}$. A result proved independently by Handel–Mosher and Guirardel [HM07, Gui05] is that $\varphi$ is geometric if and only if both $T_{+}$ and $T_{-}$ are geometric in the sense that they are dual to a measured lamination on a 2-complex. The class of fully irreducible nongeometric automorphisms is divided into the subclasses parageometric and ageometric in terms of the atracting tree $T_{+}$, see [HM07]. The parageometric outer automorphisms are the nongeometric outer automorphisms that have geometric attracting trees. The ageometric automorphisms are the fully irreducible nongeometrics that are not parageometric. Note that by the result mentioned above, the inverse of a parageometric automorphism is ageometric.

We follow conventions of [BH92] for this section, with some modifications as in [DKL15] for the inclusion of valence two vertices. We consider connected finite graphs $\Gamma$ without valence one vertices. Denote by $V(\Gamma),E(\Gamma)$ the vertex set and edge set of $\Gamma$, respectively. A graph map $f:\Gamma\to\Gamma$ sends vertices to vertices and maps edges across edge paths.

A train track map $f:\Gamma\to\Gamma$ is a graph map such that $f^{n}$ is locally injective on the interior of all edges $e\in E(\Gamma)$ for all $n\geq 1$. An alternative description of this condition is in terms of illegal turns, which we now describe. A direction at a vertex $v\in\Gamma$ is a germ of an oriented edge incident to $v$. The map $f$ induces a map on directions $Df$; we call a direction periodic if it is fixed by some power of $Df$. A turn at $v$ is an unordered pair of directions; a turn is degenerate if it consists of the same direction twice. We call a turn $\{e_{i},e_{j}\}$ illegal if $\{Df^{n}(e_{i}),Df^{n}(e_{j})\}$ is degenerate for some $n\geq 1$. A graph map $f$ is a train track map if it does not map the interior of any edge across an illegal turn. We call an edge path in $\Gamma$ legal if does not cross an illegal turn.

Identify $F_{N}$ with $\pi_{1}(R_{N})$, where $R_{N}$ is the graph with one vertex and $N$ edges. A marked graph is a graph $\Gamma$ with a homotopy equivalence $\iota:R_{N}\to\Gamma$ which we call a marking. We say a train track map $f:\Gamma\to\Gamma$ represents an automorphism $\varphi:F_{N}\to F_{N}$ if $f$ induces $\varphi$ under the marking (this is only defined up to conjugacy in $F_{N}$, so we typically consider representatives of outer automorphisms).

If we enumerate the edges of $\Gamma$ by $e_{1},\dots,e_{k}$, we can associate to a graph map $f$ a transition matrix $A$. The entry $a_{ij}$ is the number of times $e_{i}$ crosses over $e_{j}$ under $f$, not considering orientations. If for any pair $(i,j)$ there exists $n$ such that the $(i,j)$-entry of $A^{n}$ is positive, we say $A$ is irreducible. A train track map is irreducible if its transition matrix is irreducible. Geometrically, for each $i,j$, there is an $n$ such that $f^{n}(e_{i})$ maps over $e_{j}$.

Suppose $f:\Gamma\to\Gamma$ is an irreducible train track map, the transition matrix $A$ has a unique positive (left) eigenvector $(x_{i})$ up to scaling that corresponds to the largest eigenvalue $\lambda$ of $A$. We call $\lambda$ the stretch factor of $f$. If $\lambda>1$, $f$ is called an expanding irreducible train track map. Any train track map representing a fully irreducible automorphism is expanding irreducible.

Let $f:\Gamma\to\Gamma$ be a train track map. We denote by $\Gamma_{0}$ the subdivision of $\Gamma$ obtained by adding vertices to each point of $f^{-1}(V(\Gamma))$. We refer to $\Gamma_{0}$ as the Stallings subdivision of $\Gamma$. We label each oriented subdivided edge $e\in E(\Gamma_{0})$ by the oriented image of $e$ under $f$.

Suppose two edges $e_{1},e_{2}\in E(\Gamma_{0})$ have a common initial vertex $v$ and share the oriented edge label $a$. That is, $f(e_{1})=f(e_{2})=a$ and in particular $\{e_{1},e_{2}\}$ is an illegal turn. In this case we say $e_{1}$ and $e_{2}$ ‘carry the same edge label.’ Then we can define a quotient map $q_{1}:\Gamma_{0}\to\Gamma_{1}$ which identifies $e_{1}$ and $e_{2}$ preserving the edge labels. We call such a map a fold. The Stallings fold decomposition of $f$ [Sta83] is the factorization

where each $q_{i}$ is a single fold, $h:\Gamma_{k}\to\Gamma$ is a homeomorphism, and $\pi:\Gamma\to\Gamma_{0}$ is the ‘identity’ map to the subdivision. Explicitly, $f=hq_{k}\cdots q_{1}\pi$. Note that the sequence of folds $q_{i}$ is in general not unique.

A metric graph is a graph $\Gamma$ with a path metric whose edges are locally isometric images of intervals $I\subseteq\mathbb{R}$; alternatively, it is an assignment of a positive length to each edge of $\Gamma$, encoded by a map $\ell:E(\Gamma)\to\mathbb{R}_{+}$. The volume of $\Gamma$ is the sum of the edge lengths. Outer space, denoted $CV_{N}$, is the space of equivalence classes of marked metric graphs $\Gamma$ of volume one, we denote the marking $\iota:R_{N}\to\Gamma$ (recall $\iota$ is a homotopy equivalence). Alternatively, by taking universal covers, $CV_{N}$ is sometimes thought of as a space of actions of $F_{N}$ on metric simplicial trees. Outer space was introduced by Culler and Vogtmann in [CV86], see also the survey article by Vogtmann [Vog15]. The space $CV_{N}$ has a nice simplicial structure (with missing faces). Each marked topological graph $\Gamma$ without valence two vertices defines a simplex of dimension $|E(\Gamma)|-1$, where the correspondence is via the possible volume one length functions $\ell:E(\Gamma)\to\mathbb{R}_{+}$. The faces of a simplex correspond to an assignment of length 0 to some edges of $\Gamma$ and the ‘missing faces’ are those which would have assigned length 0 to a loop of $\Gamma$.

If $f:\Gamma\to\Gamma$ is a train track map representing a fully irreducible automorphism $\varphi$, then $f$ induces a natural metric structure on $\Gamma$. If $(x_{i})$ is a positive eigenvector of the transition matrix $A$, then we assign the length $x_{i}$ to the edge $e_{i}$ and scale so that the volume of $\Gamma$ is one. We call this metric the eigenmetric induced by $f$. On $\Gamma$ with this metric we can take $f$ to expand legal paths uniformly according to the stretch factor $\lambda$. When $\Gamma$ has been endowed with this metric and $f$ expands edges uniformly by $\lambda$ we call $f$ a metric train track map.

A common metric studied on $CV_{N}$ is the Lipschitz Metric [Whi91, FM11], which is non-symmetric. If $(\Gamma,\iota),(\Gamma^{\prime},\iota^{\prime})\in CV_{N}$, then the Lipschitz distance between them is $d(\Gamma,\Gamma^{\prime})=\inf_{f}\log(L(f))$, where $f:\Gamma\to\Gamma^{\prime}$ satisfies $f\iota=\iota^{\prime}$ and $L(f)$ is the Lipschitz constant of $f$. The outer space $CV_{N}$ with the Lipschitz metric is geodesic; one class of geodesics of $CV_{N}$ with this metric are the folding paths or fold lines. Locally, traveling along a fold line means identifying increasingly long segments of certain edges at the same vertex together; these arise naturally in the case that those edges form an illegal turn of a train track representative $f$ of an automorphism $\varphi:F_{N}\to F_{N}$. In this case, a fold line can be seen as a sort of continuous version of a Stallings fold decomposition.

The group $\mathrm{Out}(F_{N})$ has a right action on $CV_{N}$ by change of markings. If $[\varphi]\in\mathrm{Out}(F_{N})$, we can abuse notation by considering $\varphi$ to be a graph map $R_{N}\to R_{N}$. Then the action is defined by $(\Gamma,\iota)\cdot[\varphi]=(\Gamma,\iota\varphi)$. This action is an isometric action with respect to the Lipschitz metric, and in particular it preserves the fold lines. In the trees characterization of $CV_{N}$, the nonsimplicial trees $T_{+}$ and $T_{-}$ can be considered to live in $\partial CV_{N}$ [CM87, Pau89].

For a nongeometric fully irreducible $\varphi\in\mathrm{Out}(F_{N})$, let $TT(\varphi)\subseteq CV_{N}$ be the set of marked metric graphs on which there exists a metric train track representative of $\varphi$. The following is one of three equivalent definitions for the axis bundle introduced in [HM11].

Another characterization of $\mathcal{A}_{\varphi}$ given in [HM11] is the union of all fold lines in $CV_{N}$ that limit to $T_{+},T_{-}\in\partial CV_{N}$ in forwards and backwards time, respectively. Under that framework, one can see that $\mathcal{A}_{\varphi}$ is actually a collection of geodesic axes invariant under the action of $\varphi$.

An automorphism $\varphi$ is said to have a lone axis if $\mathcal{A}_{\varphi}$ is homeomorphic to $\mathbb{R}$. Handel and Mosher prove in [HM11] that parageometric automorphisms do not have lone axes, so the study of lone axis automorphisms is reduced to the ageometric setting.

In [MP16], Mosher and Pfaff give necessary and sufficient conditions for an automorphism to have a lone axis in terms of the ‘ideal Whitehead graph.’ We now describe this graph and these conditions and will use them to verify that the automorphism of Example 2.1 has a lone axis.

Let $\varphi:F_{N}\to F_{N}$ be a nongeometric fully irreducible automorphism with train track representative $f:\Gamma\to\Gamma$. A Nielsen path in $\Gamma$ is an edge path $\rho$ such that $f(\rho)$ is homotopic to $\rho$ rel endpoints; an indivisible Nielsen path is a Nielsen path that cannot be written as a nontrivial concatenation of Nielsen paths. Likewise, a periodic Nielsen path is an edge path that is a Nielsen path of $f^{n}$ for some $n$, as in $f^{n}(\rho)$ is homotopic rel endpoints to $\rho$, and a periodic indivisible Nielsen path is a periodic Nielsen path that cannot be written as a nontrivial concatenation of periodic Nielsen paths. There are only finitely many periodic indivisible Nielsen paths in $\Gamma$ [BFH00, Lemmas 4.2.5-6], so after a possible subdivision of $\Gamma$, we assume that each such endpoint is in $V(\Gamma)$. After such a subdivision, we call a vertex $v\in V(\Gamma)$ principal if $v$ has at least three periodic directions or $v$ is the endpoint of a periodic Nielsen path.

We will now give a description of the ideal Whitehead graph of $\varphi$ which will serve as a definition for us in the special case that $f$ admits no periodic Nielsen paths. Let $v\in V(\Gamma)$ be a principal vertex. The local Whitehead graph at $v$, $\mathcal{LW}(\Gamma,v)$, has a vertex for each direction at $v$ and an edge connecting directions when that turn is taken, that is, an edge of $\Gamma$ maps over it under $f^{n}$ for some $n\geq 1$. The local stable Whitehead graph at $v$, $\mathcal{SW}(\Gamma,v)$, is the full subgraph of $\mathcal{LW}(\Gamma,v)$ with vertices the periodic directions at $v$. The ideal Whitehead graph of $\varphi$, $\mathcal{IW}(\varphi)$, is a union of $\mathcal{SW}(\Gamma,v)$ over all $v\in V(\Gamma)$; when $f$ has no periodic Nielsen paths, the union is a disjoint union.

The above description suffices for our purposes, but we briefly recall Handel and Mosher’s definition [HM11], see also [Pfa12, Section 2.9]. Starting by ‘realizing’ the leaves of the expanding lamination [BFH97] of $\varphi$ in the attracting tree $T_{+}$, the graph $\mathcal{IW}(\varphi)$ records how leaves of the lamination cross branch points of $T_{+}$. Let $p:\tilde{\Gamma}\to\Gamma$ be the universal cover, the train track map $f:\Gamma\to\Gamma$ determines a map $f_{+}:\tilde{\Gamma}\to T_{+}$ which may identify distinct points $\tilde{v},\tilde{w}\in\tilde{\Gamma}$. If this map also identifies directions at $\tilde{v}$ and $\tilde{w}$, then the corresponding vertices of $\mathcal{SW}(\Gamma,p(\tilde{v}))$ and $\mathcal{SW}(\Gamma,p(\tilde{w}))$ are identified inside of $\mathcal{IW}(\varphi)$. Lemma 3.1 of [HM11] implies that this situation only occurs when there is an indivisible Nielsen path between $p(\tilde{v})$ and $p(\tilde{w})$.

Suppose $\mathcal{IW}(\varphi)$ has $k$ components, and the $i$th component has $m_{i}$ vertices. We define the rotationless index of $\varphi$ to be the sum

We can now state the lone axis criteria of [MP16].

A free-by-cyclic group

is a semidirect product of $F_{N}$ with $\mathbb{Z}$. We sometimes refer to such a group as the mapping torus group of $\varphi$. We now describe an associated $K(G,1)$ space which has proven fruitful in studying the splittings of free-by-cyclic groups.

Given an expanding irreducible train track map $f:\Gamma\to\Gamma$ representing $\varphi$, one can define the mapping torus of $f$, the space $\Gamma\times I/(x,1)\sim(f(x),0)$. In [DKL15], given a Stallings fold decomposition of $f$, they construct the folded mapping torus of $f$, denoted $X=X_{f}$ (typically the dependence on the folding sequence is suppressed). The space $X$ is a quotient of the mapping torus of $f$ where at each height $t$, points of $\Gamma$ are identified according to a continuous version of the Stallings fold decomposition after time $t$. The space $X$ has the structure of a 2-complex with a trapezoidal cell structure. The vertical 1-cells of the structure lie above vertices of the graph $\Gamma$, sometimes after subdivisions. There is a skew (or diagonal) 1-cell for each fold in the Stallings fold sequence. The degree of a 1-cell $\bar{v}$ is the minimal number of components of $U\setminus\bar{v}$, where $U$ is an arbitrarily small neighborhood of a point $x\in\bar{v}$. Each skew 1-cell has degree three, is the bottom edge of a unique 2-cell, and is in a top edge of two 2-cells (or possibly a single 2-cell which folds onto itself). Figures 2 and 3 depict the folded mapping torus for the automorphism of Example 2.1.

The folded mapping torus $X$ is equipped with an upward semiflow $\psi$, which descends from a suspension semiflow on the mapping torus of $f$. To analogize the cones arising from Thurston’s norm for 3-manifolds, an open cone $\mathcal{P}\subseteq H^{1}(X;\mathbb{R})=H^{1}(G;\mathbb{R})$ is defined in [DKL15] as the set of cohomology classes which have cocycle represenatives that are positive on each 1-cell of $X$ (with the trapezoidal cell structure). Each primitive integral cohomology class $\alpha\in\mathcal{P}$ is dual to a cross section $\Theta_{\alpha}$ of the flow $\psi$ (in the sense of [DKL15]), and the first return map $f_{\alpha}$ of $\psi$ is a homotopy equivalence of $\Theta_{\alpha}$. Thus each such $\alpha$ induces a splitting of $G_{\varphi}$ as $\pi_{1}(\Theta_{\alpha})\rtimes\mathbb{Z}$ and is associated to an outer automorphism $[\varphi_{\alpha}:\pi_{1}(\Theta_{\alpha})\to\pi_{1}(\Theta_{\alpha})]$, which we call the monodromy.

A larger cone $C_{X}\subseteq H^{1}(X;\mathbb{R})$ investigated in [DKL17a] is also associated to a folding mapping torus $X$. Here, primitive integral classes $\alpha\in C_{X}$ are still dual to cross sections $\Theta_{\alpha}$, but the first return maps are not necessarily homotopy equivalences. These cross sections as first return maps induce a splitting of $G$ as an ascending HNN-extension of an injective endormorphism, and it is shown that the cone $C_{X}$ is equal to a component of $\mathbb{R}_{+}\cdot\Sigma(G)$, where $\Sigma(G)$ is the BNS invariant introduced in [BNS87]. We will consider a component of the cone over $\Sigma_{s}(G)=\Sigma(G)\cap-\Sigma(G)$ which we will call $C_{s}$: $C_{s}$ is the component of $C_{X}\cap(\mathbb{R}_{+}\cdot\Sigma_{s}(G))$ that contains $\mathcal{P}$. Each integral class of $C_{s}$ induces a splitting of $G$ as a mapping torus group of an (outer) automorphism $\varphi_{\alpha}$, and so primitive integral classes in $\Sigma_{s}(G)$ have associated axis bundles.

Consider an expanding irreducible train track map $f:\Gamma\to\Gamma$ where $\Gamma$ is equipped with the eigenmetric induced by $f$. For any illegal turn $\tau=\{e_{i},e_{j}\}$ and small $\varepsilon>0$, we can define a folding path $\gamma_{\tau}:[0,\varepsilon]\to CV_{N}$ as follows: $\gamma_{\tau}(t)$ is obtained by first identifying (folding) the initial length $t$ segments of the edges $e_{i},e_{j}$, then rescaling all edge lengths so that their sum is one. The following lemma follows directly from the fold lines characterization of the axis bundle given in [HM11]. The equivalence given by Handel–Mosher of that definition and the train track based definition we use goes through a more technical third definition of the axis bundle (‘weak’ train tracks), so we will give a proof here.

It will be useful for us to be able to perform simultaneous folds while remaining in the axis bundle. If $\tau_{1},\dots,\tau_{n}$ are $n$ illegal turns at distinct vertices $v_{1},\dots,v_{n}$ and $\varepsilon>0$ is sufficiently small, we define a map $\Delta:[0,\varepsilon]^{n}\to CV_{N}$ as follows: $\Delta(t_{1},\dots,t_{n})$ is obtained by (simultaneously) folding at each turn $\tau_{i}$ by a length $t_{i}$, then rescaling the metric to be of volume one.

We now work towards proving conditions under which an automorphism $\varphi$ has many axes. The following lemma is implied by the proof of Lemma 4.3 of [MP16].

Note that the statement of Lemma 3.3 in [MP16] assumes that $\varphi$ is ageometric rather than nongeometric, because this is a small part of a larger proof. On the other hand, as already noted, it is shown in [HM11] that parageometric automorphisms do not ever have a lone axis. We now prove an analogous lemma, that if a train-track map $f:\Gamma\to\Gamma$ has many illegal turns, it is far from being lone axis in the sense that $\mathcal{A}_{\varphi}$ has large dimension. Note that as stated, it doesn’t recover Lemma 3.3.

Pfaff and Tsang [PT25] have given the axis bundle a cell structure that makes $\mathcal{A}_{\varphi}$ into what they call a ‘cubist’ complex. The embedding we describe in Lemma 3.4 lies in a face of a ‘branched cube’ in this complex, and in fact Lemma 3.4 can be viewed as a consequence of [PT25, Lemma 3.8].

We now begin to consider cross sections of the semiflow on the folded mapping torus.

Let $\varphi:F_{N}\to F_{N}$ be a nongeometric, fully irreducible automorphism and $G=F_{N}\rtimes_{\varphi}\mathbb{Z}$ the associated free-by-cyclic group. We fix some folding sequence of a train track representative $f:\Gamma\to\Gamma$ and use this to define a folded mapping torus $X$ of $f$ with upward semiflow $\psi$. Let $\mathcal{P}$ denote the positive cone of [DKL15] associated to $X$.

Given an integral class $\alpha\in\mathcal{P}$ and a positive cocycle $z$ representing $\alpha$, we obtain a ‘fibration’ map $\eta_{z}:X\to S^{1}$ à la [DKL15]. Then, for an appropriate $y\in S^{1}$, the set $\Theta_{z}=\eta_{z}^{-1}(y)$ is a graph which is a cross section to the semiflow $\psi$, and $\Theta_{z}$ has a vertex structure so that the first return map of $\psi$ is an expanding irreducible train track map. The graph $\Theta_{z}$ is connected if and only if $\alpha$ is primitive integral [DKL15, Theorem A]. The number of intersections of $\Theta_{z}$ with a 1-cell $e$ is bounded below by $\lfloor z(e)\rfloor$, because $z(e)$ measures how much $e$ wraps around $S^{1}$ under $\eta_{z}$.