TTrain Track Maps and CTs on Graphs of Groups
Rylee LymanFebruary 8, 2021
Abstract
In this paper we develop the theory of train track maps on graphs of groups. Weprove the existence of CTs representing outer automorphisms of free products. Wegeneralize an index inequality due to Feighn–Handel to the graph of groups setting,sharpening a result of Martino.
A homotopy equivalence f : G → G of a graph G is a train track map when the restrictionof any iterate of f to an edge of G yields an immersion. (Relative) train track maps wereintroduced in [BH92]; they are perhaps the main tool for studying outer automorphismsof free groups. Choosing a basepoint (cid:63) in G and a path from (cid:63) to f ( (cid:63) ) determines anautomorphism f (cid:93) : π ( G, (cid:63) ) → π ( G, (cid:63) ) and a lift of f to the universal covering tree Γ of G .The map ˜ f : Γ → Γ is equivariant in the sense that for g ∈ π ( G, (cid:63) ) and x ∈ Γ, we have˜ f ( g.x ) = f (cid:93) ( g ) . ˜ f ( x ) . The lift ˜ f : Γ → Γ also satisfies the definition of a train track map. This formulation oftrain track maps as equivariant maps of trees can be adapted in a straightforward way toautomorphisms of groups acting on trees. However, one is primarily interested in using traintrack maps to study outer automorphisms, for which it would be more convenient to be ableto work directly in the quotient graph of groups. This is one purpose of this paper.
Theorem A.
Let G be a graph of groups. If an outer automorphism ϕ of π ( G ) admits atopological representative, then there exists a relative train track map f : G (cid:48) → G (cid:48) representing ϕ . If ϕ is irreducible, then the relative train track map is a train track map. This is not the first construction of relative train track maps on graphs of groups, see[CT94], [FM15] and [Syk04], but it is the most general, allowing in particular for infiniteedge groups.One class of groups acting on trees of particular interest are free products, which actwith trivial edge stabilizers. This greater flexibility allows for stronger results. The otherpurpose of this paper is to extend Feighn–Handel’s completely split relative train track maps,
CTs [FH11], to outer automorphisms of free products.Let F = A ∗ · · · ∗ A n ∗ F k be the free product of the groups A i with a free group ofrank k represented as the fundamental group of a graph of groups G , where the A i are thenontrivial vertex groups of G , the edge groups of G are trivial, and the ordinary fundamentalgroup of the (underlying graph of) G is free of rank k . Theorem B.
Every ϕ ∈ Out( F ) which is represented by a rotationless relative train trackmap f : G → G on a G -marked graph of groups G is represented by a CT (see Section 6 forthe definition) f (cid:48) : G (cid:48) → G (cid:48) on a G -marked graph of groups. In [FH18], Feighn and Handel define an index j ( ϕ ) for an outer automorphism ϕ ∈ Out( F n ) (see Section 7 for a definition) and prove it satisfies an inequality strengthening aninequality due to Gaboriau, Jaeger, Levitt and Lutsig [GJLL98]. Our final result extendsthis definition and inequality to the free product setting, strengthening a result of Martino[Mar99]. 1 a r X i v : . [ m a t h . G R ] F e b heorem C. If ϕ ∈ Out( F ) is represented on a G -marked graph of groups, then j ( ϕ ) ≤ n + k − . Part of this work originally appeared in the author’s thesis [Lym20]. The author wouldlike to thank Lee Mosher for many helpful conversations.The strategy of the proof is to find the correct equivariant perspective so that the orig-inal arguments in [BH92] and [FH11], as well as their algorithmic counterparts in [FH18],can be adapted without too much extra effort. We build up this perspective in Section 1.This done, the proof of Theorem A follows the outline in [BH92]; we give a brisk treatmentfor completeness in Section 2 and Section 3. Attracting laminations for free products areconsidered in Section 4. We then turn to the analogue of [FH11, Theorem 2.19] in Sec-tion 5 and the construction of CTs for outer automorphisms of free products in Section 6.Theorem C is proven in Section 7.
The purpose of this section is to develop tools for working with maps of graphs of groups.Let us take up the discussion from the introduction: let F be a group acting on a (simplicial)tree T , let Φ : F → F be an automorphism, and suppose there is a map ˜ f : T → T which isequivariant in the sense that we have for all g ∈ F and x ∈ T ˜ f ( g.x ) = Φ( g ) . ˜ f ( x ) . The map ˜ f is equivariantly homotopic to a map ˜ f (cid:48) which sends vertices to vertices andeither collapses edges to vertices or linearly expands edges to edge paths. It is maps of thiskind and homotopies between them that we would like to represent on the quotient graphof groups G . Call a map of graphs a morphism when it maps vertices to vertices and eithermaps edges to edges or collapses them to vertices. The kind of map of trees we are interestedin becomes a morphism after subdividing edges of the domain tree into finitely many edges.In [Bas93], a notion of morphism of a graph of groups is defined and a correspondenceis established between equivariant morphisms of Bass–Serre trees in the sense above andmorphisms of the quotient graph of groups. To anticipate the definition, which we explainbelow, recall that an identification of the universal cover of G with T , unlike the case ofthe universal cover of a graph, depends additionally on a choice of fundamental domain forthe action of F = π ( G ) on T . Given a morphism ˜ f : T → T , the corresponding morphism f : G → G needs to account for the failure of ˜ f to take the fundamental domain in the sourceto the fundamental domain in the target. Morphisms of Graphs of Groups.
Here is the definition; it depends on a slightlyunusual description of what a graph of groups is. A curious reader is referred to [Bas93]or [Lym20, Chapter 1] for more details. A graph G determines a small category G withobjects the vertices of the first barycentric subdivision of G and nonidentity arrows frombarycenters of edges to the incident vertices. A graph of groups G is a graph G and afunctor G : G → Group mono from the small category G to the category of groups witharrows injective homomorphisms.Thus as usual there are vertex groups G v and edge groups G e . If τ ( e ) denotes the terminalvertex of the edge e in some choice of orientation and ¯ e denotes e with its orientation reversed,we have injective homomorphisms ι e : G e → G τ ( e ) and ι ¯ e : G e → G τ (¯ e ) . We have G e = G ¯ e .A morphism of graphs of groups f : G → G (cid:48) is a morphism f : G → G (cid:48) of underlyinggraphs together with a pseudonatural transformation of functors f : G = ⇒ G (cid:48) f . This com-prises two kinds of data: for v a vertex and e an edge of G there are homomorphisms f v : G v → G (cid:48) f ( v ) and f e : G e → G (cid:48) f ( e ) , and for each oriented edge e there is an element2 e ∈ G (cid:48) τ ( f ( e )) such that the following diagram commutes G e G (cid:48) f ( e ) G τ ( e ) G (cid:48) f ( τ ( e )) f e ι e ad( δ e ) ι f ( e ) f τ ( e ) g ∈ G e f e ( g ) ι e ( g ) f τ ( e ) ι e ( g ) = δ e ι f ( e ) f e ( g ) δ − e . Here ad( δ e ) is the inner automorphism g (cid:55)→ δ e gδ − e . If the edge e is collapsed to a vertexby the morphism f , then ι f ( e ) should be replaced with the identity of G f ( e ) An edge path γ in a graph of groups G is a finite sequence γ = e (cid:48) g e g · · · g k − e k − g k − e (cid:48) k , where e , . . . , e k − are edges of Γ, e (cid:48) and e (cid:48) k are terminal and initial segments of edges e and e k , respectively, where g i ∈ G v i , and where v i = τ (¯ e i ) = τ ( e i − ). We allow the case where e (cid:48) and e (cid:48) k are empty, in which case they will be dropped from the notation. A path is nontrivial if it contains (a segment of) an edge. There is a notion of homotopy (rel endpoints) for edgepaths: it is generated by replacing a segment of the form eι e ( h ) with ι ¯ e ( h ) e , where e is anedge and h ∈ G e , and by adding or removing segments of the form e ¯ e for an edge e . Anedge path γ is tight if the number of edges in γ cannot be shortened by a homotopy.Let p be a point of G . The fundamental group π ( G , p ) is the group of homotopy classesof edge paths that form loops based at p . The fundamental theorem of Bass–Serre theoryasserts the existence of a tree Γ and an action of π ( G , p ) on Γ such that the quotient graphof groups is naturally identified with G .Suppose γ = e (cid:48) g e g · · · g k − e k − g k − e (cid:48) k is an edge path in G , where g i ∈ G v i . A morphism f : G → G (cid:48) acts on γ , sending it to f ( γ ) = f ( e (cid:48) ) δ − e f v ( g ) δ ¯ e f ( e ) δ − e f v ( g ) · · · f ( e k − ) δ − e k − f v k − ( g k − ) δ ¯ e k f ( e (cid:48) k ) . The pseudonaturality condition ensures that f preserves homotopy classes of paths, andthus induces a homomorphism f (cid:93) : π ( G , p ) → π ( G (cid:48) , f ( p )). Example 1.1.
The most important example of a morphism between graphs of groups is thenatural projection π : Γ → G , where Γ is the Bass–Serre tree for G . The definition dependson a choice of basepoint p ∈ G , a lift ˜ p ∈ Γ, and a fundamental domain containing ˜ p —thusfixing an action of π ( G , p ) on Γ. Since the vertex and edge groups of Γ are trivial, for π todefine a morphism of graphs of groups, we need to define δ ˜ e for each oriented edge ˜ e of Γ.Recall [Bas93, 1.12] that given a choice, for each oriented edge e , a set S e of left cosetrepresentatives for G τ ( e ) /ι e ( G e ) containing 1 ∈ G τ ( e ) , there is a normal form for edge pathsin G . For each vertex ˜ v ∈ Γ, there is a unique path γ starting at p in G in the normal form γ = e (cid:48) s e · · · s k − e k , where s i ∈ S ¯ e i for each i , which lifts to a tight path ˜ γ from ˜ p to ˜ v . If ˜ e is an oriented edgeof Γ, let ˜ γ be the unique tight path in Γ connecting ˜ p to τ (˜ e ). The path ˜ γ corresponds toa unique path γ in G of the above form. If π (˜ e ) = e k is the last edge appearing in γ , define δ ˜ e = s k − . Otherwise define δ ˜ e = 1. Observe that under this definition, we have π (˜ γ ) = γ as a path in G .Given a vertex v in G , let st( v ) denote the set of oriented edges e of G with terminalvertex τ ( e ) = v . Recall that for each lift ˜ v ∈ π − ( v ) there is a G v -equivariant bijectionst(˜ v ) ∼ = (cid:97) e ∈ st( v ) G v /ι e ( G e ) × { e } . In fact, the morphism π induces this bijection: if π (˜ e ) = δ ˜ e eδ − e , the above map is ˜ e (cid:55)→ ([ δ ˜ e ] , e ), where [ δ ˜ e ] denotes the left coset of ι e ( G e ) in G v represented by δ ˜ e .3 roposition 1.2. Let f : G → G (cid:48) be a morphism, p ∈ G and q = f ( p ) . Write Γ and Γ (cid:48) forthe Bass–Serre trees of G and G (cid:48) , respectively. There is a unique morphism ˜ f : Γ → Γ (cid:48) suchthat the following diagram commutes (Γ , ˜ p ) (Γ (cid:48) , ˜ q )( G , p ) ( G (cid:48) , q ) ˜ fπ Γ π Γ (cid:48) f where π Γ and π Γ (cid:48) are the natural projections, and ˜ p and ˜ q are the distinguished lifts of p and q , respectively.Proof. If ˜ x is a point of Γ, there is a unique tight path γ from ˜ p to x . The path f π Γ ( γ ) hasa (unique) lift ˜ γ to Γ (cid:48) beginning at ˜ q such that π Γ (cid:48) (˜ γ ) differs from f π Γ ( γ ) only possibly byan element of G (cid:48) fπ Γ (˜ x ) when ˜ x is a vertex. In particular, the terminal endpoint ˜ γ (1) is well-defined, and depends only on the homotopy class rel endpoints of f π Γ ( γ ). Define ˜ f (˜ x ) =˜ γ (1). It is easy to see that ˜ f : Γ → Γ (cid:48) defines a morphism, and that the diagram commutes.For uniqueness, observe that any morphism ˜ f (cid:48) making the above diagram commute mustsatisfy π Γ (cid:48) ˜ f (cid:48) (˜ η ) = f π Γ (˜ η )for any path ˜ η in Γ, so specializing to paths connecting ˜ p to ˜ x , we see that ˜ f (cid:48) (˜ x ) = ˜ f (˜ x ) forall ˜ x ∈ Γ.The map ˜ f is f (cid:93) -equivariant, again in the sense that for g ∈ π ( G , p ) and x ∈ Γ we have˜ f ( g.x ) = f (cid:93) ( g ) . ˜ f ( x ) . Proposition 1.3 (cf. 4.1–4.5 of [Bas93]) . Let f (cid:93) : π ( G , p ) → π ( G (cid:48) , q ) be a homomorphism,and let ˜ f : (Γ , ˜ p ) → (Γ (cid:48) , ˜ q ) be an f (cid:93) -equivariant morphism of Bass–Serre trees in the senseabove. There is a morphism f : ( G , p ) → ( G (cid:48) , q ) which induces f (cid:93) and ˜ f and such that thefollowing diagram commutes (Γ , ˜ p ) (Γ (cid:48) , ˜ q )( G , p ) ( G (cid:48) , q ) . ˜ fπ Γ π Γ (cid:48) f Proof.
As a morphism of graphs f is easy to describe. By f (cid:93) -equivariance, the map π Γ (cid:48) ˜ f yields a well-defined map on π ( G , p )-orbits; this is the map f : G → G (cid:48) as a morphism ofgraphs.Identify the graph of groups structures G and G (cid:48) with those induced by the actions ofthe respective fundamental groups on Γ and Γ (cid:48) . For Γ this involves a choice of fundamentaldomain T ⊂ Γ containing ˜ p . Each edge e ∈ G has a single preimage ˜ e ∈ T . If e is notcollapsed by f , define δ e for the morphism f as δ ˜ f (˜ e ) for the morphism π Γ (cid:48) . If e is collapsed,define δ e = 1. Thus for ˜ γ a path in T , π Γ (cid:48) ˜ f (˜ γ ) = f π Γ (˜ γ ).Let v be a vertex of G and write w = f ( v ). To define f v : G v → G (cid:48) w , recall that underthe identification of graph of groups structures, G v is the stabilizer of a vertex ˜ v ∈ π − ( v ) ∩ T . Similarly, G (cid:48) w is identified with the stabilizer of some vertex ˜ w ∈ Γ (cid:48) . The stabilizersof ˜ f (˜ v ) and ˜ w are conjugate in π ( G (cid:48) , q ) via some element g w such that g w . ˜ f (˜ v ) = ˜ w .Furthermore recall that there is a preferred translate for the fundamental domain for theaction of π ( G (cid:48) , q ) containing ˜ f (˜ v ) and another preferred translate containing ˜ w . Namely,the translate containing the edges corresponding to those in { ([1] , e ) : e ∈ st( w ) } f (˜ v )) and st( ˜ w ) with (cid:97) e ∈ st( w ) G w /ι e ( G e ) × { e } . We require g w to take the preferred translate for ˜ f (˜ v ) to the preferred translate for ˜ w . Therestriction of h (cid:55)→ g w f (cid:93) ( h ) g − w to the stabilizer of ˜ v defines a homomorphism f v : G v → G (cid:48) w .Now let e be an edge of G and write a = f ( e ). The story is similar: the stabilizerof the preimage ˜ e in T is identified with G e , and some element g a ∈ π ( G (cid:48) , q ) takes ˜ f (˜ e )to the preferred preimage ˜ a . If a is a vertex, we again require g a to match up preferredfundamental domains. The homomorphism f e : G e → G (cid:48) a is h (cid:55)→ g a f (cid:93) ( h ) g − a . Whether e is an edge of G or G (cid:48) , the monomorphism ι e has a uniform description. If v = τ ( e ) and τ (˜ e ) = ˜ v , then the monomorphism ι e is the inclusion of the stabilizer of ˜ e intothe stabilizer of ˜ v . If not, there is some element t e with t e .τ (˜ e ) = ˜ v matching up preferredfundamental domains. In this latter case, ι e is the map h (cid:55)→ t e ht − e . In the former case, write t e = 1. In the case where f collapses e to a vertex a , let ι a = 1 G a and t a = 1. Tracing an element h ∈ G e around the pseudonaturality square, we have h g a f (cid:93) ( h ) g − a t e ht − e g w f (cid:93) ( t e ht − e ) g − w = δ e t a g a f (cid:93) ( h ) g − a t − a δ − e . f e ι e ad( δ e ) ι a f v Equality holds: to see this, we only need to check the case where e is not collapsed. In thiscase, note that δ e ∈ G (cid:48) w was defined to take ( t a g a ) . ˜ f (˜ e ), whose image under the correspon-dence st( ˜ w ) ∼ = (cid:97) e ∈ st( w ) G (cid:48) w /ι e ( G (cid:48) e ) × { e } is ([1] , e ) to ( g w f (cid:93) ( t e )) . ˜ f (˜ e ). This completes the definition of f as a morphism of graphs ofgroups. Checking that f induces ˜ f and f (cid:93) is straightforward; we leave it to the reader.The correspondence established above is not quite perfect: the following two operationson f : ( G , p ) → ( G (cid:48) , q ) do not change f (cid:93) nor ˜ f .1. Let v (cid:54) = p be a vertex of G and g ∈ G (cid:48) f ( v ) . Replace f v with ad( g ) ◦ f v and replace δ e with gδ e for each oriented edge e with τ ( e ) = v .2. Let e be an edge of G not containing p and g ∈ G (cid:48) f ( e ) . Replace δ e and δ ¯ e with δ e ι f ( e ) ( g )and δ ¯ e ι f ( e ) ( g ), respectively.We shall consider two morphisms f and f (cid:48) equivalent fixing p if f can be transformed into f (cid:48) by a finite sequence of the above operations. If one ignores the stipulations around thebasepoint, we say the two morphisms are equivalent. We are only interested in morphismsup to equivalence. 5 opological Representatives.
The previous propositions give us a method for transfer-ring between morphisms f : G → G (cid:48) and equivariant pairs ˜ f : Γ → Γ (cid:48) and f (cid:93) : π ( G , p ) → π ( G (cid:48) , q ), and more generally, those maps which become morphisms after subdividing eachedge in the domain into finitely many edges. A map f : G → G (cid:48) is a homotopy equivalence ifthere exists a map g : G (cid:48) → G such that each double composition gf and f g is homotopic tothe identity—one can see this homotopy as maps of graphs of groups or by lifting to mapsof Bass–Serre trees which are equivariantly homotopic to the identity. A homotopy equiv-alence f : G → G is a topological representative if it maps vertices to vertices and edges tonontrivial tight edge paths. In the following sections, we prove Theorem A and Theorem Bby performing a number of operations on topological representatives.Given a graph of groups G , The collection of outer automorphisms of F = π ( G ) thatadmit a topological representative f : G → G forms a subgroup of Out( F ). We suggest thename modular group or mapping class group of G for this group. In some cases this subgroupis all of Out( F ). One case where this happens is when F is virtually free and vertex groupsof G are finite. An outer automorphism ϕ ∈ Out( F ) belongs to this subgroup if for anyconjugacy class [ g ] in F , the conjugacy class ϕ ([ g ]) is elliptic in the Bass–Serre tree for G ifand only if [ g ] is elliptic.Another case where this “modular group” is all of Out( G ) is when F is a free product A ∗ · · · ∗ A n ∗ F k where F k is a free group and A , . . . , A n are freely indecomposable andnot infinite cyclic. In this case one may take the graph of groups G to be the thistle with n prickles and k petals. This is a graph of groups with one vertex (cid:63) with trivial vertex group, n vertices with vertex group each of the A i , and n + k edges. The first n edges connectvertices with nontrivial vertex group to (cid:63) , and the remaining k edges form loops based at (cid:63) . Example 1.4.
Consider F = C ∗ C ∗ C ∗ C = (cid:104) a, b, c, d | a = b = c = d = 1 (cid:105) the free product of four copies of the cyclic group of order two. Let Φ : F → F be theautomorphism Φ a (cid:55)→ bb (cid:55)→ cc (cid:55)→ dd (cid:55)→ cbdadbc, (notice that, e.g. c − = c ). A topological representative f : G → G of Φ on the thistle withfour prickles is depicted in Figure 1. (cid:104) a (cid:105) e (cid:104) b (cid:105) e (cid:104) c (cid:105) e (cid:104) d (cid:105) e f e (cid:55)→ e e (cid:55)→ e e (cid:55)→ e e (cid:55)→ e ¯ e de ¯ e be ¯ e ce Figure 1: The topological realization f : G → G . The purpose of this section is to prove the irreducible case of Theorem A. The strategy isa straightforward adaptation of the arguments of [BH92, Section 1] to graphs of groups.6t the end of the section we prove a proposition characterizing irreducibility for outerautomorphisms of free products.Fix once and for all a graph of groups G . A marked graph of groups is a graph of groups G together with a homotopy equivalence σ : G → G .Given a topological representative f : G → G and an ordering e , . . . , e m of the edgesof G , there is an associated m × m transition matrix M with ij th entry counting thenumber of times the f -image of the j th edge crosses the i th edge in either direction. Themap f is irreducible if the matrix M is irreducible. Associated to every irreducible matrixis its Perron–Frobenius eigenvalue λ ≥
1. An irreducible matrix with Perron–Frobeniuseigenvalue λ = 1 is a transitive permutation matrix. The transition matrix of Example 1.4is , which is irreducible and for which the Perron–Frobenius eigenvalue is the largest real rootof the polynomial x − x − x − x − λ ≈ . v of G inessential if for some oriented edge e with τ ( e ) = v , the homomor-phism ι e : G e → G v is surjective.A subgraph G of G is invariant with respect to the topological representative f : G → G if f ( G ) ⊂ G . It is a forest if each component C of G is a tree and in the induced graphof groups structure we have that π ( G| C ) acts with global fixed point on its Bass–Serre tree.In C , this means there is a choice of vertex v in C and an orientation of each edge e of C toward this vertex such that each homomorphism ι ¯ e : G e → G τ (¯ e ) is surjective. A forestis nontrivial if it contains at least one edge. An outer automorphism ϕ ∈ Out( π ( G )) is irreducible if it admits a topological representative f : G → G and if whenever G has noinessential valence-one vertices and no nontrivial invariant forests, then f is irreducible.A homotopy equivalence f : G → G taking vertices to vertices is tight if for each edge e , either f ( e ) is a tight edge path, or f ( e ) is a vertex. A homotopy equivalence may be tightened to a tight homotopy equivalence by a homotopy relative to the vertices of G . Lemma 2.1 ([BH92] p. 7) . If f : G → G is a tight homotopy equivalence, collapsing amaximal pretrivial forest in G produces a topological representative f (cid:48) : G (cid:48) → G (cid:48) . If instead f : G → G is a topological representative of an irreducible outer automorphism and G has noinessential valence-one vertices, collapsing a maximal invariant forest yields an irreducibletopological representative f (cid:48) : G (cid:48) → G (cid:48) .Proof. A forest in G is pretrivial with respect to a homotopy equivalence f : G → G ifeach edge in the forest is eventually mapped to a point. Maximal pretrivial forests are inparticular invariant. We describe how to collapse invariant forests.If f : G → G is a tight homotopy equivalence and G ⊂ G is an invariant forest, define G = G /G to be the quotient graph of groups obtained by collapsing each component C of G to a vertex. The vertex group of the vertex determined by C is π ( G| C ). Since G is a forest, this fundamental group is equal to some vertex group in C . Let π : G → G be the quotient map, and define f = πf π − : G → G . Since G was f -invariant, this iswell-defined. If e ⊂ G is an edge not in G , then the edge path for f ( e ) is obtained from f ( e ) by deleting all occurrences of edges in G . Since f was tight, if eσ ¯ e is a subpath of the f -image of some edge e (cid:48) not in G , where σ is a nontrivial path in G , then σ must be ofthe form σ (cid:48) g ¯ σ (cid:48) for some path σ (cid:48) in G and g an element of some vertex group. In f ( e (cid:48) ), thepath eσ ¯ e is replaced by eg ¯ e . This implies that f : G → G is tight. The transition matrixfor f : G → G is obtained from the transition matrix for f : G → G by deleting the rowsand columns associated to the edges of G . 7ecall we write st( v ) for the set of oriented edges e with terminal vertex τ ( e ) = v . A turn at v is a pair of elements of the set (cid:97) e ∈ st( v ) G v /ι e ( G e ) × { e } . If f : G → G is a morphism of graphs of groups, f determines a map Df sending a turnbased at v to a turn based at f ( v ) via the rule { ([ g ] , e ) , ([ g ] , e ) } (cid:55)→ { ([ f v ( g ) δ e ] , f ( e )) , ([ f v ( g ) δ e ] , f ( e )) } . The pseudonaturality condition ensures that this map is well-defined; we have f v ( gι e ( h )) δ e = f v ( g ) f v ( ι e ( h )) δ e = f v ( g ) δ e ι f ( e ) ( f e ( h )) . If f is a topological representative instead of a morphism, there is nonetheless a well-definedmap Df of turns given by first subdividing so that f becomes a morphism and then actingvia the above rule. In Example 1.4, the vertex (cid:63) is mapped to itself by f ; the restriction of Df to (cid:63) is determined by the dynamical system e (cid:55)→ e (cid:55)→ e ↔ e .A turn is degenerate if it consists of a pair of identical elements and is nondegenerate otherwise. A turn is illegal with respect to a topological representative f : G → G if itsimage under some iterate of Df is degenerate and is legal otherwise. In Example 1.4, a turn { e i , e j } based at (cid:63) is illegal if i and j are equal mod 2, and is legal otherwise.Consider the edge path γ = g e g e · · · e k g k +1 . We say γ takes the turns { ([1] , e i ) , ([ g i +1 ] , ¯ e i +1 ) } . The path γ is legal if it takes only legalturns.A topological representative f : G → G is a train track map if f ( e ) is a legal path foreach edge e of Γ. Equivalently, f is a train track map if for each k ≥ e of Γ,we have that f k ( e ) is a tight edge path. In Example 1.4, f is not a train track map becausethe image of e takes the illegal turn { e , e } . Example 1.4 Continued.
Let us fold f at the illegal turn { e , e } . To do this, firstsubdivide e at the preimage of the vertex with vertex group (cid:104) c (cid:105) so e becomes the edgepath e (cid:48) e (cid:48)(cid:48) and identify e (cid:48)(cid:48) with e . The action of the resulting map f (cid:48) : G → G is obtainedfrom f by replacing instances of e with e (cid:48) e . Thus we have f (cid:48) ( e ) = e ¯ e ¯ e (cid:48) de (cid:48) e ¯ e be ¯ e c. Tighten f (cid:48) by a homotopy with support on e (cid:48) to remove e ¯ e , yielding an irreducible topo-logical representative f : G → G . See Figure 2.The Perron–Frobenius eigenvalue λ for f : G → G is the largest real root of thepolynomial x − x − x + x − λ ≈ . λ < λ . However, f is stillnot a train track map: Df sends the turn { (1 , e (cid:48) ) , ( b, ¯ e ) } , which is crossed by f ( e (cid:48) ) to { ( c, ¯ e ) , ( c, ¯ e ) } ; thus this turn is illegal. We cannot quite fold e and the end of ¯ e (cid:48) becausethe f -image of the latter ends with ¯ e c . Lifting to the universal cover ˜ f : Γ → Γ , it isnot the edge ˜ e (cid:48) which is folded with ˜ e but b. ˜ e (cid:48) . We may remedy the situation by changingthe fundamental domain in Γ , or equivalently by changing the marking on G by twistingthe edge e (cid:48) by b − = b . This replaces (cid:104) d (cid:105) with (cid:104) bdb (cid:105) , replaces f ( e ) with e (cid:48) be and replaces f ( e (cid:48) ) with e ¯ e b ¯ e (cid:48) de (cid:48) e ¯ e . Then we fold e (cid:48) and ¯ e . The resulting graph of groups G isabstractly isomorphic to our original graph of groups G , but the marking differs. The actionof the resulting map f (cid:48)(cid:48) : G → G on edges is obtained by replacing instances of e (cid:48) with e (cid:48)(cid:48) ¯ e . Thus we have f (cid:48)(cid:48) ( e (cid:48)(cid:48) ) = e ¯ e be e (cid:48)(cid:48) bdbe (cid:48)(cid:48) ¯ e e , a (cid:105) e (cid:104) b (cid:105) e (cid:104) c (cid:105) e (cid:104) d (cid:105) e (cid:48) f e (cid:55)→ e e (cid:55)→ e e (cid:55)→ e (cid:48) e e (cid:48) (cid:55)→ e ¯ e ¯ e (cid:48) de (cid:48) be ¯ e c Figure 2: The topological representative f : G → G . (cid:104) a (cid:105) e (cid:104) b (cid:105) e (cid:104) c (cid:105) e (cid:104) bdb (cid:105) e (cid:48)(cid:48) f e (cid:55)→ e e (cid:55)→ e e (cid:55)→ e (cid:48)(cid:48) ¯ e be e (cid:55)→ e ¯ e be ¯ e (cid:48)(cid:48) bdbe (cid:48)(cid:48) Figure 3: The topological representative f : G → G .and we may tighten to produce an irreducible topological representative f : G → G . SeeFigure 3. The Perron–Frobenius eigenvalue λ is the largest real root of x − x − x +2 x − λ ≈ . λ < λ . The restriction of Df to turns incident to (cid:63) isdetermined by the dynamical system e (cid:55)→ e ↔ e , e (cid:55)→ e . The only illegal turn in G is { e , e } , which is not crossed by the f -image of any edge, so f : G → G is a train trackmap.The main result of this section is the following theorem. Theorem 2.2.
Suppose ϕ ∈ Out( π ( G )) is irreducible. Then there exists a train track maprepresenting ϕ . The broad-strokes outline of the proof of Theorem 2.2 is much the same as the previousexample. By folding at illegal turns, we often produce nontrivial tightening, which decreasesthe Perron–Frobenius eigenvalue. By controlling the presence of valence-one and valence-two vertices, we may argue that the transition matrix lies in a finite set of matrices, thus thePerron–Frobenius eigenvalue may only be decreased finitely many times. In the remainderof this section, we make this precise by recalling Bestvina and Handel’s original analysis.The proofs are identical to the original, so we omit them.
Subdivision.
Given a topological representative f : G → G , if p is a point in the interiorof an edge e such that f ( p ) is a vertex, we may give G a new graph of groups structure bydeclaring p to be a vertex, with vertex group equal to G e . Lemma 2.3 (Lemma 1.10 of [BH92]) . If f : G → G is a topological representative and f : G → G is obtained by subdivision, then f is a topological representative. If f isirreducible, then f is too, and the associated Perron–Frobenius eigenvalues are equal. alence-One Homotopy. Recall that a valence-one vertex v with incident edge e is inessential if the monomorphism ι e : G e → G v is an isomorphism.If v is an inessential valence-one vertex with incident edge e , let G denote the subgraph ofgroups determined by G \ { e, v } , and let π : G → G be the map collapsing e . Let f : G →G be the topological representative obtained from πf | G by tightening and collapsing amaximal pretrivial forest. We say that f : G → G is obtained from f : G → G by a valence-one homotopy.
Lemma 2.4 (Lemma 1.11 of [BH92]) . If f : G → G is an irreducible topological represen-tative with Perron–Frobenius eigenvalue λ and f : G → G is obtained from f : G → G by performing valence-one homotopies on all inessential valence-one vertices of G followedby the collapse of a maximal invariant forest, then f : G → G is irreducible, and theassociated Perron–Frobenius eigenvalue λ satisfies λ < λ . Valence-Two Homotopy.
We likewise distinguish two kinds of valence-two vertices. Avalence-two vertex v with incident edges e i and e j is inessential if at least one of themonomorphisms ι e i : G e i → G v and ι e j : G e j → G v is an isomorphism, say ι e j : G e j → G v . Let π be the map that collapses e j to a point and expands e i over e j . Define a map f (cid:48) : G → G by tightening πf . Observe that no vertex of G is mapped to v . Thus we may define a newgraph of groups structure G (cid:48) by removing v from the set of vertices. Thus the edge path e i ¯ e j is now an edge, which we will call e i with edge group G e i . Let f (cid:48)(cid:48) : G (cid:48) → G (cid:48) be themap obtained by tightening f (cid:48)(cid:48) ( e i ) = f (cid:48) ( e i ¯ e j ). Finally, let f : G → G be the topologicalrealization obtained by collapsing a maximal pretrivial forest. We say that f : G → G isobtained by a valence-two homotopy of v across e j . Lemma 2.5 (Lemma 1.12 of [BH92]) . Let f : G → G be an irreducible topological repre-sentative, and suppose G has no inessential valence-one vertices. Suppose f : G → G isthe irreducible topological representative obtained by performing a valence-two homotopy of v across e j followed by the collapse of a maximal invariant forest. Let M be the transitionmatrix of f and choose a positive eigenvector (cid:126)w with M (cid:126)w = λ (cid:126)w . If w i ≤ w j , then λ ≤ λ ;if w i < w j , then λ < λ . Remark 2.6.
The statement of the lemma hides a problem: if we cannot freely choose whichedge incident to an inessential valence-two vertex to collapse via a valence-two homotopy,we may be forced to increase λ . Fortunately, such valence-two vertices are not produced inthe proof of Theorem 2.2. Folding.
Suppose some pair of edges e , e in G have the same f -image. Define a newgraph of groups G by identifying e and e to a single edge e . The map f : G → G descendsto a well-defined homotopy equivalence f : G → G . This is an elementary fold. Moregenerally if e (cid:48) and e (cid:48) are maximal initial segments of e and e with equal f -images andendpoints sent to a vertex by f , we first subdivide at the endpoints of e (cid:48) and e (cid:48) if they arenot already vertices and then perform an elementary fold on the resulting edges. Lemma 2.7 (Lemma 1.15 of [BH92]) . Suppose f : G → G is an irreducible topological rep-resentative and that f : G → G is obtained by folding a pair of edges. If f is a topologicalrepresentative, then it is irreducible, and the associated Perron–Frobenius eigenvalues satisfy λ = λ . Otherwise, let f : G → G be the irreducible topological representative obtained bytightening, collapsing a maximal pretrivial forest, and collapsing a maximal invariant forest.Then the associated Perron–Frobenius eigenvalues satisfy λ < λ .Proof of Theorem 2.2. Let f : G → G be an irreducible topological representative of ϕ . Sup-pose the Perron–Frobenius eigenvalue λ satisfies λ = 1. Then f transitively permutes theedges of G and is thus a train track map.So assume λ >
1. By performing valence-one and valence-two homotopies, we mayassume that G has no inessential valence-one or valence-two vertices except in the corner10ase where G has one (inessential) vertex and one edge that forms a loop. In this corner caseany topological representative is irreducible and the Perron–Frobenius eigenvalue satisfies λ = 1, thus we may discard this case.Call a vertex v of G essential if for all oriented edges e ∈ st( v ), the monomorphism ι e : G e → G v is not surjective. Let η ( G ) be the number of essential vertices of G , and let β ( G ) be the first Betti number of G . A homotopy equivalence of graphs of groups sendsessential vertices to essential vertices, so the quantities η ( G ) and β ( G ) are equal for anymarked graph of groups obtained from G by any of the operations described in this section.We claim that G has at most 2 η ( G ) + 3 β ( G ) − G (cid:48) from G by cyclically ordering the essential vertices of G and attaching an edge from eachessential vertex to its neighbors in the cyclic ordering. The graph G (cid:48) has no valence-one orvalence-two vertices and first Betti number η ( G ) + β ( G ). An Euler characteristic argumentreveals that G (cid:48) has at most 3( η ( G ) + β ( G )) − G follows.We will show that if f : G → G is not a train track map, then there is an irreducibletopological representative f : G → G without inessential valence-one or valence-two ver-tices such that the associated Perron–Frobenius eigenvalues satisfy λ < λ . The argumentin the previous paragraph shows that the size of the transition matrix of f is uniformlybounded. Furthermore, if M is an irreducible matrix, its Perron–Frobenius eigenvalue λ isbounded below by the minimum sum of the entries of a row of M . To see this, let (cid:126)w be apositive eigenvector. If w j is the smallest entry of (cid:126)w , λw j = ( M (cid:126)w ) j is greater than w j timesthe sum of the entries of the j th row of M .Thus if we iterate this argument reducing the Perron–Frobenius eigenvalue, there areonly finitely many irreducible transition matrices that can occur, so at some finite stage thePerron–Frobenius eigenvalue will reach a minimum. At this point, we must have a traintrack map.To complete the proof, we turn to the question of decreasing λ . Suppose f : G → G is nota train track map. Then there exists a point p in the interior of an edge such that f ( p ) is avertex, and f k is not locally injective (as a map of graph of groups) at p for some k >
1. Weassume that topological representatives act linearly on edges with respect to some metricon G . Since λ >
1, this means the set of points of G eventually mapped to a vertex is dense.Thus we can choose a neighborhood U of p so small that it satisfies the following conditions.1. The boundary ∂U is a two-point set { s, t } , where f (cid:96) ( s ) and f (cid:96) ( t ) are vertices for some (cid:96) ≥ f i | U is injective for 1 ≤ i ≤ k − f k is two-to-one on U \ { p } , and f k ( U ) is contained within a single edge.4. p / ∈ f i ( U ), for 1 ≤ i ≤ k .First we subdivide at p . Then we subdivide at f i ( s ) and f i ( t ) for 1 ≤ i ≤ (cid:96) − p has valence two; denote the incident edgesby e and e (cid:48) . Observe that f k − ( e ) and f k − ( e (cid:48) ) are single edges that are identified by f .Thus we may fold. The resulting map f (cid:48) : G (cid:48) → G (cid:48) may be a topological realization, in whichcase the Perron–Frobeniues eigenvalue λ (cid:48) satisfies λ (cid:48) = λ . In this case f (cid:48) k − ( e ) and f (cid:48) k − ( e (cid:48) )are single edges that are identified by f . In the contrary case, nontrivial tightening occurs.After collapsing a maximal pretrivial forest and a maximal invariant forest, the resultingirreducible topological representative f (cid:48)(cid:48) : G (cid:48)(cid:48) → G (cid:48)(cid:48) has Perron–Frobenius eigenvalue λ (cid:48)(cid:48) satisfies λ (cid:48)(cid:48) < λ .Repeating this dichotomy k times if necessary, we have either decreased λ , or we havefolded e and e (cid:48) so that p is now an inessential valence-one vertex.We remove inessential valence-one and valence-two vertices by the appropriate homo-topies. Since valence-one homotopy always decreases the Perron–Frobenius eigenvalue, theresulting irreducible topological representative f : G → G has Perron–Frobenius eigen-value λ satisfying λ < λ . 11 emark 2.8. As in the original, the proof of Theorem 2.2 provides in outline an algorithmthat takes as input a topological representative of an irreducible outer automorphism andreturns a train track map.A reduction for an outer automorphism ϕ ∈ Out( π ( G )) is a topological representative f : G → G which has no inessential valence-one vertices and no invariant forests but has anontrivial invariant subgraph. If ϕ has a reduction, then it is reducible —i.e. not irreducible.Let F = A ∗ · · · ∗ A n ∗ F k be a free product, represented as the fundamental group of agraph of groups G with trivial edge groups, where the A i are vertex groups of G and theordinary fundamental group of G is free of rank k . For example the A i might be freelyindecomposable and not infinite cyclic, in which case G is a Grushko splitting of F . Definethe complexity of F relative to G to be the quantity n + 2 k −
1. If F (cid:48) is a free factor of F relative to this free product decomposition, we may define the complexity of F (cid:48) relative to G analogously. The final result of this section is the following characterization of reducibilityfor outer automorphisms ϕ ∈ Out( F ) represented on G -marked graphs of groups. Proposition 2.9.
Let F be a free product. An outer automorphism ϕ ∈ Out( F ) is reduciblerelative to G if and only if there are free factors F , . . . , F m of F with positive complexitysuch that F ∗ · · · ∗ F m is a free factor of F and ϕ cyclically permutes the conjugacy classesof the F i .Proof. Suppose first that ϕ is reducible relative to G ; let f : G → G be a reduction and let G i = f i ( G ), 0 ≤ i ≤ m − f -invariantsubgraph. Then each π ( G| G i ) determines a free factor F i with positive complexity suchthat F ∗ · · · ∗ F m is a free factor of F and such that ϕ cyclically permutes the conjugacyclasses of the F i .Conversely, suppose F , . . . , F m are free factors with positive complexity as in the state-ment of the proposition. Take F m +1 a free factor so that F = F ∗ · · · ∗ F m ∗ F m +1 . Supposethat n i and k i are the data determining the complexity of F i for 1 ≤ i ≤ m + 1. Let G i bethe thistle with n i prickles and k i petals (if n m +1 = k m +1 = 0, then G m +1 is a vertex) anddistinguished vertex (cid:63) i . For each i satisfying 1 ≤ i ≤ m choose automorphisms Φ i : F → F representing ϕ such that Φ( F i ) = F i +1 , with indices taken mod m , and let f i : G i → G i +1 be the corresponding topological representatives taking (cid:63) i to (cid:63) i +1 . Define G to be the unionof the G i for 1 ≤ i ≤ m + 1 together with, for 1 ≤ i ≤ m , an oriented edge E i connecting (cid:63) i to (cid:63) m +1 .Collapsing the E i to a point yields a homotopy equivalence G → G , where G is thethistle with n prickles and k petals. Identifying the image of π ( G i , (cid:63) i ) with G i will serve as(the inverse of) a marking. We will use Φ to create a topological representative f : G → G for ϕ . Define f ( G i ) = f i ( G i ) for 1 ≤ i ≤ m . By assumption there exist c i ∈ F such thatΦ ( x ) = c i Φ i ( x ) c − i . Choose γ i a closed tight edge path based at (cid:63) m +1 representing c i (so γ is the trivial path) and define f ( E i ) = γ i E i +1 with indices taken mod m . Finally define f ( G m +1 ) by Φ and the marking on (Γ , G ).The topological representative f : G → G is a reduction for ϕ unless G has an invariantcontractible forest. Since thistles have contractible subgraphs, there are a few possibilities.If there is a family of edges e , . . . , e m with e i ∈ G i and f ( e i ) = e i +1 with indices mod m ,we may collapse each of these edges. Likewise if some edge of G m +1 is sent to itself, we maycollapse it. If each c i = 1 ∈ F , then the E i also form an invariant forest that is contractibleif the subgraph they span contains at most one vertex with vertex group some A i . After allthese forest collapsings, the only worry is that F m +1 has complexity zero and the E i wouldbe collapsed, leaving G as the only f -invariant subgraph. In this case, choose A an edgeof G sharing an initial vertex with E , and change f via a homotopy with support in E so that f ( E ) = f ( A ) f ( ¯ A ) E , then fold the initial segment of E mapping to f ( A ) with allof A . The resulting graph is combinatorially identical to G but the markings differ. Now f ( E ) = f ( ¯ A ) E and f ( E k ) = ¯ AE , so the E i no longer form an invariant forest.12 Relative Train Track Maps
The purpose of this section is to prove the general case of Theorem A. The strategy is toadapt arguments in [BH92, Section 5] and [FH18, Section 2]. At the end of the sectionwe prove that given an outer automorphism ϕ of a free product and a nested sequence of ϕ -invariant free factor systems, there is a relative train track map such that the free factorsystems are realized by filtration elements. Filtrations. A filtration on a marked graph of groups G with respect to a topologicalrepresentative f : G → G is an increasing sequence ∅ = G ⊂ G ⊂ · · · ⊂ G m = G of f -invariant subgraphs. The subgraphs are not required to be connected. Strata.
The r th stratum of G is the subgraph H r containing those edges of G r not con-tained in G r − . An edge path has height r if it is contained in G r and meets the interior of H r . If both edges of a turn T are contained in a stratum H r , then T is a turn in H r . If apath has height r and contains no illegal turns in H r then it is r -legal. Transition Submatrices.
Relabeling the edges of G and thus permuting the rows andcolumns of the transition matrix M so that the edges of H i precede those of H i +1 , M becomes block upper-triangular, with the i th block M i equal to the square submatrix of M containing those rows and columns corresponding to edges in H i .A filtration is maximal when each M i is either irreducible or the zero matrix. If M i is irreducible, call H i an irreducible stratum and a zero stratum otherwise. If H i is irre-ducible, M i has an associated Perron–Frobenius eigenvalue λ i ≥
1. If λ i >
1, then H i is an exponentially-growing stratum. Otherwise λ i = 1, we say H i is non-exponentially-growing and M i is a transitive permutation matrix. Eigenvalues.
Let H r , . . . , H r k be the exponentially-growing strata for f : G → G . Wedefine PF( f ) to be the sequence of associated Perron–Frobenius eigenvalues λ r , . . . , λ r k innonincreasing order. We order the set { PF( f ) | f : G → G is a topological representative } lexicographically; thus if PF( f ) = λ , . . . , λ k and PF( f (cid:48) ) = λ (cid:48) , . . . , λ (cid:48) (cid:96) , then PF( f ) < PF( f (cid:48) )if there is some j with λ j < λ (cid:48) j and λ i = λ (cid:48) i for i satisfying 1 ≤ i < j , or if k < (cid:96) and λ i = λ (cid:48) i for i satisfying 1 ≤ i ≤ k . Relative Train Track Maps.
Throughout the paper, we will assume our filtrations aremaximal unless otherwise specified. Given σ a path in G , let f (cid:93) ( σ ) denote the tight pathhomotopic rel endpoints to f ( σ ). We will denote the filtration associated to f : G → G as ∅ = G ⊂ · · · ⊂ G m = G . A topological representative f : G → G is a relative train trackmap if for every exponentially-growing stratum H r , we have(EG-i) Turns in H r are mapped to turns in H r by Df ; every turn with one edge in H r andthe other in G r − is legal.(EG-ii) If σ ⊂ G r − is a nontrivial path with endpoints in H r ∩ G r − , then f (cid:93) ( σ ) is as well.(EG-iii) If σ ⊂ Γ r is a legal path, then f ( σ ) is a (tight) r -legal path.The main result of this section is Theorem 3.1.
There is an algorithm that takes as input a topological representative f : G →G and improves it to a relative train track map f (cid:48) : G (cid:48) → G (cid:48) .
13e sketch the outline of the proof: we begin with a topological representative that is bounded, a term which will be defined below. We use two new operations, described inLemma 3.3 and Lemma 3.4 so that the resulting topological representative satisfies (EG-i)and (EG-ii). If (EG-iii) is not satisfied, as in [BH92] and [FH18], we modify the algorithmin the proof of Theorem 2.2 to reduce PF( f ), the set of Perron–Frobenius eigenvalues forthe exponentially-growing strata of f : G → G , while remaining bounded. The boundednessassumption ensures that we will hit a minimum value after a finite number of moves, atwhich point (EG-iii) will be satisfied.
Bounded Representatives.
As we observed in the proof of Theorem 2.2, if G is a markedgraph of groups without inessential valence-one or valence-two vertices, then G has at most2 η ( G ) + 3 β ( G ) − G . Our assumption that ϕ was irreducible allowed us to remove valence-two vertices, but we cannot always do this in the general case. Instead, call a topologicalrepresentative f : G → G bounded if there are at most 2 η ( G ) + 3 β ( G ) − H r , the associated Perron–Frobenius eigenvalue λ r is also the Perron–Frobenius eigenvalue of a matrix with at most2 η ( G )+3 β ( G ) − f : G → G is bounded,the set of PF( f (cid:48) ) for f (cid:48) : G (cid:48) → G (cid:48) a bounded representative of ϕ satisfying PF( f (cid:48) ) ≤ PF( f )is finite, so operations decreasing PF( f ) will eventually reach a minimum, which we willdenote PF min . Elementary Moves Revisited.
In [BH92, Lemmas 5.1–5.4], Bestvina and Handel revisitthe four elementary moves subdivision, valence-one homotopy, valence-two homotopy and folding to analyze their impact on PF( f ). All of these moves except valence-two homotopyproduce a topological representative f (cid:48) : G (cid:48) → G (cid:48) such that the associated Perron–Frobeniuseigenvalues satisfy PF( f (cid:48) ) ≤ PF( f ). In the case of valence-two homotopy within a singleexponentially-growing stratum H r , it may happen that λ r is replaced by some number ofeigenvalues λ (cid:48) that all satisfy λ (cid:48) ≤ λ r , so it is possible that PF( f (cid:48) ) > PF( f ). Nonetheless,we have the following result. Call an elementary move safe if performing it on a topologicalrepresentative f : G → G yields a new topological representative f (cid:48) : G (cid:48) → G (cid:48) with PF( f (cid:48) ) ≤ PF( f ). Lemma 3.2 ([BH92] Lemma 5.5) . If f : G → G is a bounded topological representative and f (cid:48) : G (cid:48) → G (cid:48) is obtained from f by a sequence of safe moves with PF( f (cid:48) ) < PF( f ) , then thereis a bounded topological representative f (cid:48)(cid:48) : G (cid:48)(cid:48) → G (cid:48)(cid:48) with PF( f (cid:48)(cid:48) ) < PF( f ) . The idea of the proof is the following: given f (cid:48) : G (cid:48) → G (cid:48) , perform all valence-one ho-motopies and safe valence-two homotopies. Then perform valence-two homotopies until theresulting topological representative f (cid:48)(cid:48) : G (cid:48)(cid:48) → G (cid:48)(cid:48) is bounded. We have PF( f (cid:48) ) ≤ PF( f (cid:48)(cid:48) ) < PF( f ). Invariant Core Subdivision.
We recall the construction of the invariant core subdivision of an exponentially-growing stratum H r . Assume that a topological representative f : G → G linearly expands edges over edge paths with respect to some metric on G . If f ( H r ) is notentirely contained in H r , then the set I r := { x ∈ H r | f k ( x ) ∈ H r for all k > } is an f -invariant Cantor set. The invariant core of an edge e in H r is the smallest closedsubinterval of e containing the intersection of I r with the interior of e . The endpoints ofinvariant cores of edges in H r form a finite set which f sends into itself. Declaring elementsof this finite set to be vertices is called invariant core subdivision. The following lemma says that invariant core subdivision can be used to create topolog-ical representatives whose exponentially-growing strata satisfy (EG-i).14 emma 3.3 ([BH92] Lemma 5.13) . If f (cid:48) : G (cid:48) → G (cid:48) is obtained from f : G → G by aninvariant core subdivision of an exponentially-growing stratum H r , then PF( f (cid:48) ) = PF( f ) ,and the map Df (cid:48) maps turns in the resulting exponentially-growing stratum H (cid:48) r to itself,so H (cid:48) r satisfies (EG-i). If H j is another exponentially-growing stratum for f : G → G thatsatisfies (EG-i) or (EG-ii), then the resulting exponentially-growing stratum H (cid:48) j for f (cid:48) : G (cid:48) →G (cid:48) still satisfies those properties. In fact, invariant core subdivision affects only edges in H r . If new vertices are created,then one or more non-exponentially-growing strata are added to the filtration below H r . Collapsing Inessential Connecting Paths.
The following lemma says that an appli-cation of operations already defined may be used to construct topological representativeswhose exponentially-growing strata satisfy (EG-ii).
Lemma 3.4 ([BH92] Lemma 5.14) . Let f : G → G be a topological representative withexponentially-growing stratum H r . If α ⊂ G r − is a path with endpoints in H r ∩ G r − suchthat f (cid:93) ( α ) is trivial, we construct a new topological representative f (cid:48) : G (cid:48) → G (cid:48) such that if H (cid:48) r is the stratum of G (cid:48) determined by H r , then H (cid:48) r ∩ G (cid:48) r − has fewer points than H r ∩ G r − . The new topological representative f (cid:48) : G (cid:48) → G (cid:48) is constructed by subdividing at thepreimages of vertices in α and repeatedly folding, followed by tightening and collapsing apretrivial forest. As such, PF( f (cid:48) ) ≤ PF( f ). If H j is an exponentially-growing stratum for f : G → G with j > r that satisfies (EG-i) or (EG-ii), the resulting exponentially-growingstratum H (cid:48) j still satisfies these properties. If H r satisfies (EG-i), then so does H (cid:48) r . Lemma 3.5 ([FH18] Lemma 2.4) . There is an algorithm that checks whether a topologicalrepresentative f : G → G is a relative train track map.Proof.
Since (EG-i) is a finite property, we may assume that each exponentially-growingstratum satisfies (EG-i). Suppose H j is an exponentially-growing stratum. A connectingpath for H j is a tight path α in G r − with endpoints in H r ∩ G r − . Let C be a component of G r − . If C is contractible, then there are only finitely many tight paths in C with endpointsat vertices, so checking (EG-ii) for connecting paths in C is a finite property.So suppose C is noncontractible. We claim that (EG-ii) for C is equivalent to theproperty that each vertex in H j ∩ C is periodic. If this latter property fails, there is some k > v and w in H j ∩ C with f k ( v ) = f k ( w ). In this case, there is some pathconnecting f k − ( v ) and f k − ( w ) whose f (cid:93) -image is trivial (consider what a homotopy inversefor f does to f k ( v )), so (EG-ii) fails. If each vertex in H j ∩ C is periodic, each connectingpath α for H j contained in C either has distinct endpoints or determines a nontrivial loopin the fundamental group of G based at its endpoint. Both of these properties are preservedby f (cid:93) , so (EG-ii) holds.Finally, (EG-iii) for H j is equivalent to checking that f ( e ) is j -legal for each edge e ∈ H j ,so is a finite-property. Proof of Theorem 3.1.
We begin with a topological representative and applying valence-one and valence-two homotopies until the resulting topological representative f : G → G isbounded. Consider the highest exponentially-growing stratum H r of G . We check whether H r satisfies (EG-i) and (EG-ii) using Lemma 3.5. If not, apply Lemma 3.3 and Lemma 3.4to create a new topological representative, still called f : G → G such that the resultingexponentially-growing stratum H r satisfies (EG-i) and (EG-ii). Repeat with the next highestexponentially-growing stratum until all exponentially-growing strata satisfy these properties.Check whether the resulting topological representative, which we still call f : G → G , satisfies(EG-iii). If it does, we are done.If not, then there is some edge e in an exponentially-growing stratum H r such that f ( e )is not r -legal. We apply the algorithm in the proof of Theorem 2.2: there is a point P in H r where f k is not injective at P for some k >
1. We subdivide and then repeatedly fold.15ither we have reduced the eigenvalue for H r or produced a valence-one vertex. We removeall valence-one vertices via homotopies and perform all valence-two homotopies which do notincrease PF( f ). At this point we have created a new topological representative f : G (cid:48) → G (cid:48) with PF( f (cid:48) ) < PF( f ), but f (cid:48) may not be bounded. Apply Lemma 3.2 to produce a newbounded topological representative f (cid:48)(cid:48) : G (cid:48)(cid:48) → G (cid:48)(cid:48) with PF( f (cid:48)(cid:48) ) < PF( f ). If (EG-i) and(EG-ii) are not satisfied by f (cid:48)(cid:48) , we may restore these properties without increasing PF( f (cid:48)(cid:48) ).Because PF( f ) can only be decreased finitely many times before reaching PF min , eventuallythis process terminates, yielding a relative train track map. Corollary 3.6. If f : G → G is a bounded topological representative satisfying (EG-i) andwith
PF( f ) = PF min , then the exponentially-growing strata of f satisfy (EG-iii). Free Factor Systems.
Let F = A ∗ · · · ∗ A n ∗ F k be a free product, represented as thefundamental group of a graph of groups G with trivial edge groups, where the A i are vertexgroups of G , and the ordinary fundamental group of G is free of rank k . In Section 2,we showed that if ϕ ∈ Out( F ) is represented on a G -marked graph of groups, then ϕ isirreducible relative to G if it preserved the conjugacy class of no free factor relative to G of positive complexity. If F i is a free factor of F relative to G , let [[ F i ]] denote itsconjugacy class. If F , . . . , F m are free factors of F relative to G with positive complexityand F ∗ · · · ∗ F m is a free factor of F , the collection { [[ F ]] , . . . , [[ F m ]] } is a free factorsystem. Example 3.7. If f : G → G is a topological representative where G is a G -marked graph ofgroups and G r ⊂ G is an f -invariant subgraph with noncontractible connected components C , . . . , C k , then the conjugacy classes [[ π ( G| C i )]] of the fundamental groups of the C i arewell-defined. We define F ( G r ) := { [[ π ( G| C )]] , . . . , [[ π ( G| C k )]] } . Notice that each π ( G| C i ) has positive complexity. We say that G r realizes F ( G r ).The subgroup of Out( F ) representable on a G -marked graph of groups acts on the setof conjugacy classes of free factors of F relative to G . If ϕ is such an outer automorphismand F (cid:48) a free factor, [[ F (cid:48) ]] is ϕ -invariant if ϕ ([[ F (cid:48) ]]) = [[ F (cid:48) ]]. In this case there is someautomorphism Φ : F → F representing ϕ such that Φ( F (cid:48) ) = F (cid:48) and Φ | F (cid:48) is well-definedup to an inner automorphism of F (cid:48) , so it induces an outer automorphism ϕ | F (cid:48) ∈ Out( F (cid:48) ),which we will call the restriction of ϕ to F (cid:48) .There is a partial order (cid:64) on free factor systems: we say [[ F ]] (cid:64) [[ F ]] if F is conjugateto a free factor of F . We say that F (cid:64) F for free factor systems F and F if for each[[ F i ]] ∈ F , there exists [[ F j ]] ∈ F such that [[ F i ]] (cid:64) [[ F j ]]. Proposition 3.8.
Suppose ϕ ∈ Out( F ) is representable on a G -marked graph of groups. If F (cid:64) · · · (cid:64) F d is a nested sequence of ϕ -invariant free factor systems relative to G , thenthere is a relative train track map f : G → G representing ϕ such that each free factor systemis realized by some element of the filtration ∅ = G ⊂ · · · ⊂ G m = G .Proof. The first step is to construct a topological representative f : G → G and an associatedfiltration such that each F i is realized by a filtration element. We proceed by induction on d , the length of the nested sequence of ϕ -invariant free factor systems. The case d = 1 isaccomplished in Proposition 2.9. Let F d = { [[ F ]] , . . . , [[ F (cid:96) ]] } . For each j with 1 ≤ j ≤ d − i satisfying 1 ≤ i ≤ (cid:96) , write F ij for the set of conjugacy classes of free factors in F j that are conjugate into F i . Then for each i , F i (cid:64) · · · (cid:64) F id − is a nested sequence of ϕ -invariant free factor systems. By induction, there are graphs ofgroups G i and topological representatives f i : G i → G i representing the restriction ϕ | F i of16 to F i together with associated (not necessarily maximal!) filtrations G i ⊂ · · · G id − suchthat G ij realizes F ij . Inductively, we may assume that f i fixes some vertex v i of G i , and that G i has no inessential valence-one or valence-two vertices.As in the proof of Proposition 2.9, take a complementary free factor F (cid:96) +1 so that F = F ∗ · · · ∗ F (cid:96) ∗ F (cid:96) +1 , and an associated thistle G (cid:96) +1 . The graph of groups G is constructedas follows: begin with the disjoint union of the G i . Glue G (cid:96) +1 to G by identifying thevertex (cid:63) of G (cid:96) +1 with the fixed point v . Then attach an edge connecting v to v i for2 ≤ i ≤ (cid:96) . The resulting graph of groups has no inessential valence-one or valence-twovertices. Define f : G → G from the f i as in the proof of Proposition 2.9 and observe that f is bounded. For 1 ≤ j ≤ d −
1, define G j = (cid:83) (cid:96)i =1 G ij , and define G d = (cid:83) (cid:96)i =1 G i . Then ∅ = G ⊂ · · · ⊂ G d ⊂ G d +1 = G is an f -invariant filtration, and each F i for i satisfying1 ≤ i ≤ d is realized by G i . Complete this filtration to a maximal filtration; this completesthe first step.The next step is to promote our topological representative to a relative train track map.Note that (cf. the proof of [BFH00, Lemma 2.6.7]) that the moves described in Section 2and this section all preserve the property of realizing free factors. More precisely, suppose C and C are disjoint noncontractible components of some filtration element G r and that f (cid:48) : G (cid:48) → G (cid:48) is obtained from f : G → G by collapsing a pretrivial forest, folding, subdivi-sion, invariant core subdivision, valence-one homotopy or (properly restricted) valence-twohomotopy. If p : G → G (cid:48) is the identifying homotopy equivalence, then p ( C ) and p ( C ) aredisjoint, noncontractible subgraphs of G (cid:48) . Thus we may work freely and apply the proof ofTheorem 3.1 to produce a relative train track map. The purpose of this section is to construct the attracting laminations associated to an outerautomorphism of a free product. We follow [BFH00, Section 3] in our treatment, and wedefer to that paper for proofs and additional details.Let F = A ∗ · · · ∗ A n ∗ F k be a free product represented by a graph of groups G withtrivial edge groups, where the A i are vertex groups of G , and the ordinary fundamentalgroup of G is free of rank k . Let Γ be the Bass–Serre tree for G . A line in Γ is a proper,linear embedding σ : R → Γ. We are interested in the unoriented image of σ more than themap (cf. [BFH00, 2.2]). The Space of Lines.
There is a space of lines in Γ , denoted ˜ B (Γ). It is equipped with a“compact–open” topology: if ˜ γ ⊂ Γ is an edge path (say with endpoints at vertices), define N (˜ γ ) ⊂ ˜ B (Γ) to be the set of lines that contain ˜ γ as a subpath. The sets N (˜ γ ) form a basisfor the topology on ˜ B (Γ). Let π : Γ → G be the natural projection. By declaring preimagesof vertices of vertices of Γ to be vertices, we may give R a graph structure and consider π ◦ σ : R → G as a morphism of graphs of groups (see Example 1.1). This is a line in G .There is a space of lines in G , denoted B ( G ) and a projection ˜ B (Γ) → B ( G ). We give B ( G )the quotient topology. Given an edge path γ in G , define N ( γ ) to be the set of those linesin G that contain γ as a subpath. The sets N ( γ ) form a basis for the topology on B ( G ). Abstract Lines.
Bestvina–Feighn–Handel define spaces of abstract lines, ˜ B and B , usingthe Gromov boundary of the free group. In the case of a free product, the Gromov boundaryof Γ is not compact in general. If one wanted a compact space, one could use the boundaryGuirardel and Horbez define in [GH19]. Let ∂ Γ denote the Gromov boundary of Γ. If f : G → G is a homotopy equivalence and ˜ f : Γ → Γ (cid:48) is any lift to the Bass–Serre trees, that˜ f induces a homeomorphism ∂f : ∂ Γ → ∂ Γ (cid:48) . A line σ in Γ is determined by its endpoints,an unordered pair of distinct points { α, ω } in ∂ Γ. Conversely, given a pair of distinct points { α, ω } in ∂ Γ, there is a unique line σ in Γ connecting them. Thus we have a homeomorphismbetween ˜ B (Γ) and ˜ B = ( ∂ Γ × ∂ Γ \ ∆) / Z / Z , where ∆ denotes the diagonal and Z / Z acts17y interchanging the factors. This homeomorphism is equivariant, taking the action of F onlines in Γ to the diagonal action of F on ∂ Γ × ∂ Γ, thus it yields a homeomorphism between thequotient of ˜ B by the action of F , which we denote B , and B ( G ). If f : G → G is a homotopyequivalence, the homeomorphism ∂f : ∂ Γ → ∂ Γ (cid:48) yields a homeomorphism ˜ f (cid:93) : ˜ B (Γ) → ˜ B (Γ (cid:48) )and a homeomorphism f (cid:93) : B ( G ) → B ( G ). If β ∈ B corresponds to γ ∈ B ( G ), we say γ realizes β in G .An element of F is peripheral in a G -marked graph of groups G if it is conjugate intosome vertex group. The conjugacy class of a nonperipheral element determines a periodic bi-infinite line in B ( G ). A line β ∈ B is carried by the conjugacy class of a free factor [[ F i ]] if it isin the closure of the periodic lines in B determined by the conjugacy classes of nonperipheralelements of F i . If G is a marked graph of groups and K ⊂ G is a connected subgraph suchthat [[ π ( G| K )]] = [[ F i ]], then β is carried by [[ F i ]] if and only if the realization of β in G iscontained in K .Let ϕ ∈ Out( F ) be representable on G . We say β (cid:48) ∈ B is weakly attracted to β ∈ B under the action of ϕ if ϕ k(cid:93) ( β (cid:48) ) → β (note that B is not Hausdorff). A subset U ⊂ B is an attracting neighborhood of β ∈ B for the action of ϕ if ϕ (cid:93) ( U ) ⊂ U and if { ϕ k(cid:93) ( U ) : k ≥ } isa neighborhood basis for β ∈ B . A line σ : R → G in G is birecurrent if every subpath of σ occurs infinitely often as a subpath of each end of σ . Lemma 4.1 ([BFH00] Lemma 3.1.4) . If some realization of β ∈ B in a G -marked graph ofgroups is birecurrent, then every realization of β in a marked graph of groups is birecurrent.If β is birecurrent, then ϕ (cid:93) ( β ) is birecurrent for every ϕ ∈ Out( F ) that is representable on G . A closed subset Λ + ⊂ B is an attracting lamination for ϕ if it is the closure of a singlepoint β such that:1. The line β is birecurrent.2. The line β has an attracting neighborhood for the action of some iterate of ϕ .3. The line β is not carried by any F or Z / Z ∗ Z / Z free factor.The line β is said to be generic for Λ + . The set of attracting laminations for ϕ is denoted L ( ϕ ). Lemma 4.2 ([BFH00] Lemma 3.1.6) . L ( ϕ ) is ϕ -invariant. A nonnegative integral matrix M is aperiodic if there is some power k such that M k hasall positive entries. Aperiodic matrices are irreducible. Suppose f : G → G is a relative traintrack map with filtration ∅ = G ⊂ G ⊂ · · · ⊂ G m = G and that H r is an exponentially-growing stratum. We say that H r is aperiodic if its transition matrix is aperiodic, and that f : G → G is eg-aperiodic if each exponentially-growing stratum is aperiodic. Lemma 4.3 ([BFH00] Lemma 3.1.9) . Suppose that f : G → G is a relative train track mapand that H r is an aperiodic exponentially-growing stratum. There is an attracting lamination Λ + with generic leaf β such that H r is the highest stratum crossed by the realization λ of β in G .Proof. The strategy of Bestvina–Feighn–Handel’s proof is to look at weak limits of f k ( E )for E an edge of H r . They show that λ is not a periodic line, thus it cannot be carried byany F nor Z / Z ∗ Z / Z free factor. Lemma 4.4 ([BFH00] Lemma 3.1.10) . Assume that β ∈ B is a generic line of some Λ + ∈L ( ϕ ) , that f : G → G and ∅ = G ⊂ G ⊂ · · · ⊂ G m = G are a relative train track map andfiltration representing ϕ and that λ is the realization of β in G . Then the highest stratum H r crossed by λ is exponentially-growing and λ is r -legal. roof. We argue by induction on the structure of F . The base cases of F = A , F and Z / Z ∗ Z / Z are trivial, so we assume the result holds for outer automorphisms of proper freefactors of F . The rest of the proof follows from Bestvina–Feighn–Handel’s arguments. Corollary 4.5 ([BFH00] Corollary 3.1.11) . Assume that f : G → G and ∅ = G ⊂ G ⊂· · · ⊂ G m = G are a relative train track map and filtration representing ϕ , and that H r isan aperiodic exponentially-growing stratum. Assume further that β ∈ B is Λ + -generic forsome Λ + ∈ L ( ϕ ) and that H r is the highest stratum crossed by the realization of β in G . Allsuch β have the same closure. Lemma 4.6 ([BFH00] Lemma 3.1.13) . L ( ϕ ) is finite. Lemma 4.7 ([BFH00] Lemma 3.1.14) . The following are equivalent:1. Each element of L ( ϕ ) is ϕ -invariant.2. Each element of L ( ϕ ) has an attracting neighborhood for ϕ (cid:93) .3. Every relative train track map f : G → G representing ϕ is eg-aperiodic.4. Some relative train track map f : G → G representing ϕ is eg-aperiodic. As a step towards the existence of CTs, Feighn and Handel construct in [FH11, Theorem2.19] relative train track maps that satisfy a number of extra properties. The goal of thissection is to prove the existence of such relative train track maps for outer automorphismsof free products. To state the theorem we require some more terminology.
Nielsen Paths.
A path σ is a periodic Nielsen path with respect to a topological repre-sentative f : G → G if σ is nontrivial and f k(cid:93) ( σ ) = σ for some k ≥
1. The minimal such k is the period of σ , and σ is a Nielsen path if it has period 1. A periodic Nielsen path is indivisible if it cannot be written as a concatenation of nontrivial periodic Nielsen paths.
Non-Exponentially-Growing Strata. If Hr is a non-exponentially growing but not pe-riodic stratum for a topological representative f : G → G , each edge e has a subinterval whichis eventually mapped back over e , so the subinterval contains a periodic point. After declar-ing all of these periodic points to be vertices, reordering, reorienting, and possibly replacing H r with two non-exponentially-growing strata, we may assume the edges E , . . . , E k of H r satisfy f ( E i ) = E i +1 u i , where indices are taken mod k and u i is a path in G r − . Henceforthwe always adopt this convention. Note that in the case where g is an element of τ ( ¯ E i +1 )and E i satisfies f ( E i ) = gE i +1 u i , we still perform a subdivision.If H r is a non-exponentially-growing stratum with edges E , . . . , E k such that f ( E i ) = E i +1 g i for some element g ∈ G τ ( E i +1 ) with indices taken mod k , we say that H r is an almostperiodic stratum and each E i is an almost periodic edge. If H r is an almost periodic stratumconsisting of a single edge E r , the stratum and the edge are almost fixed. For a topological representative f : G → G , let Per( f ) denote the set of f -periodic pointsin G . The subset of points with period 1 is Fix( f ). A subgraph C ⊂ G is wandering if f k ( C ) ⊂ G \ C for all k ≥ non-wandering otherwise.The core of a subgraph C ⊂ G is the minimal subgraph (of groups) K of C such thatthe inclusion is a homotopy equivalence. 19 nveloped Zero Strata. Suppose that f : G → G is a topological representative, that u < r and that the following hold.1. The stratum H u is irreducible.2. The stratum H r is exponentially-growing and all components of G r are noncon-tractible.3. For each i satisfying u < i < r , the stratum H i is a zero stratum that is a componentof G r − , and each vertex of H i has valence at least two in G r .Then we say that each H i is enveloped by H r , and write H zr = (cid:83) rk = u +1 H k .Finally, recall from the end of Section 3 our discussion on free factor systems, the partialorder (cid:64) on free factor systems, and the free factor system F ( C ) realized by a subgraph ofgroups C .Let F = A ∗ · · · ∗ A n ∗ F k be a free product represented as π ( G ), where G is a graphof groups with trivial edge groups, vertex groups the A i , and ordinary fundamental groupfree of rank k . Theorem 5.1 ([FH11] Theorem 2.19) . Given an outer automorphism ϕ ∈ Out( F ) repre-sentable on a G -marked graph of groups, there is a relative train track map f : G → G ona G -marked graph of groups and filtration ∅ = G ⊂ G ⊂ · · · ⊂ G m = G representing ϕ satisfying the following properties:(V) The endpoints of all indivisible periodic Nielsen paths are vertices.(P) If a periodic or almost periodic stratum H m is a forest, then there exists a filtrationelement G j such that F ( G j ) (cid:54) = F ( G (cid:96) ∪ H m ) for any filtration element G (cid:96) .(Z) Each zero stratum H i is enveloped by an exponentially-growing stratum H r . Eachvertex of H i is contained in H r and meets only edges in H i ∪ H r .(NEG) The terminal endpoint of an edge in a non-periodic, non-exponentially-growing stratum H i is periodic, and is contained in a filtration element G j with j < i that is its owncore.(F) The core of a filtration element is a filtration element.Moreover, if F (cid:64) · · · (cid:64) F d is a nested sequence of ϕ -invariant free factor systems, wemay assume that each free factor system is realized by some filtration element.Proof. We adapt the proof of [FH11, Theorem 2.19, pp. 56–62]. Begin with a relative traintrack map f : G → G which has no inessential valence-one vertices.
Property (V).
To prove (V), we need to collect more information about Nielsen paths.
Lemma 5.2 (cf. [FH11] Lemma 2.11) . Suppose f : G → G is a relative train track map and H r is an exponentially-growing stratum.1. There are only finitely many indivisible periodic Nielsen paths of height r .2. If σ is an indivisible periodic Nielsen path of height r , then σ contains exactly oneillegal turn in H r . Feighn and Handel use the foregoing to show that in fact,
Lemma 5.3 ([FH11] Lemma 2.12) . If f : G → G is a relative train track map, there are onlyfinitely many points in G that are the endpoints of an indivisible periodic Nielsen path. Ifthese points are not already vertices, they lie in the interior of exponentially-growing strata. E is a periodic edge, then it is a periodic Nielsen path, but it is not indivisible.This latter lemma implies that (V) can be accomplished by declaring the periodic points inthe statement to be vertices. The resulting topological representative is still a relative traintrack map. Sliding.
The move sliding was introduced in [BFH00, Section 5.4, p. 579]. Suppose H i isa non-periodic, non-exponentially-growing stratum that satisfies our convention: the edges E , . . . , E k of H i satisfy f ( E j ) = E j +1 u j where indices are taken mod k and u j is a pathin G i − . We will call the edge of H i we focus on E . Let α be a path in G i − from theterminal vertex of E to some vertex of G i − . Define a new graph of groups G (cid:48) by removing E from G and gluing in a new edge E (cid:48) with initial vertex equal to the initial vertex of E and terminal vertex the terminal vertex of the path α . See Figure 4. Define homotopyequivalences p : G → G (cid:48) and p (cid:48) : G (cid:48) → G by sending each edge other than E and E (cid:48) toitself, and defining p ( E ) = E (cid:48) ¯ α and p (cid:48) ( E (cid:48) ) = E α . Define f (cid:48) : G (cid:48) → G (cid:48) by tightening pf p (cid:48) : G (cid:48) → G (cid:48) . If G r is a filtration element of G , define G (cid:48) r = p ( G r ). The G (cid:48) r form thefiltration for f (cid:48) : G (cid:48) → G (cid:48) . E E (cid:48) α Figure 4: Sliding E along α Lemma 5.4 ([FH11] Lemma 2.17) . Suppose f (cid:48) : G (cid:48) → G (cid:48) is obtained from f : G → G by sliding E along α as described above. Let H i be the non-exponentially-growing stratum of G containing E , and let k be the number of edges of H i .1. f (cid:48) : G (cid:48) → G (cid:48) is a relative train track map if f : G → G was.2. f (cid:48) | G (cid:48) r − = f | G r − .3. If k = 1 , then f (cid:48) ( E (cid:48) ) = E (cid:48) [¯ αu f ( α )] , where [ γ ] denotes the path obtained from γ bytightening.4. If k (cid:54) = 1 , then f (cid:48) ( E k ) = E (cid:48) [¯ αu k ] , f (cid:48) ( E (cid:48) ) = E [ u f ( α )] and f (cid:48) ( E j ) = E j +1 u j for ≤ j ≤ k − .5. For each exponentially-growing stratum H r , p (cid:93) defines a bijection between the set ofindivisible periodic Nielsen paths in G of height r and the indivisible Nielsen paths in G (cid:48) of height r (cid:48) . Lemma 5.5 ([FH11] Lemma 2.10) . Let H r be an exponentially-growing stratum of a relativetrain track map f : G → G and v a vertex of H r . Then there is a legal turn in G r based at v . In particular, the lemma implies that if v has valence one in G r , then v has nontrivialvertex group. 21 roperty (NEG) Part One. We will first show that the terminal vertex of an edge ina non-periodic, non-exponentially-growing stratum H i is either periodic or has valence atleast three. Let E , . . . , E k be the edges of H i . As usual, assume f ( E j ) = E j +1 u j , whereindices are taken mod k and u j is a path in G i − . Suppose the terminal vertex v of E isnot periodic and has valence two. Then v has trivial vertex group. If E is the other edgeincident to v , then E does not belong to an exponentially-growing stratum by the previouslemma.We perform a valence-two homotopy of ¯ E over E . If v is a vertex of an exponentially-growing stratum, f ( v ) (cid:54) = v , so before collapsing the pretrivial forest, properties (EG-i)through (EG-iii) are preserved. The pretrivial forest is inductively constructed as follows:any edge which was mapped to E is added, then any edge which is mapped into the pretrivialforest is added. Thus the argument above shows that no vertex of an exponentially-growingstratum is incident to any edge in the pretrivial forest, so the property of being a relativetrain track map is preserved. After repeating this process finitely many times, the terminalvertex of E is either periodic or has valence at least three in G .Finally, we arrange that v is periodic: the component of G i − containing v is nonwan-dering (because f k − ( u ) is contained in it), so contains a periodic vertex w . Choose apath α in G i − from v to w and slide E along α . No valence-one vertices are createdbecause v was assumed to have valence at least three. Repeating this process for each edgein a non-periodic, non-exponentially-growing stratum, we establish the first part of (NEG),namely the following.(NEG*) The terminal vertex of each edge in a non-periodic, non-exponentially-growing stratumis periodic. Property (Z) Part One.
Property (Z) has several parts. Let H i be a zero stratum. For H i to be enveloped, let H u the first irreducible stratum below H i and H r the first irreduciblestratum above. One condition we need is that no component of G r is contractible. Following[FH11, p. 58], we postpone that step and merely show here that each component is non-wandering.First we arrange that if a filtration element G i has a wandering component, then H i isa wandering component. Suppose that G i has wandering components. Call their union W and their complement N . Since N is f -invariant, it is contained in a union of strata. If N is not precisely equal to a union of strata, the difference is that N contains part but not allof a zero stratum, so we may divide this zero stratum to arrange so that N is a union ofstrata. Thus W is a union of zero strata. Since N is f -invariant, we may push all strata in W higher than all strata in N . We define a new filtration. Strata in N and higher than G i remain unchanged. The strata that make up W will be the components of W . If C and C (cid:48) are such components, C (cid:48) will be higher than C if C (cid:48) ∩ f k ( C ) = ∅ for all k ≥
0. We completethis to an ordering on the components of W , yielding the new filtration.Now we work toward showing that zero strata are enveloped by exponentially-growingstrata. Suppose that K is a component of the union of all zero strata in G , that H i is thehighest stratum that contains an edge of K and that H u is the highest irreducible stratumbelow H i . We aim to show that K ∩ G u = ∅ . So assume K ∩ G u (cid:54) = ∅ . By the previousparagraph, because H u is irreducible, each component of G u is non-wandering, so K meets G u in a unique component C of G u . If each vertex of K has valence at least two in C ∪ K ,then each edge of K belongs to a tight path in K with endpoints in C , and we may closethis path up in C to form a tight loop in K ∪ C . But some iterate of f : G → G maps K ∪ C into C , so this is a contradiction. Therefore some vertex v ∈ K has valence one in K ∪ C .This valence-one vertex v is not periodic, so by (NEG*), v is not an endpoint of anedge in a non-exponentially-growing stratum. We saw in Section 3 that (EG-ii) for anexponentially-growing stratum H r is equivalent to the condition that vertices of H r ∩ G r − contained in in non-wandering components of G r − are periodic. Thus v is not the endpointof an edge in an exponentially-growing stratum above H i . By construction, v is also not22he endpoint of an edge of another zero stratum. Thus v has valence one in G , but wehave not produced any inessential valence-one vertices so far. This contradiction shows that G u ∩ K = ∅ .The lowest edge in K is mapped either to another zero stratum or into G u . In any case,by connectivity, K is wandering, so we can reorganize zero strata so that K = H i . Repeatingthis for each component of the union of all zero strata, we have arranged that if H i is a zerostratum and H r is the first irreducible stratum above H i , then H i is a component of G r − .Let H r be the first irreducible stratum above H i . Because H r is irreducible, no compo-nent of G r is wandering, so the component that contains H i intersects H r . No vertex of H i is periodic, so (NEG*) implies H r is exponentially-growing, and the argument above showsthat vertices of H i meet only edges of H i and H r and every edge of H i has valence at leasttwo in G r . This satisfies every part of the definition of the zero stratum H i being envelopedby H r , except that we have not shown that all components of G r are non-contractible, onlythat they are non-wandering.Note also that H i is contained in the core of G r : one obtains the core of G r by repeatedlyremoving from G r any edge incident to a valence-one vertex with trivial vertex group. Eachvertex of H i has valence at least two in G r , and Lemma 5.5 says that each valence-onevertex of H r has nontrivial vertex group. Tree Replacement.
The final part of property (Z) we will establish now is that everyvertex of H i is contained in H r . We do so by Feighn and Handel’s method of tree replacement. Replace H i with a tree H (cid:48) i whose vertex set is exactly H i ∩ H r . We may do so for everyzero stratum at once, (with a priori different exponentially-growing strata H r , of course)and call the resulting graph of groups G (cid:48) . Let X denote the union of all irreducible strata.There is a homotopy equivalence p (cid:48) : G (cid:48) → G that is the identity on edges in X and sendseach edge in a zero stratum H (cid:48) i to the unique path in H i with the same endpoints. Choosea homotopy inverse p : G → G (cid:48) that also restricts to the identity on edges in X and mapseach zero stratum H i to the corresponding tree H (cid:48) i . Define f (cid:48) : G (cid:48) → G (cid:48) by tightening pf p (cid:48) : G (cid:48) → G (cid:48) . The map f (cid:48) still satisfies (EG-i). Because p (cid:93) and p (cid:48) (cid:93) send nontrivial pathswith endpoints in X to nontrivial paths with endpoints in X , (EG-ii) is preserved as well.Because PF( f ) = PF( f (cid:48) ), Corollary 3.6 implies f (cid:48) satisfies (EG-iii) as well. Nothing wehave done so far changes the realization of free factor systems by filtration elements as well.We replace f : G → G by f (cid:48) : G (cid:48) → G (cid:48) in what follows. Property (P).
Let F (cid:64) · · · (cid:64) F d be our chosen nested sequence of ϕ -invariant free factorsystems. If none was specified, instead define this sequence to be the sequence determinedby F ( G r ) as G r varies among the filtration elements of G . We will show that if H m is aperiodic or almost periodic forest, then there is some F i that is not realized by H m ∪ G (cid:96) for any filtration element G (cid:96) . Assume that this does not hold for some almost periodicforest H m . Then for all i satisfying 1 ≤ i ≤ d , there is some filtration element G (cid:96) such that F ( H m ∪ G (cid:96) ) = F i . In this case we will collapse an invariant forest containing H m , reducingthe number of non-exponentially-growing strata. Iterating this process establishes (P).Let Y be the set of all edges in G \ H m eventually mapped into H m by some iterate of f .Each edge of Y is thus contained in a zero stratum. We want to arrange that if α is a tightpath in a zero stratum with endpoints at vertices that is not contained in Y , then f (cid:93) ( α ) isnot contained in Y ∪ H m . If there is such a path α , let E i be an edge crossed by α and notcontained in Y . Perform a tree replacement as above, removing E i and adding in an edgewith endpoints at the endpoints of α . By our preliminary form of property (Z), if a vertexincident to E i has valence two, then it is an endpoint of α , so this process does not createinessential valence-one vertices. The image of the new edge is contained in Y ∪ H m , so weadd it to Y . Because there are only finitely many paths in zero strata with endpoints atvertices, we need only repeat this process finitely many times if necessary.Let G (cid:48) be the graph of groups obtained by collapsing each component of H m ∪ Y to apoint, and let p : G → G (cid:48) be the quotient map. Identify the edges of G \ ( Y ∪ H m ) with the23dges of G (cid:48) and define f : G (cid:48) → G (cid:48) on each edge E of the complement by tightening pf ( E ).By construction, f (cid:48) : G (cid:48) → G (cid:48) is a topological representative of ϕ , and f (cid:48) ( E ) is obtainedfrom f ( E ) by removing all occurrences of edges in Y ∪ H m . If p ( H r ) is not collapsed to apoint, the stratum H r and p ( H r ) are thus of the same type, and we see that f (cid:48) has one lessnon-exponentially-growing stratum and possibly fewer zero strata. The previous properties,(NEG*) and our preliminary form of (Z) are still satisfied.Let H r be an exponentially-growing stratum. As remarked above, checking (EG-ii) for p ( H r ) is equivalent to checking that each vertex of p ( H r ) ∩ C is periodic for each non-wandering component C of p ( G r − ). Let v (cid:48) be such a vertex. By assumption, there is avertex v ∈ H r such that p ( v ) = v (cid:48) . If v is periodic, we are done. If not, the componentof G r − containing v is wandering, contradicting the assumption that C is non-wandering.This verifies (EG-ii). it is easy to see that (EG-i) is still satisfied, and that PF( f ) = PF( f (cid:48) ),so Corollary 3.6 implies (EG-iii) is still satisfied.It remains to check that our family of free factor systems is still realized. Let F j besuch a free factor system. By assumption on H m , there is G (cid:96) such that F ( G (cid:96) ∪ H m ) = F j .Each non-contractible component of G (cid:96) ∪ H m is mapped into itself by some iterate of f .Since Y is eventually mapped into H m , some iterate of f induces a bijection between thenon-contractible components of G (cid:96) ∪ H m ∪ Y and those of G (cid:96) ∪ H m , so p ( G (cid:96) ) realizes F j . Repeating this process decreases the number of non-exponentially-growing strata, soeventually property (P) is established. Property (Z).
Suppose that C is a non-wandering component of some filtration element.We will show that C is non-contractible. The lowest stratum H i containing an edge of C iseither exponentially-growing or periodic. If H i is exponentially-growing, Lemma 5.5 showsthat each vertex of H i has valence at least two in H i or has nontrivial vertex group, showingthat C is non-contractible. If instead H i is periodic, we show that (P) implies H i is not aforest. If it were, (P) says in particular that there is some filtration element G j such that F ( G j ) (cid:54) = F ( G j ∪ H i ). Thus j < i . This is only possible if H i is not disjoint from G j , but itis by assumption. Therefore H i must not be a forest, and in particular C is not contractible.This proves that (Z) follows from the form of (Z) we have already established. Property (NEG).
Let E be an edge in a non-periodic, non-exponentially-growing stra-tum H i . Let C be the component of G i − containing the terminal vertex v of E ; it is non-wandering by our work proving (NEG*). By the argument in the previous paragraph, if H (cid:96) is the lowest stratum containing an edge of C , then H (cid:96) is either periodic or exponentially-growing, and the argument in the previous paragraph shows that H (cid:96) is non-contractible. Infact, H (cid:96) is its own core. In the exponentially-growing case this follows since the argumentabove shows in either case that every valence-one vertex of H (cid:96) has nontrivial vertex group.To see this in the periodic case, observe that if some edge of H (cid:96) is incident to a vertex withtrivial vertex group and valence one in H (cid:96) , then every edge has this property and H (cid:96) is aforest, in contradiction to (P).Choose a periodic vertex w in H (cid:96) and a path γ from v to w . Slide E along γ . Theresult is a relative train track map which still realizes our sequence of free factor systemsand still satisfies (Z). Working up through the filtration repeating this process establishes(NEG). This time sliding may have introduced inessential valence-one vertices, but (NEG),(Z) and Lemma 5.5 imply that only valence-one vertices are mapped to the valence-onevertices created. We perform valence-one homotopies to remove each of these vertices. Ifproperty (P) is not satisfied, restore it using the process above. Since the number of non-exponentially-growing strata decreases, this process terminates. Property (F).
We want to show that the core of each filtration element is a filtrationelement. If H (cid:96) is a zero stratum, then F ( G (cid:96) ) = F ( G (cid:96) − ), so assume that H (cid:96) is irreducible,and thus G (cid:96) has no contractible components. If a vertex v with trivial vertex group has24alence one in G (cid:96) , then Lemma 5.5 and (Z) imply the incident edge E belongs to a non-exponentially-growing stratum H i . If H i were periodic, it would be a forest, because everyedge would be incident to a vertex of valence one in H i , so (P) implies that some and henceevery valence-one vertex of H i is contained in some lower filtration element. This exhauststhe possibilities: v must be the initial endpoint of a non-periodic non-exponentially-growingedge. All edges in such a stratum have initial vertex a valence-one vertex of G (cid:96) , and no vertexof valence at least two in G (cid:96) maps to them. Thus we may push all such non-exponentially-growing strata H i above G (cid:96) \ H i . After repeating this process finitely many times, F ( G (cid:96) ) isrealized by a filtration element that is its own core. Working upwards through the strata,(F) is satisfied. In this section we prove the existence of CTs for free products.Let G be a graph of groups with trivial edge groups and Γ its Bass–Serre tree. Let x ∈ G be a point and ˜ x ∈ Γ a lift of x . A direction at x is a component of the complement Γ \ ˜ x .If x is in the interior of an edge, there are two directions at x , and if x is a vertex, the setof directions at x is in one-to-one correspondence with the set (cid:97) e ∈ st( x ) G x × { e } . Just as a topological representative f : G → G acts on the set of turns based at a vertex v ,there is an action of the map f sending the set of directions at x to the set of directions at f ( x ).Let f : G → G be a relative train track map. A periodic point x ∈ Per( f ) is principal (cf. [FH18, Definition 3.5]) if none of the following conditions hold.1. x is not an endpoint of a nontrivial periodic Nielsen path and there are exactly twoperiodic directions at x , both of which are contained in the same exponentially-growingstratum.2. x is contained in a component C of Per( f ) that is either a topological circle or aninterval with each endpoint a vertex with nontrivial vertex group and each point of C has exactly two periodic directions.3. x has infinite vertex group G x and there are no periodic directions at x .If each principal periodic vertex is fixed and if each periodic direction based at a principalperiodic vertex is fixed, we say f is rotationless. As before, let F = A ∗ · · · ∗ A n ∗ F k be a free product represented as π ( G ), where G isa graph of groups with trivial edge groups, vertex groups the A i , and ordinary fundamentalgroup free of rank k . Given a ∈ F , let [ a ] u be the unoriented conjugacy class of a , i.e. [ a ] u =[ b ] u if and only if b is conjugate to a or a − . If σ is a closed path, we let [ σ ] u be the unoriented conjugacy class determined by σ . Non-Exponentially-Growing Strata.
Suppose that f : G → G is a rotationless relativetrain track map satisfying the conclusions of Theorem 5.1. The initial endpoint of an edgein a non-periodic non-exponentially-growing stratum H i is principal, so each such stratumconsists of a single edge E i satisfying f ( E i ) = E i u i for some closed path u i ⊂ G i − . If u i is a Nielsen path, E i is a linear edge , and we define the axis for E i to be [ w i ] u , where w i isroot-free and u i = w d i i for some d i (cid:54) = 0.If E i and E j are linear edges and there exist m i , m j > w such that u i = w m i and u j = w m j , then a path of the form E i w p ¯ E j for p ∈ Z iscalled an exceptional path. The map f (cid:93) sends exceptional paths to exceptional paths.25 educed Filtrations. A filtration ∅ = G ⊂ G ⊂ · · · ⊂ G m = G is reduced with respectto ϕ ∈ Out( F ) if it satisfies the following property: if F is a free factor system is ϕ k -invariantfor some k > F ( G r − ) (cid:64) F (cid:64) F ( G r ), then either F = F ( G r − ) or F = F ( G r ). Taken Paths, Splittings. If σ is a maximal subpath of f k(cid:93) ( E ) in a zero stratum, where k > E is an edge of an irreducible stratum H r , we say σ is ( r -)taken. A decompositionof a path σ into subpaths σ = σ · · · σ n is a splitting if f (cid:93) ( σ ) = f (cid:93) ( σ ) · · · f (cid:93) ( σ n ): that is, f (cid:93) ( σ ) is obtained from f ( σ ) by tightening each f ( σ i ) and then concatenating. For ourpurposes, merely performing multiplication in a vertex group does not count as tightening.A nontrivial path is completely split if it has a splitting, called a complete splitting intosubpaths, each of which is either a single edge in an irreducible stratum (possibly togetherwith a vertex group element), an indivisible Nielsen path, an exceptional path, or a path ina zero stratum H i with endpoints in an exponentially-growing stratum that is both maximal(i.e. not contained in a larger such subpath of σ ) and taken.A relative train track map f : G → G is completely split if1. f ( E ) is completely split for each edge E of an irreducible stratum.2. if σ is a taken path in a zero stratum with endpoints in an exponentially-growingstratum, then f (cid:93) ( σ ) is completely split. CTs.
A relative train track map f : G → G and filtration ∅ = G ⊂ G ⊂ · · · ⊂ G m = G is a CT if it satisfies the following properties (cf. [FH11, Definition 4.7]).1. ( Rotationless ) The map f : G → G is rotationless.2. (
Completely Split ) The map f : G → G is completely split.3. (
Filtration ) The filtration is reduced. The core of a filtration element is a filtrationelement.4. (
Vertices ) The endpoints of all indivisible periodic (hence fixed by [FH11, Lemma3.28]) Nielsen paths are principal vertices. The terminal endpoint of each non-fixednon-exponentially-growing edge which is not almost fixed (see item 7) is principal andhence fixed.5. (
Periodic Edges ) Each periodic edge is fixed and each endpoint of a fixed edge isprincipal. If a fixed stratum H r is disjoint from G r − , it is noncontractible. If not,then G r − is a core graph and any endpoint of the unique edge E r not contained in G r − has nontrivial vertex group.6. ( Almost Periodic Edges ) The initial endpoint of an almost periodic edge is prin-cipal, so an almost periodic edge is almost fixed. If an almost fixed stratum H r isdisjoint from G r − , it is noncontractible. If not, then G r − is a core graph and anyendpoint of the unique edge E r not contained in G r − has nontrivial vertex group.7. ( Zero Strata ) If H i is a zero stratum, then H i is enveloped by an exponentially-growing stratum H r , each edge in H i is r -taken and each vertex in H i is contained in H r and meets only edges in H i ∪ H r .8. ( Linear Edges ) For each linear edge E i there is a closed root-free Nielsen path w i such that f ( E i ) = E i w d i i for some d i (cid:54) = 0. If E i and E j are distinct linear edges withthe same axis then w i = w j and d i (cid:54) = d j .9. ( NEG Nielsen Paths ) If the highest edges in an indivisible Nielsen path σ belongto a non-exponentially-growing stratum then either(a) there is a linear edge E i with w i as in (Linear Edges) and there exists k (cid:54) = 0 suchthat σ = E i w ki ¯ E i , or 26b) there is an almost fixed edge E i with f ( E i ) = E i g i for some g i ∈ G τ ( E i ) , and thereexists some nontrivial h ∈ G τ ( E i ) such that f τ ( E i ) ( h ) = g − i hg i and σ = E i h ¯ E i .10. ( EG Nielsen Paths ) If H r is an exponentially-growing stratum and σ is an indivisibleNielsen path of height r , then f | G r = θ ◦ f r − ◦ f r where:(a) the map f r : G r → G is a composition of proper extended folds defined byiteratively folding σ (see [FH11, p. 61]);(b) the map f r − : G → G is a composition of folds involving edges in G r − ; and(c) the map θ : G → G r is a homeomorphism.Feighn and Handel develop a theory of principal automorphisms Φ ∈ Aut( F n ) repre-senting ϕ ∈ Out( F n ), and define (forward) rotationless outer automorphisms based on thedynamics of the action of principal automorphisms on the Gromov boundary of F n . Inthe case of a free product, a parallel theory is possible using the boundary ∂ Γ defined byGuirardel–Horbez and discussed in Section 4. The development of this theory follows en-tirely from Section 3 of [FH11], so we omit it, directing the interested reader to [FH11].Let us remark that in the corner cases of F = A and F = Z / Z ∗ Z / Z one should say(for convenience) that every element and the identity element of Out( F ) are rotationless,respectively. For our purposes, the relevant result is [FH11, Proposition 3.29] which in thispaper would say that if f : G → G represents ϕ ∈ Out( F ) and satisfies the conclusions ofTheorem 5.1, then f : G → G is rotationless if and only if ϕ is forward rotationless. Every ϕ ∈ Out( F ) has a rotationless power.The main result of this section is the following theorem. Theorem 6.1.
Suppose ϕ ∈ Out( F ) is represented by a rotationless relative train trackmap f : G → G and that C is a nested sequence of ϕ -invariant free factor systems. Then ϕ isrepresented by a CT f (cid:48) : G (cid:48) → G (cid:48) and filtration ∅ = G ⊂ G ⊂ · · · ⊂ G m = G (cid:48) that realizes C .Proof. We follow Feighn—Handel’s proof [FH11, p. 87–93]. We may assume without lossof generality that the nested sequence C is of maximal length with respect to (cid:64) so thatany filtration that realizes C is reduced. By Theorem 5.1 and item (3) of [FH11, Lemma2.20], we may construct a relative train track map f : G → G representing ϕ whose filtrationrealizes C such that each contractible component of a filtration element is a union of zerostrata and such that the endpoints of all indivisible Nielsen paths of exponentially-growingheight are vertices. Property (EG Nielsen Paths).
We will argue that if (EG Nielsen Paths) is not satisfied,then there is a relative train track map f (cid:48) : G (cid:48) → G (cid:48) representing ϕ with N ( f (cid:48) ) < N ( f ),where N ( f ) denotes the number of indivisible Nielsen paths (with respect to f : G → G ) ofexponentially-growing height.First a word about terminology: every indivisible Nielsen path σ of exponentially-growingheight may be folded. The fold at the illegal turn of σ may be full or partial, and a full foldmay be proper or improper. Our first lemma says that if the fold at σ is partial, then N ( f )may be decreased. Lemma 6.2 ([FH11] Lemma 4.29) . Suppose that H r is an exponentially-growing stratum ofa relative train track map f : G → G and that σ is an indivisible Nielsen path of height r . Ifthe fold at the illegal turn of σ is partial then there is a relative train track map f (cid:48) : G (cid:48) → G (cid:48) satisfying N ( f (cid:48) ) < N ( f ) .Proof. The lemma follows from the proofs of Lemmas 5.2.3 and 5.2.4 of [BFH00]. In theproof of Lemma 5.2.4, if σ = αβ is the decomposition of σ into maximal r -legal subpaths,it is argued that if α and β are single edges, then N ( f ) may be reduced by folding them.In order to perform this fold, we must know that the initial vertices of α and ¯ β do not both27ave nontrivial vertex group. If they did, the free product of the two vertex groups woulddetermine a free factor of F invariant under ϕ . This contradicts our assumption that thefiltration is reduced unless α and β are the only edges in H r and the terminal vertex v of α has trivial vertex group. But in this case there is only one turn in H r based at v , and thisturn is illegal. This contradicts Lemma 5.5. Therefore we may indeed fold α and β . Lemma 6.3 ([FH11] Lemma 4.30) . Suppose that H r is an exponentially-growing stratumof a relative train track map f : G → G and that σ is an indivisible Nielsen path of height r . Suppose that the fold at the illegal turn of σ is proper, and let f (cid:48) : G (cid:48) → G (cid:48) be the relativetrain track map obtained from f : G → G by folding σ . Then N ( f (cid:48) ) = N ( f ) and there is abijection H s → H (cid:48) s between the exponentially-growing strata of f and those of f (cid:48) such that H s and H (cid:48) s have the same number of edges for all s . Lemma 6.4 ([FH11] Lemma 4.31) . Suppose that H r is an exponentially-growing stratumof a relative train track map f : G → G and that σ is an indivisible Nielsen path of height r .Suppose that the fold at the illegal turn of σ is improper. Then there exists a relative traintrack map f (cid:48) : G (cid:48) → G (cid:48) and a bijection H s → H (cid:48) s between the exponentially-growing strata of f and those of f (cid:48) with the following properties.1. N ( f ) = N ( f (cid:48) ) .2. H (cid:48) r has fewer edges than H r .3. If s > r then H (cid:48) s and H s have the same number of edges. Lemma 6.5 ([FH11] Corollary 4.33) . Suppose that H r is an exponentially-growing stratumof a relative train track map f : G → G and that σ is an indivisible Nielsen path of height r . Then the fold at the illegal turn of each indivisible Nielsen path obtained by iterativelyfolding σ is proper if and only if H r satisfies (EG Nielsen Paths). If some exponentially-growing stratum of f : G → G does not satisfy (EG Nielsen Paths),let H r be the highest such stratum. By the previous lemma there is a sequence of properfolds leading to a relative train track map and an indivisible Nielsen path with either apartial or improper fold. We may apply Lemma 6.2 or Lemma 6.4, respectively. In theformer situation N ( f ) decreases, while in the latter the number of edges of H r decreases.Since both quantities are finite, eventually all further folds are proper, at which point theprevious lemma implies that H r satisfies (EG Nielsen Paths).In the following steps, the number of edges in each exponentially-growing strata and thenumber of indivisible Nielsen paths of exponentially-growing height are not increased. Ifafter some step (EG Nielsen Paths) fails, then we may return to this step and restore thisproperty. By the above argument this process terminates. Applying Theorem 5.1.
Apply the proof of Theorem 5.1 to produce a new relative traintrack map f : G → G satisfying the conclusions of that theorem. Note that in the processof the proof, the number of edges in each exponentially-growing stratum and the numberof indivisible Nielsen paths of exponentially-growing height is unchanged. As noted in theprevious paragraph, we may assume that (EG Nielsen Paths) remains satisfied.
Properties (Rotationless), (Filtration) and (Zero Strata).
Properties (Rotation-less) and (Filtration) follow from [FH11, Proposition 3.29] and property (F) of Theorem 5.1.By property (Z) of Theorem 5.1, to prove property (Zero Strata) it suffices to arrange thatevery edge in a zero stratum H i is r -taken. Each edge E in H i is contained in an r -takenpath σ ⊂ H i . If E itself is not r -taken, perform a tree replacement, replacing E by a paththat has the same endpoints as σ and is marked by σ .28 roperty (Periodic Edges) and (Almost Periodic Edges). First suppose that nocomponent C of Per( f ) is topologically a circle or an interval with both endpoints verticeswith nontrivial vertex group, with each point in C having exactly two periodic directions.Then the endpoints of any periodic edge are principal, each periodic edge is fixed and eachperiodic stratum H r is a single edge E r . Similarly, the initial endpoint of an almost periodicedge is principal, so each almost periodic edge is almost fixed and each almost periodicstratum H r is a single edge E r . If the conclusions of (Periodic Edges) or (Almost PeriodicEdges) do not hold for the edge E r , then one could collapse E r without changing the freefactor systems realized by the filtration elements. This would violate (P), so (PeriodicEdges) and (Almost Periodic Edges) are satisfied in this case.In the general case we will reduce the number of components of Per( f ) that are circles orintervals with both endpoints vertices with nontrivial vertex group, with every point havingexactly two periodic directions until we are in the former case.By [FH11, Lemma 3.30(1)], if C is such a component, then C is f -invariant. If C isan interval, then C is fixed. If C is a circle, then g = f | C is orientation-preserving. Inthe case where C is a circle there are two steps: first arranging that C ⊂ Fix( f ), and then“untwisting” near C to add another edge to Fix( f ). In the case where C is an interval onlythe latter step is necessary.By (Zero Strata) and the fact that there are no periodic directions based in C andpointing out of C , every edge not in C that has an endpoint in C is non-periodic andnon-exponentially-growing and intersects C in its terminal endpoint. Since all non-periodicvertices are contained in exponentially-growing strata, no vertex in the complement of C maps into C . By (NEG), C is a component of some filtration element.For the first step, suppose C is a circle. Extend the rotation g − : C → C to a map h : G → G that has support on a small neighborhood of C , that is homotopic to the identityand such that h ( E j ) ⊂ E j ∪ C for each non-periodic non-exponentially-growing edge E j thathas terminal endpoint in C . Redefine f on each edge E to be h (cid:93) f (cid:93) ( E ). Edges in C are nowfixed. If f ( E j ) = E j u j , then the new path u j and the old u j differ only by possible initialand terminal segments in C ; the f -image of all other edges is unchanged. The map f : G → G is a relative train track map that has all the properties we have already established, withthe possible exception of (P) which fails if one or more edge with terminal vertex in C isnow a fixed edge that should be collapsed as in the proof of Theorem 5.1.Let E m be the first non-periodic non-exponentially-growing edge that has terminal end-point in C . Then f ( E m ) = E m u , where u is a closed tight path that wraps around C —whichmay now be either a circle or an interval with endpoints vertices with nontrivial vertexgroup—some number of times. We will alter f so that E m becomes a fixed edge. Choose h (cid:48) : G → G that is the identity on C , that satisfies h (cid:48) ( E j ) = E j ¯ u for each E j with terminalvertex in C and is the identity otherwise. The map h (cid:48) is homotopic to the identity. Rede-fine f on each edge to be h (cid:93) f (cid:93) ( E ). Now E m is fixed, so C no longer forms a problematiccomponent. If necessary to restore (P), collapse fixed edges with endpoint in C and repeatthis step. Induction: The NEG Case.
We establish the rest of the properties by induction up thefiltration. Let M be the number of irreducible strata in the filtration, and for 0 ≤ m ≤ M let G i ( m ) be the smallest filtration element containing the first m irreducible strata. Weprove by induction on m that one can modify f so that f | G i ( m ) is a CT. The base case m = 0is trivial. So assuming for r = i ( m ) that f | G r is a CT, we will arrange that f | G s is a CTfor s = i ( m + 1). In this step we assume that H s is non-exponentially-growing and hence isa single edge E s satisfying f ( E s ) = E s u s for some path u s ⊂ G s − .By (Zero Strata), r = s −
1. By choosing a path τ ∈ G s − , we may slide E s along τ so that after sliding we have f ( E s ) = E s [¯ τ u s f ( τ )], where [ γ ] denotes the path obtainedfrom γ by tightening. As remarked in the (EG Nielsen Paths) step, we may assume slidingpreserves (EG Nielsen Paths).Suppose first that there is some path τ so that after sliding E s along τ we have E s ⊂ f ). This is equivalent to [¯ τ u s f ( τ )] being trivial, and hence to f (cid:93) ( E s τ ) = E s [ u s f ( τ )] = E s τ ; that is, to E s τ being a Nielsen path. In this case if the initial vertex of E s hasnontrivial vertex group or is contained in G s − , then (Periodic Edges) is satisfied, as are allthe previous properties; the remaining properties of a CT follow by induction.If on the other hand the initial vertex of E s is not contained in G s − and has trivialvertex group, then collapse E s to a point as in the (Periodic Edges) step. None of ourpreviously achieved properties are lost, and the remaining properties of a CT follow byinduction.Now suppose that there is no choice of τ such that E s τ is a Nielsen path. If f ( E s ) = E s g s for some g s ∈ G τ ( E s ) , i.e. E i is almost fixed, then f | G s satisfies (Almost Fixed Edges) andthe remaining properties of a CT, follow by induction, so we are done with the inductivestep. So suppose E i is not almost fixed. Proposition 6.6 ([FH11] Proposition 4.35) . Suppose that1. f : G → G is a relative train track map that satisfies (EG Nielsen Paths),2. f | G s − is a CT, and3. H s is a non-exponentially-growing stratum with single edge E s for which there doesnot exist µ ⊂ G s − such that E s µ is a Nielsen path.Then there exists a path τ ⊂ G s − along which we may slide E s such that after sliding, thefollowing conditions hold.1. f ( E s ) = E s · u s is a nontrivial splitting.2. If σ is a circuit or path with endpoints at vertices and if σ has height s , then thereexists k ≥ such that f k(cid:93) ( σ ) splits into subpaths of the following type.(a) E s or ¯ E s ,(b) an exceptional path of height s ,(c) a subpath of G s − .3. u s is completely split and its initial vertex is principal.4. f | G s satisfies (Linear Edges). After performing the slide described in the proposition, since u s is nontrivial, f | G s satis-fies (Periodic Edges) and all the previously established properties. Properties (CompletelySplit), (Vertices), (Almost Fixed Edges), (NEG Nielsen Paths), and (Linear Edges) for f | G s follow from the proposition and the inductive hypothesis. This completes the inductive stepin the case that H s is non-exponentially-growing. Induction: The EG Case.
Suppose now that H s is exponentially-growing. Properties(Vertices), (Almost Fixed Edges), (NEG Nielsen Paths) and (Linear Edges) for f | G s followfrom these properties for f | G s − . Thus it suffices to establish (Completely Split).For each edge E ⊂ H s , there is a decomposition f ( E ) = µ · ν · µ · · · ν (cid:96) − · µ (cid:96) , where the ν i are maximal subpaths in G r . Let { ν i } be the collection of all such paths that occur as E varies over the edges of H s . By (Zero Strata) and (EG-ii), we have that f k(cid:93) ( ν i ) is nontrivialfor each k and i . By [FH11, Lemma 4.25], since f | G r is a CT, we may choose k so largethat each f k(cid:93) ( ν i ) is completely split. We may also assume that the endpoints of f k(cid:93) ( ν i ) areperiodic and hence principal. There are finitely many paths σ with endpoints in H s ∩ G s − contained in the strata between G r and H s . Each f ( σ ) is either a homotopically nontrivialpath with endpoints in H s ∩ G s − or a path in G r with fixed endpoints. We therefore mayassume that f k(cid:93) ( σ ) is completely split for each such σ . After applying the following move k times with j = r , we have that f | G s is completely split, completing the induction step andthe proof of the theorem. 30 hanging the Marking. Suppose that f : G → G is a rotationless relative train trackmap satisfying the conclusions of Theorem 5.1 with respect to the filtration ∅ = G ⊂ G ⊂ · · · ⊂ G m = G , that 1 ≤ j ≤ m , that every component of G j is noncontractible andthat f fixes every vertex in G j that meets edges of G \ G j . Define a homotopy equivalence g : G → G by g | G j = f | G j and g | ( G \ G j ) equal to the identity.Define a new marked graph of groups G (cid:48) from G by changing the marking via g . That is,as a graph of groups we have G = G , and if τ : G → G is the marking for G , then gτ : G → G (cid:48) is the marking for G (cid:48) .Define f (cid:48) : G (cid:48) → G (cid:48) by f | G (cid:48) j = f | G j and f (cid:48) ( E ) = ( gf ) (cid:93) ( E ) for all edges E higher than G j . We say that f (cid:48) : G (cid:48) → G (cid:48) is obtained from f : G → G by changing the marking on G j via f . Lemma 6.7 ([FH11] Lemma 4.27) . Suppose that f (cid:48) : G (cid:48) → G (cid:48) is obtained from f : G → G by changing the marking on G j via f . The following hold.1. f (cid:48) | G j = f | G j ;2. for every path σ ⊂ G with endpoints at vertices and every k > , we have g (cid:93) f k(cid:93) ( σ ) =( f (cid:48) ) k(cid:93) g (cid:93) ( σ ) ;3. f (cid:48) : G (cid:48) → G (cid:48) is a homotopy equivalence that represents the same element of Out( F ) as f : G → G ;4. there is a one-to-one correspondence between Nielsen paths for f and Nielsen pathsfor f (cid:48) ; and5. f (cid:48) : G (cid:48) → G (cid:48) is also a rotationless relative train track map satisfying the conclusions ofTheorem 5.1 with respect to the same filtration as f . Gaboriau, Jaeger, Levitt and Lustig define in [GJLL98, Theorem 4] an index for (outer)automorphisms of free groups and show that it satisfies an inequality that reproves andsharpens Bestvina–Handel’s proof of the Scott conjecture. Martino [Mar99] proved theanalogous result for free products. We remark that Martino’s result is stated for the Grushkodecomposition of the free product, but that the proof does not use this assumption in anyessential way. More recently Feighn–Handel sharpen this inequality [FH11, Proposition15.14]. The purpose of this section is to extend this latter inequality to the setting of freeproducts.As usual, let F = A ∗ · · · ∗ A n ∗ F k be represented by the graph of groups G with vertexgroups the A i , trivial edge groups, and ordinary fundamental group free of rank k . Supposethat Φ ∈ Aut( F ) admits a topological representative on a G -marked graph of groups. Recallfrom Section 4 that if Γ is the Bass–Serre tree of G , then Φ acts on the Gromov boundary ∂ Γ. The subgroup Fix(Φ) inherits a splitting as a free product B ∗ · · · ∗ B m ∗ F (cid:96) from G .Define the rank (or G -rank ) of Fix(Φ) to be the quantity m + (cid:96) . Let a (Φ) denote the numberof Fix(Φ)-orbits of attracting fixed points of Φ in its action on ∂ Γ, and let b (Φ) denote thenumber of Fix(Φ)-orbits of attracting fixed points of Φ associated to NEG rays (see belowfor a definition). Define the quantity j (Φ) as j (Φ) = max (cid:26) , rank(Fix(Φ)) + 12 a (Φ) + 12 b (Φ) − (cid:27) . Two automorphisms Φ and Φ (cid:48) are isogredient if there exists w ∈ F such that Φ (cid:48) =ad( w )Φ ad( w ) − , where ad( w ) is the inner automorphism g (cid:55)→ wgw − . Note that iso-gredience is an equivalence relation and that if Φ and Φ (cid:48) are isogredient, then j (Φ) = j (Φ (cid:48) ).31iven ϕ ∈ Out( F ) an automorphism that admits a topological representative on a G -markedgraph of groups, define the quantity j ( ϕ ) as j ( ϕ ) = (cid:88) j (Φ) , where the sum is taken over representatives of the isogredience classes of automorphisms Φrepresenting ϕ . If j (Φ) is positive, then Φ is a principal automorphism (see [FH11, Definition3.1]). By [FH11, Remark 3.9], there are only finitely many isogredience classes of principalautomorphisms, so this sum has only finitely many nonzero terms.The main result of this section is the following theorem. Theorem 7.1. If ϕ ∈ Out( F ) admits a topological representative on a G -marked graph ofgroups, then j ( ϕ ) ≤ n + k − . The strategy of the proof of Theorem 7.1 is to firstly show that if ψ = ϕ K is a rotationlessiterate of ϕ , then j ( ϕ ) ≤ j ( ψ ). Then, using a CT f : G → G representing ψ , to constructa graph of groups S N ( f ), invariants of which calculate j ( ψ ), and argue by induction upthrough the filtration that j ( ψ ) ≤ n + k − Rays and Attracting Fixed Points
We recall [FH18, Definition 3.9]: let f : G → G bea CT representing ϕ ∈ Out( F ), and let E denote the set of oriented, non-fixed, non-linearedges whose initial vertex is principal and which support a fixed direction at that vertex. If E is an edge of E , such that the direction gE is fixed, there is a path u such that f k(cid:93) ( gE )has a splitting of the form gE · u · f (cid:93) ( u ) · · · · · f k − (cid:93) ( u ) for all k ≥ | f k(cid:93) ( u ) | → ∞ as k tends to infinity. The limit of the increasing sequence gE ⊂ f ( gE ) ⊂ f (cid:93) ( gE ) ⊂ · · · is a ray R E in G .If Γ is the Bass–Serre tree for G , each lift of R E to Γ has a well-defined endpoint in ∂ Γ, so R E determines an F -orbit of points in ∂ Γ. If ˜ E is a lift of E to Γ and ˜ f : Γ → Γis the lift of f sending ˜ E to ˜ E ˜ u , then the lift ˜ R E of R E with initial edge ˜ E determines anattracting fixed point for ˜ f . The lift ˜ f is a principal lift (see [FH11, Definition 3.1]). By[FH18, Lemma 3.10], every F -orbit of attracting fixed points for every principal lift of f isrepresented by some R E for E ∈ E .If P is an attracting fixed point for the automorphism Φ determined by the principal lift˜ f which is represented by the lifted ray ˜ R E , we say P is an NEG ray for Φ if the stratum of f : G → G containing E is non-exponentially growing. By [FH18, Lemma 15.4] and [HM20,Part II, Definitions 2.9 and 2.10 and Lemma 2.11], this definition is independent of thechoice of CT representing ϕ ∈ Out( F ). If ϕ is not rotationless (and so not represented bya CT), then we say P is an NEG ray for Φ if it is an NEG ray for Φ K , where ϕ K is arotationless iterate of ϕ . This definition is independent of the choice of iterate by [FH18,Remark 15.6]. Lemma 7.2. If ψ = ϕ K is a rotationless iterate of ϕ , then j ( ϕ ) ≤ j ( ψ ) .Proof. Let Φ , . . . , Φ N be a set of representatives of isogredience classes of principal auto-morphisms representing ϕ . Feighn and Handel prove in [FH18, Lemma 15.8] that the sum (cid:80) Ni =1 a (Φ i ) (respectively, (cid:80) Ni =1 b (Φ i )) is equal to the number of F -orbits of attracting fixedpoints of Φ (respectively, F -orbits of NEG rays for Φ) as Φ varies over all principal auto-morphisms representing ϕ . Then they observe in [FH18, Lemma 15.9] that by definition if P is an attracting fixed point (respectively an NEG ray) for the principal automorphismΦ, then P is also an attracting fixed point (respectively an NEG ray) for the principal au-tomorphism Φ K representing ψ . Therefore if we write a ( ϕ ) = (cid:80) a (Φ) and b ( ϕ ) = (cid:80) b (Φ)32here the sum is over representatives of isogredience classes of principal automorphisms Φrepresenting ϕ , we see that a ( ϕ ) ≤ a ( ψ ) and b ( ϕ ) ≤ b ( ψ ).Recall that for Φ an automorphism representing ϕ , Fix(Φ) inherits a splitting as a freeproduct from G . By the main result of [CT94], the free product is of the formFix(Φ) = B ∗ · · · ∗ B m ∗ F (cid:96) (i.e. there are finitely many factors). Define ˆ r (Φ) = max { , m + (cid:96) − } , and define ˆ r ( ϕ ) = (cid:80) ˆ r (Φ), where the sum is over representatives of isogredience classes of principal automor-phisms Φ representing ϕ . To finish the proof, we show that ˆ r ( ϕ ) ≤ ˆ r ( ψ ).So suppose ˆ r ( ϕ ) >
0; we argue as in [FH18, Lemma 15.11]. Let Φ , . . . , Φ s be therepresentatives of isogredience classes of principal automorphisms for which ˆ r (Φ i ) >
0. Foreach i satisfying 1 ≤ i ≤ s , there exists a principal automorphism Ψ j representing ψ suchthat Φ Ki is isogredient to Ψ j . By replacing Φ i within its isogredience class we may assumethat Φ Ki = Ψ j . Let Ψ , . . . , Ψ t denote the representatives of isogredience classes of principalautomorphisms for which there exists some Φ i with Φ Ki = Ψ j . The assignment i (cid:55)→ j definesa function p : { , . . . , s } → { , . . . , t } . It suffices to show that s (cid:88) i =1 ˆ r (Φ i ) ≤ t (cid:88) j =1 ˆ r (Ψ j ) , which will hold if we can show that for each j satisfying 1 ≤ j ≤ t , we have (cid:88) i ∈ p − ( j ) ˆ r (Φ i ) ≤ ˆ r (Ψ j ) . Fix j , write F = Fix(Ψ j ) and take i ∈ p − ( j ). We have Φ Ki = Ψ j , so F = Fix(Φ Ki ). Since ψ is rotationless (see [FH11, Definition 3.13]), we have that Φ i ( F ) = F , that Fix(Φ i ) ⊂ F ,and that Φ | F is a finite order automorphism.We claim that if i (cid:48) ∈ p − ( j ), then Fix(Φ i ) and Fix(Φ i (cid:48) ) are not conjugate in F . For ifFix(Φ i ) = h Fix(Φ i (cid:48) ) h − , then we have Fix(Φ i ) = Fix(ad( h )Φ i (cid:48) ad( h − )), where ad( h ) is theinner automorphism g (cid:55)→ hgh − . These automorphisms differ by an inner automorphismad( k ), where k centralizes Fix(Φ i ), hence k = 1, so they are equal, in contradiction to theassumption that Φ i and Φ i (cid:48) are not isogredient.There are two cases: either F is infinite dihedral or it is not. If F is infinite dihedral, theonly possibility for the finite order automorphism Φ i | F is the identity, and by the argumentin the previous paragraph we conclude that p − ( j ) = { i } , and the displayed inequality forˆ r (Ψ j ) holds. If F is not infinite dihedral, by [CT94], it inherits a free product decompositionas F = B ∗ · · · ∗ B m ∗ F (cid:96) from G (i.e. there are finitely many factors). The argument in theproof of [FH18, Lemma 3.12] applies to show that F is self-normalizing, so the restriction ϕ | F is well-defined and has finite order.Martino [Mar99] defines an index i (Φ) for Φ an automorphism of a free product F representing the outer automorphism ϕ as i (Φ) = max (cid:26) , rank(Fix(Φ)) + 12 a (Φ) − (cid:27) , and defines i ( ϕ ) = (cid:80) i (Φ) where the sum is over representatives of isogredience classes ofautomorphisms Φ representing ϕ . If F = A ∗· · ·∗ A n ∗ F k is represented as a graph of groups G with nontrivial vertex groups the A i , trivial edge groups, and ordinary fundamental groupfree of rank k and ϕ may be represented on a G -marked graph of groups, he shows that i ( ϕ ) ≤ n + k −
1. (In fact, as remarked above, Martino’s result is stated for the Grushkodecomposition of F , but this is not necessary in the proof.) If Φ : F → F represents ϕ | F ,then i (Φ) = ˆ r (Φ), since there are no attracting fixed points. Thus Martino’s result appliedto ϕ | F shows that (cid:88) i ∈ p − ( j ) ˆ r (Φ i ) ≤ i ( ϕ | F ) ≤ m + (cid:96) − r (Ψ j ) . ielsen Paths in CTs. Let f : G → G be a CT representing ψ ∈ Out( F ). Each Nielsenpath with endpoints at vertices is a composition of fixed edges and indivisible Nielsen paths.There are three kinds of indivisible Nielsen paths. The first possibility is that E is an almostfixed edge, that is, if f ( E ) = Eg for some group element g ∈ G τ ( E ) , then for any nontrivialgroup element h ∈ G τ ( E ) such that f τ ( E ) ( h ) = g − hg , Eh ¯ E is an indivisible Nielsen path.The second is that E is a linear edge, so there exists a closed root-free Nielsen path w E such that f ( E ) = Ew dE for some d (cid:54) = 0. Then Ew kE ¯ E is an indivisible Nielsen path for any k (cid:54) = 0. By (NEG Nielsen Paths), these are the only possibilities for indivisible Nielsen pathsof non-exponentially-growing height. The final possibility is an indivisible Nielsen path µ ofexponentially-growing height r . By [FH11, Lemma 4.19], µ and ¯ µ are the only indivisibleNielsen paths of height r , and the initial edges of µ and ¯ µ are distinct. An Euler Characteristic for Graphs of Groups. If G is a finite graph of groupswith trivial edge groups, we define the (negative) Euler characteristic χ − ( G ) of G to be thenumber of edges of G minus the number of vertices with trivial vertex group. Note that G defines a splitting of its fundamental group as a free product B ∗ · · · ∗ B m ∗ F (cid:96) , where the B i are the nontrivial vertex groups of G and the ordinary fundamental group of G is freeof rank (cid:96) . The ordinary negative Euler characteristic of G calculates the quantity (cid:96) −
1. Inour definition, the m vertices with nontrivial vertex group are not counted, so we see that χ − ( G ) = m + (cid:96) − The Core Filtration. If f : G → G is a CT, then the core of a filtration element is afiltration element. The core filtration ∅ = G = G (cid:96) ⊂ G (cid:96) ⊂ · · · ⊂ G (cid:96) M = G m = G is the coarsening of the filtration for f : G → G defined by including only those elementsthat are their own core. By (Periodic Edges) and (Almost Fixed Edges), we have (cid:96) = 1.The i th stratum of the core filtration, H c(cid:96) i is H c(cid:96) i = (cid:96) i (cid:91) j = (cid:96) i − +1 H j . The change in negative Euler characteristic is ∆ i χ − = χ − ( G (cid:96) i ) − χ − ( G (cid:96) i − ). The Index of a Finite-Type Graph of Groups.
Let f : G → G be a CT, and supposethat H is a graph of groups equipped with an immersion H → G . (Thus edge groups of H are trivial.) We say H is of finite type if it has finitely many connected components, each ofwhich is a finite graph of groups together with a finite number of infinite rays; in particularthere are only finitely many nontrivial vertex groups in each component. Let H , . . . , H t bethe components of H . Let a ( H i ) denote the number of ends (i.e. infinite rays) of H i , andlet b ( H i ) denote the number of rays mapping to NEG rays in G . Define the index of H i tobe the quantity j ( H i ) = rank( π ( H i )) + 12 a ( H i ) + 12 b ( H i ) − . Define j ( H ) = (cid:80) ti =1 j ( H i ). The Graph of Groups S N ( f ) . Let f : G → G be a CT representing ψ ∈ Out( F ). In theproof of Theorem 7.1, we build a finite-type graph of groups S N ( f ) consideed by Feighn–Handel in [FH18, Section 12] equipped with an immersion S N ( f ) → G such that j ( S N ( f )) = j ( ψ ), by [FH18, Lemma 12.4]. Each Nielsen path (containing an edge) and each ray R E for E ∈ E lifts to S N ( f ). In the proof we build up S N ( f ) in stages by induction up through thecore filtration. Here we describe its construction without reference to the filtration.34egin with the subgraph of groups of G consisting of all principal vertices and all fixededges. For each vertex v , set its vertex group in S N ( f ) to be Fix( f v ). We add a piece to S N ( f ) for indivisible Nielsen paths and edges in E in the following way.Suppose E is an almost fixed edge, say f ( E ) = Eg , such that there exists an indivisibleNielsen path of the form Eh ¯ E for h ∈ G τ ( E ) . To the principal initial vertex of E in S N ( f ) weattach an edge mapping to E . Although E may form a loop in G , the edge we attach doesnot; we set the vertex group of its terminal vertex to be the subgroup of G τ ( E ) consisting ofthose h such that f v ( h ) = g − hg . We call the newly attached subgraph of groups the pin associated to E .Suppose now that E is a linear edge, say f ( E ) = Ew d for some closed, root-free Nielsenpath w . To the principal initial vertex of E in S N ( f ) we attach a lollipop: a graph consistingof two edges sharing a vertex, one of which forms a loop. The loop maps to w , and theother edge to E .Suppose now that µ is an indivisible Nielsen path of exponentially-growing height. At-tach an edge mapping to µ to the principal endpoints of µ in S N ( f ).Suppose finally that E ∈ E . If E belongs to a non-exponentially-growing stratum or E belongs to an exponentially-growing stratum and is not the initial edge of an indivisibleNielsen path, attach to its principal initial vertex an infinite ray mapping to R E . If E isthe initial edge of an indivisible Nielsen path µ and E (cid:48) is the initial edge of ¯ µ , then R E and R E (cid:48) have a common terminal subray R E,E (cid:48) (see [FH18, Section 12] for more details). Recallthat we may write µ = αβ , where α and β are the maximal legal segments of µ . In this casesubdivide the edge mapping to µ into two edges, one mapping to α and the other to β , andattach R E,E (cid:48) at the newly added vertex. This completes the construction of S N ( f ). Proof of Theorem 7.1.
We follow the outline of [FH18, Proposition 15.14]. Let f : G → G be a CT representing ψ ∈ Out( F ). We construct S N ( f ) by inducting up through the corefiltration of G . For i satisfying 0 ≤ i ≤ M , let S N ( i ) denote the subgraph of groups of S N ( f )constructed by the i th term of the core filtration G (cid:96) i . We shall prove that j ( S N ( i )) ≤ χ − ( G (cid:96) i ) , from which the theorem follows. We prove this inequality by induction. The case i = 0 holdstrivially, since S N (0) and G (cid:96) are empty. Let ∆ k j = j ( S N ( k )) − j ( S N ( k − k −
1. We prove that ∆ k j ≤ ∆ k χ − , so that the inequality holds for k as well. The proof has two cases. Case 1.
Suppose that H c(cid:96) k contains no exponentially-growing strata. We claim that oneof the following occurs.(a) We have (cid:96) k = (cid:96) k − + 1. The unique edge in H c(cid:96) k is disjoint from G (cid:96) k − and is fixed oralmost fixed. Either it forms a loop, or both incident vertices have nontrivial vertexgroup.(b) We have (cid:96) k = (cid:96) k − + 1. The unique edge in H c(cid:96) k either has both endpoints containedin G (cid:96) k − or one endpoint is contained in G (cid:96) k − and the other has nontrivial vertexgroup.(c) We have (cid:96) k = (cid:96) k − + 2. The two edges in H c(cid:96) k share an initial endpoint, are notfixed nor almost fixed, and have terminal endpoints in G (cid:96) k − . (Note that the initialendpoint has trivial vertex group, otherwise we would be in case (b).)The proof is similar to [FH09, Lemma 8.3]. Because H c(cid:96) k contains no exponentially-growingstrata, it contains no zero strata, so each H j is a single edge E j for each j satisfying (cid:96) k − + 1 ≤ j ≤ (cid:96) k . If some E j is fixed or almost fixed, then either (a) or (b) holds by(Periodic Edges) or (Almost Periodic Edges), respectively. If each E j is not fixed or almostfixed, and (b) does not hold, then E j adds an inessential valence-one vertex to G (cid:96) k − , and35he terminal vertex of E j belongs to G (cid:96) k − by [FH11, Lemma 4.22]. Each new valence-onevertex must be an endpoint of E (cid:96) k , so (c) holds.We analyze each subcase.(a) Suppose first that the edge E (cid:96) k is fixed. To S N ( k ) we add a new component containingthe subgraph spanned by E (cid:96) k . If E (cid:96) k forms a loop with trivial incident vertex group,we see that ∆ k j = ∆ k χ − = 0. Otherwise ∆ k j ≤ ∆ k χ − = 1.If instead E (cid:96) k is almost fixed, then its initial vertex v is principal and we add itto S N ( k ) If we add a pin corresponding to E (cid:96) k , then ∆ k j = ∆ k χ − = 1, otherwise∆ k j = 0 and ∆ k χ − = 1.(b) In this case we always have ∆ k χ − = 1. If E (cid:96) k is fixed, we add it to S N ( k ) as in item(a). If both endpoints are contained in G (cid:96) k − or the valence-one vertex v has Fix( f v )nontrivial, then ∆ k j = 1. Otherwise ∆ k j = 0.If instead E (cid:96) k is almost fixed and we add a pin, then ∆ k j = 1, otherwise ∆ k j = 0. Ifthe initial vertex of E (cid:96) k is not contained in G (cid:96) k − , then E (cid:96) k is added, if at all, as a newcomponent to S N ( k ). In this case we have ∆ k j = 1 if the pin is added and Fix( f v ) isnontrivial for v the initial vertex of E (cid:96) k ; otherwise ∆ k j = 0.Suppose then that E (cid:96) k is linear; to S N ( k ) we add a lollipop. If the initial vertex of E (cid:96) k is contained in G (cid:96) k − , then ∆ k j = 1. If not, then the lollipop is added as a newcomponent to S N ( k ); we have ∆ k j = 1 if Fix( f v ) is nontrivial for v the initial vertexof E (cid:96) k , and ∆ k j = 0 if not.Finally suppose E (cid:96) k is nonlinear. Then to the initial vertex v of E (cid:96) k in S N ( k ) weattach the NEG ray R E (cid:96)k . We have ∆ k j = 1 if v ∈ G (cid:96) k − or Fix( f v ) (cid:54) = 1, and ∆ k j = 0otherwise.(c) In this case we have ∆ k χ − = 1. The vertex in G (cid:96) k not contained in G (cid:96) k − is principal,so determines a new component of S N ( k ). To this new vertex we attach a lollipop foreach linear edge, and an NEG ray for each nonlinear edge, for a total of three possiblecombinations. In all cases ∆ k j = 1. Case 2.
The argument again is similar to [FH09, Lemma 8.3]. Suppose that H c(cid:96) k containsan exponentially-growing stratum H r . Since the core of a filtration element is a filtrationelement and G r − does not carry the attracting lamination associated to H r , we see that G r is its own core. Thus H (cid:96) k is the unique exponentially-growing stratum in H c(cid:96) k . We claimthat there exists some u k satisfying (cid:96) k − ≤ u k < (cid:96) k such that both of the following hold.(a) For j satisfying (cid:96) k − < j ≤ u k , the stratum H j is a single edge which is not fixed noralmost fixed whose terminal vertex is in G (cid:96) k − and whose initial vertex is an inessentialvalence-one vertex in G u k .(b) For j satisfying u k < j < (cid:96) k , the stratum H j is a zero stratum.The existence of u k and that (b) holds follows from (Zero Strata). That (a) holds followsfrom (Periodic Edges), (Almost Periodic Edges) and [FH11, Lemma 4.22].To calculate χ − , note that χ − ( G ) is equal to the sum over the vertices v of G of valence( v ) − v has trivial vertex group and valence( v ) if not. Note that each edgecontributes valence to two vertices.For j as in item (a) above, the contribution to S N ( k ) is the addition of a new component,with either a lollipop or an NEG ray. Thus in each case ∆ k j = 0, while ∆ k χ − = 0. Thiscompletes the analysis up to G u k .For each vertex v ∈ H (cid:96) k , let ∆ k j ( v ) and ∆ k χ − ( v ) be the contributions to ∆ k j and ∆ k χ from v not already considered. If v is principal, let κ ( v ) denote the number of oriented edgesof H (cid:96) k incident to v that do not support a fixed direction at v . If v ∈ H (cid:96) k is not principal,let κ ( v ) = valence( v ) − v has trivial vertex group and κ ( v ) = valence( v ) otherwise.36uppose first that there are no indivisible Nielsen paths of height (cid:96) k . If the vertex v is notprincipal, then ∆ k j ( v ) = 0. By [FH18, Lemma 3.8], either v is a new vertex or it has infinitevertex group. In either case, ∆ k χ − ( v ) = κ ( v ) and we have ∆ k χ − ( v ) − ∆ k j ( v ) = κ ( v ) ≥ v is principal, let L ( v ) denote the number of oriented edges basedat v in E ∩ H (cid:96) k . If v has already been added to S N ( k ), then ∆ k j ( v ) = L ( v ) and ∆ k χ − = ( L ( v ) + κ ( v )). If v has nontrivial vertex group, then ∆ k χ − ( v ) = ( L ( v ) + κ ( v )) and∆ k j ( v ) = L ( v ) if Fix( f v ) (cid:54) = 1 and L ( v ) − v is new and has trivialvertex group, then ∆ k j ( v ) = L ( v ) − k χ − ( v ) = ( L ( v ) + κ ( v )) −
1. In all cases wesee that ∆ k χ − ( v ) − ∆ k j ( v ) ≥ κ ( v ) ≥ µ of height (cid:96) k . By [FH11,Lemma 4.24], we have (cid:96) k = u k + 1. Let w and w (cid:48) be the endpoints of µ . By [BFH00, Lemma5.1.7], if w (cid:54) = w (cid:48) , then at least one of w and w (cid:48) is a new vertex. By [FH11, Corollary 4.19]if w = w (cid:48) , then w is new. Let e and e (cid:48) be the initial edges of µ and ¯ µ respectively.Let V be the set of vertices of H (cid:96) k that are not endpoints of µ . Each vertex of V maybe handled as in the case without indivisible Nielsen paths, so we conclude (cid:88) v ∈V ∆ k χ − ( v ) − (cid:88) v ∈V ∆ k j ( v ) ≥ (cid:88) v ∈V κ ( v ) ≥ . Let ∆ k j ( µ ) and ∆ k χ − ( µ ) be the contributions to ∆ k j and ∆ k χ − coming from theendpoints of µ and not already considered. There are three subcases to consider accordingto whether w = w (cid:48) and whether w and w (cid:48) are new.1. Suppose µ is a closed path based at the new vertex w . In S N ( k ), we add a new vertex w , a loop at w mapping to µ , a ray for each E ∈ E incident to w other than e and e (cid:48) ,and then one ray corresponding e and e (cid:48) . Therefore,∆ k j ( µ ) = 1 + 12 ( L ( w ) − − L ( w ) − w has trivial vertex group or Fix( f w ) = 1 and ∆ k j ( µ ) = L ( w ) + otherwise.Similarly we have ∆ k χ − ( µ ) = 12 ( L ( w ) + κ ( w )) − w has trivial vertex group and ∆ k χ − ( µ ) = ( L ( w ) + κ ( w )) otherwise. Thus∆ k χ − ( µ ) − ∆ k j ( µ ) ≥ κ ( w ) − . Since there is always at least one illegal turn in H (cid:96) k —for instance the one in µ —andsince illegal but nondegenerate turns are determined by distinct edges, there must beat least one vertex of H (cid:96) k with κ ( v ) (cid:54) = 0, so we conclude that ∆ k j ≤ ∆ k χ − as desired.2. Suppose that w is new and w (cid:48) is old. In S N ( k ) we add the new vertex w , an edgeconnecting w to w (cid:48) , one ray for each E ∈ E based at w or w (cid:48) other than e and e (cid:48) andone ray corresponding to e and e (cid:48) . Thus we have∆ k j ( µ ) = 12 ( L ( w ) + L ( w (cid:48) )) − w has trivial vertex group or Fix( f w ) = 1 and ∆ k j ( µ ) = ( L ( w ) + L ( w (cid:48) )) + otherwise. Similarly,∆ k χ − ( µ ) = 12 ( L ( w ) + L ( w (cid:48) ) + κ ( w ) + κ ( w (cid:48) )) − w has trivial vertex group, and ∆ k χ − ( µ ) = ( L ( w )+ L ( w (cid:48) )+ κ ( w )+ κ ( w (cid:48) )) otherwise.In each case, the argument concludes as in the previous subcase.37. Suppose w and w (cid:48) are distinct and both new. The change in S N ( k ) is the addition oftwo new vertices, an edge connecting them, one ray for each E ∈ E based at w or w (cid:48) other than e and e (cid:48) and one ray corresponding to e and e (cid:48) . The calculation has severalpossibilities depending on whether w or w (cid:48) have trivial or nontrivial vertex group, butas in the previous two cases, we see that ∆ k χ − ( µ ) − ∆ k j ( µ ) ≥ ( κ ( w ) + κ ( w (cid:48) )) − ,and we conclude the argument as before. References [Bas93] Hyman Bass. Covering theory for graphs of groups.
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