EVERY INFINITE ORDER MAPPING CLASS HAS ANINFINITE ORDER ACTION ON THE HOMOLOGY OF

SOME FINITE COVER

ASAF HADARI

Abstract. We prove the following well known conjecture: let Σ be anoriented surface of finite type whose fundamental group is a nonabelianfree group. Let φ ∈ Mod(Σ) be a an infinite order mapping class. Thenthere exists a finite solvable cover Σ0 → Σ, and a lift φ0 of φ such that theaction of φ0 on H1(Σ0,Z) has infinite order. Our main tools are the theoryof homological shadows, which was previously developed by the author, andFourier transforms.

1. Introduction

Let Σ be an oriented surface of finite type, and let Mod(Σ) be its mappingclass group. The finite dimensional representation theory of Mod(Σ) has beenthe subject of much study, and much about it remains mysterious. For in-stance, it is not generally known whether the groups Mod(Σ) are linear. Onthe other hand, these groups have extensive collections of finite dimensionalrepresentations.

The largest and most tractable collection of representations are the homo-logical representations which are associated to finite covers of Σ. The firstof these is the standard homological representation Mod(Σ)→ GL(H1(Σ,Z))given by the action on first homology. The kernel of this representation iscalled the Torelli group.

More generally, pick a base point β ∈ Σ and let Σ′ be the surface Σ punc-tured at β. If π : Σ0 → Σ is a finite cover, there is a finite index subgroupM0 ≤ Mod(Σ′, β) of mapping classes fixing β that lift to Σ0. This gives arepresentation M0 ↪→ Mod(Σ0) → GL(H1(Σ0,Z)), that can be induced to arepresentation ρπ of Mod(Σ′). Thus, for each finite cover of Σ, we construct afinite dimensional representation of Mod(Σ′).

Homological representations are a very rich family of representations. Forexample, in [9] Grunewald, Larsen, Lubotzky and Malestein show that themapping class group of a once punctured surface has several infinite families of

Date: November 7, 2017.1

2 ASAF HADARI

arithmetic groups as images of homological representation (In [8], Grunewaldand Lubotzky show a similar result for homological representations of Aut(Fn),which are constructed in a similar way). Looijenga proves a similar result forabelian covers in [14].

These representations contain a great deal of information about the mappingclass group. For instance in [21] Putman and Wieland exhibit a beautiful con-nection between properties of homological representations and the virtual firstBetti number of mapping class group. In [18] McMullen studies the structureof genus 0 homological representations extensively and explores their strongconnections to the study of moduli spaces of Rieman surfaces.

Furthermore, due to the fact that homological representations are the onlyclass of representations of Mod(Σ) whose images are well understood, theyoften come up in applications where representations of mapping class groupwith large images are needed. One such application is the study of propertiesof random elements of subgroups of Mod(Σ) (see for example [15], [16], [17],[22]).

Aside from containing information about the group as a whole, these rep-resentations contain information about individual elements. It is a simple ex-ercise to show that for any nontrivial φ ∈ Mod(Σ) there is a cover π in whichρπ(φ) is non trivial. Koberda and later Koberda and Mangahas showed thathomological representations can detect the Nielsen-Thurston classification ofa mapping class ([12], [13]).

Despite being a well studied family, homological representations still remainquite mysterious and many surprisingly basic and natural questions aboutthem remain unanswered. One such question is the following.

Question 1.1. Suppose φ ∈ Mod(Σ) has infinite order. Is it possible to finda cover π such that ρπ(φ) has infinite order?

McMullen considered a stronger version of question 1.1. Let φ be a pseudo-Anosov mapping class with dilatation λ. For any finite cover π, let σπ be thespectral radius of ρπ(φ). It is easy to show that log(σπ) ≤ λ. For pseudo-Anosovs with orientable foliations, it’s true that λ = log σI , where I is thetrivial cover. For some time it was conjectured that for any pseudo-Anosovmap λ = sup log σπ, where the supremum is taken over all finite covers. In [19],McMullen showed that this is not the case for a large class of pseudo-Anosovmaps. McMullen then asked the following question.

Question 1.2. In the notation above, is sup log σπ > 0? In other words, canone always find a finite cover where the action on homology has an eigenvalueoff of the unit circle?

In this paper we provide a positive answer to question 1.1 for mappingclasses of punctured surfaces.

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 3

Theorem 1.3. Let Σ be an oriented surface of finite type whose fundamentalgroup is free. Let φ ∈ Mod(Σ) be an infinite order mapping class. Then thereexists a finite characteristic solvable cover Σ0 → Σ, and a lift φ0 of φ suchthat the action of φ0 on H1(Σ0,Z) has infinite order.

We also prove a related theorem for automorphisms of free groups.

Theorem 1.4. Let φ ∈ Out(Fn) be a fully irreducible automorphism. Thenthere exists a representative f of φ in Aut(Fn) and an f -invariant finite indexcharacteristic subgroup K � Fn such that the action of f |K on H1(K,Z) hasinfinite order, and Fn/K is solvable.

The restriction that Σ have a free fundamental group is rather technical.We conjecture that our proof can be strengthened to cover the case of closedsurfaces.

Acknowledgements. We wish to thank Benson Farb, Thomas Koberda, andJustin Malestein for helpful discussions and comments about earlier versionsof this paper.

1.1. Sketch of proof. We sketch the proof in the case where f ∈ Aut(π1(Σ))is induced by a pseudo-Anosov diffeomorphism, and acts trivially on H1(Σ,Z).Part of the difficulty in the proof arises from the fact that we are searching fora finite cover with certain properties, and that the set of all such finite coversis extremely large. To circumvent this problem, we will always try to look forabelian covers, which can be parametrized as rational points on a torus. Wewill end up taking a sequence of such covers.

Step 1 - passing to abelian covers. We begin by considering an infiniteabelian cover that contains information about all finite abelian covers. LetR =

∨n1 S

1 be a wedge of n circles. We have that π1(R) = Fn. Pick a

continuous map ϕ : R→ R inducing the automorphism f . Let R→ R be theuniversal abelian cover that is the cover corresponding to [Fn, Fn]. The space

R is an n-dimensional grid.

Let G = C1(R,Z) be the space of 1-chains in R. This is the space of formal

combinations of edges in R with coefficient in Z. The reason we work with1-chains as opposed to homology classes is that the space G has a particularlynice algebraic description - if we set H = H1(Fn,Z), then G ∼= Z[H]n.

The map ϕ can be lifted to a map ϕ that acts on G. This action also hasa particularly nice algebraic description - it is given by multiplication by amatrix Aϕ ∈ Mn(Z[H]). This matrix is the image of f under the Magnusrepresentation, which was originally defined by Satoh (see [23] for a survey).

The matrix Aϕ contains information about the action of f on finite abeliancovers. Suppose ψ : H → Cn is a homomorphism with finite image. We can

4 ASAF HADARI

think of ψ as a homomorphism from Fn → Cn. Let r = rψ = #ψ(Fn) be thesize of the image of ψ. Denote by Rψ the cover corresponding to the kernel ofthe natural map Fn → H1(Fn,Z/nZ). Note that this is always a characteristiccover.

The map ψ can be thought of as an assignment of complex values to aset of generators of H. Since the coordinates of Aϕ are polynomials in thesegenerators, we can evaluate Aϕ at ψ by plugging in the values into thesepolynomials. In this way we get a matrix ψ(Aϕ) ∈ Mn(C). This matrix givesthe action of a lift of ϕ on a subspace of C1(Rψ,Z), which can be identifiedwith the twisted chain space C1(R,Cψ).

We show that if this action has eigenvalues off of the unit circle, then sodoes the action of a lift of some power of f on H1(Rψ,Z). Thus, it is enough toshow that for some ψ, the matrix ψ(Aϕ) has eigenvalues off of the unit circle.One way to do this is to find a ψ such |ψ(Tr(Aϕ))| > n. This is the contentof Proposition 2.5. We will find such a ψ by studying the geometry of thesupport of Tr(Aϕ).

Step 2 - homological shadows. We study the geometry of the support ofTr(Aϕ) using the technology of homological shadows. Given a word w ∈ Fn, the

homological shadow of w or Sw is the set of vertices of R through which a lift

of w passes. Since the vertices of R correspond to the elements of H1(Fn,Z),we can think of Sw as a subset of H ⊗ R ∼= Rn. In [10] we showed that thereexists an n-dimensional convex polytope Sf ∈ Rn such that for any word wwith infinite f orbit, limk→∞

1kSfk(w) = Sf . This technology is discussed in

Section 2.3.

Step 3 - stable covers. Let v be a vertex of Sf . Suppose Rv → R is a finitecover to which f lifts, such that the action of f on H1(Rv,Z) is trivial andand a term projecting to v appears in Tr(Aϕv), where ϕv denotes a lift of ϕto the cover Rv, (we will actually use a slightly more technical definition forstable vertices which will be better suited for our needs). In this case we saythat v is stable in Rv. If all the vertices are stable, then the cover is calledstable. In Proposition 2.18 we use Fourier transform techniques to show thatif we find a stable cover to which f lifts, then we can find a ψ that satisfiesthe condition of Proposition 2.5, and thus find a cover where the homologicalaction of f has infinite order.

Step 4 - stabilizing vertices. The remainder of the proof reduces to thefollowing problem - given a vertex v of Sf , find a cover in which it is stable.We do this in two stages, which are covered in Section 2.5

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 5

To every vertex v we assign an integer measuring its complexity. Roughlyspeaking, this integer detects how deep into the lower central series of Fn weneed to go before being able to detect the vertex v. A vertex is stable if weassign it the integer 1. The assignment of this integer is the content of Lemma2.30.

We then show that for every vertex v, there is a finite abelian cover thateither reduces its complexity or has an infinite order homological action by alift of f . This is done in Lemma 2.31. Repeated application of this processallows us to find a cover where the vertex is stable.

2. Proof

The main difficulty in proving Theorem 1.3 is to prove it for pseudo-Anosovmapping classes. To accomplish this, we will prove a slightly more generalform of Theorem 1.4. After some initial reductions, the proof of Theorem 1.3will be almost the same as this case.

Theorem 2.1. Let φ ∈ Out(Fn) be an outer automorphism that has a traintrack representative with Perron-Frobenius transition matrix. Then there existsa representative f of φ in Aut(Fn) and a characteristic finite index subgroupK � Fn such that the action of f on H1(K,Z) has infinite order and Fn/K issolvable.

Note that a pseudo-Anosov diffeomorphism ϕ ∈ Mod(S) induces a well de-fined element φ∗ ∈ Out(π1(S)), which has a train track representative withPerron-Frobenius transition matrix (see [2] for definitions). We now proceedto prove Theorem 2.1.

Remark 2.2. Throughout the proof, we will often find it necessary to replacef with a power of itself. Since the finite covers we will be considering arealways characteristic, both f and any power of f lift to these covers. If theaction on homology of some power of f has infinite order, then so does theaction of f . Thus, it is enough to prove Theorem 2.1 for some power of f .

Remark 2.3. If the action of f on H1(Fn,Z) has infinite order, then Theorem2.1 is trivial. If the action of f on H1(Fn,Z) has finite order, we can replacef with a power of itself that acts trivially. Thus, we will always assume thatf is in the Torelli group, so f acts trivially on H1(Fn,Z).

2.1. Notation and preliminaries. Pick an automorphism f that projectsto φ. Let (Γ, β) be a graph with fundamental group Fn (which we will specifylater on in the proof) together with a chosen base point β ∈ V (Γ). Denotem = #E(Γ). Fix, once and for all, an orientation on each of the edges of Γ.

6 ASAF HADARI

Let ϕ : (Γ, β) → (Γ, β) be a continuous function that induces the auto-morphism f . This map can be taken to send any edge to a concatenation ofedges.

Let Γ→ Γ be the universal abelian cover of Γ, that is, the cover associatedto the subgroup [Fn, Fn]. Denote H = H1(Γ) (which we will think of as a

multiplicative group), and G = C1(Γ,Z), the group of 1-chains in Γ. The

action of H on Γ by deck transformations gives an action of H on G, whichturns it into a Z[H] module.

In fact, we have the Z[H]-module isomorphism G ∼= Z[H]E(Γ). We constructsuch an isomorphism explicitly. Pick once and for all an orientation on each

edge of Γ. Label the edges of Γ e1, . . . , em. Every edge in the graph Γ projects

to some edge ei of Γ. Our choice of a base point β of Γ gives a correspondencebetween the lifts of the edge ei and the elements of H. Thus, to each such liftwe can uniquely assign an element of H. Given such an edge η, we denote theelement of h assigned to it by h(η).

Any chain η ∈ G can be written as a formal sum of edges of Γ:

η =m∑i=1

∑j

aijηij

where each edge ηij is a lift of ei. The function:

h(η) =

∑

j a1jh(η1j)...∑

j amjh(ηmj)

∈ (Z[H])m

gives the isomorphism described above.

2.1.1. The lifted action of ϕ on G. The automorphism f acts on H. This ex-tends to a ring endomorphism of Z[H] which we always denote by f∗. Through-out the proof we assume that f is in the Torelli subgroup, so f∗ = In.

Pick a lift β of β, and lift ϕ to a continuous map ϕ : (Γ, β) → (Γ, β). Themap ϕ induces a homomorphism of G ∼= Z[H]E(Γ) given by

ϕ∗

h1...hm

= Aϕ

h1...hm

for some matrix Aϕ ∈Mm(Z[H]). The assignment f → Aϕ is called the Mag-nus representation of the Torelli group.

We need an explicit description of the he matrix Aϕ. This is usually doneusing Fox calculus (see [24] for details). Since our proof uses several variants

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 7

of this matrix, we choose to describe the construction in a somewhat indirectway via a directed graph (or digraph) called the transition graph of ϕ.

2.1.2. The transition graph of ϕ and the matrix Aϕ. Let T = Tϕ be the follow-ing directed graph. Set V (T ) = E(Γ). Let e ∈ V (T ). Write ϕ(e) = e1 . . . es,where each ei is an edge of Γ. Add s edges to T emanating from e, where theith edge, which we denote ηi, connects e to the vertex ei.

Let h : G → Z[H]E(Γ) be the isomorphism specified above. For any edgee let xe be the pre image under h of the element with 1 in the e-coordinate,and 0 in all others. Let e1 . . . es be the path ϕxe. Write this path as wgi

±1u,where gi is the edge corresponding to ei, traversed in the positive direction.We associate the following objects to the edge ηi.

(a) The sign of ηi, or s(ηi) ∈ {1,−1}, defined to be 1 if gi = ei and −1 oth-erwise. That is, the sign of ηi denotes whether or not the correspondingedge of ϕ(e) is traversed in the positive or negative direction.

(b) All but one of the coordinates of the vector (gi) are 0. We denotethe remaining coordinate t(ηi), and call it the translation of ηi. Thetranslation t(ηi) is the element in H corresponding to the beginning (orend, as determined by the sign) of the lift of the ith edge of ϕ(e).

To a cycle γ = η1 . . . ηk in T we associate the following objects.

(a) The sign of γ, or s(γ) =∏k

i=1 s(ηi).

(b) The translation of γ or t(γ) =∏k

i=1 t(ηi) and the normalized translationof γ or tn(γ) = 1

kt(γ), where the latter is thought of as an element of

H⊗R. In the cases we will be considering, we get the simpler expressiont(γ) =

∏ki=1 t(ηi).

Note that throughout this paper, we employ the word cycle in a somewhatnonstandard way to mean a sequence of edges originating and terminating atthe same point, not up to cyclical re-ordering of the edges. Our cycles willalways be based cycles. While this does not affect the definitions given above,it will come up in the sequel.

We are now able to give a description of Aϕ, that follows directly form thedefinitions. Given e, e′ ∈ E(Γ), we have that (Aϕ)e,e′ =

∑s(η)t(η), where the

sum is taken over all edges η of T connecting e to e′. Notice that in this sum,we are viewing H as a subset of Z[H] in the obvious way.

2.1.3. Matrices associated to subgraphs of T and to covers. We will requiretwo variants of the matrix Aϕ. The first variant is obtained by consideringsubgraphs of T . Let T ′ be a subgraph of T . We construct a matrix Aϕ[T ′] bysetting:

8 ASAF HADARI

(Aϕ[T ′])e,e′ =∑

η=(e,e′)∈E(T ′)

s(η)t(η)

The second variant involves covers. Let π0 : (Γ0, β0) → (Γ, β) be a finitecover to which ϕ can be lifted, and let ϕ0 be a lift of ϕ such that ϕ0(β0) = β0.Let T0 be the directed graph assigned to ϕ0. Then the graph T0 is a finitecover of T , where the covering map respects the directionality of edges. If T ′is a subgraph of T , let T ′0 be its lift to T0. We will use the notation Aϕ[T ′, π0]for the matrix associated to T ′0 .

2.2. Abelian covers and Tr(Aϕ).

2.2.1. The polynomial ∆ and its specializations. Let:

∆ = ∆ϕ(t) = det(tI − Aϕ) ∈ Z[H][t]

be the characteristic polynomial of Aϕ. Given a homomorphism ψ : H → S1,we define the specialization of ∆ at ψ, or ∆ψ by taking the image of ∆ underthe map Z[H][t] → Z[ψ(H)][t]. Similarly, we define (Aϕ)ψ ∈ Mm(C) to bethe image of Aϕ under this map. We use the letter ∆ in analogy with theAlexander polynomial (which is often also denoted ∆), whose specializationsare useful for the studying the action of lifts of a map on the homology ofabelian covers (See [20] for details.)

Suppose that ψ : H → S1 ⊂ C∗ has finite image. We abuse notation andview ψ as a map from Fn to C. Let rψ = #ψ(Fn) and let Γψ be the covercorresponding to the kernel of the natural map Fn → H1(Fn,Z/nZ). Notethat this is a characteristic cover.

The group Fn acts on C1(Γψ,C) via its action on Γψ by deck transformations.This factors through an action of H on C1(Γψ,C). Consider the subspaceCψ ⊂ C1(Γψ,C) defined by

Cψ = {x ∈ C1(Γψ,C) | a(x) = ψ((a)x,∀a ∈ Fn}The space Cψ is an m-dimensional subspace of C1(Γψ,C). Indeed, supposee1, . . . , em are the edges of Γ. For each i, choose a lift ei of e1 in Γψ. Then thespace Cψ is spanned by vectors of the form∑

a∈H

ψ(a)a(ei).

If ϕψ is a lift of ϕ to Γψ, then Cψ is ϕψ invariant. Then the action ofthe lift of ϕ on the space Cψ by multiplication by the matrix (Aϕ)ψ, and itscharacteristic polynomial is ∆ψ.

It’s possible to view the space Cψ somewhat differently (though we won’tuse this identification). Then the complex C∗(Γψ,Z), and the space C are both

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 9

Z[H] modules (where the action on C is given by ψ). Let C∗(Γψ,Cψ) be thecomplex C∗(Γψ,Z)⊗Z[H] C. The space C1(Γψ,Cψ) can be naturally identifiedwith the subspace of Cψ.

2.2.2. Fourier transforms on Abelian groups. Before we proceed, we need torecall some standard facts about the Fourier transform Zn. Note that in thedefinitions that follow, all L-spaces are spaces of functions into C. Recall thatT = (S1)n is the group of characters of Zn. Let f ∈ L1(Zn). We define a

function f ∈ L1(T ) by setting:

f(ξ) =∑x∈Zn

f(x)ξ(x)

We summarize the facts we need in the following proposition.

Proposition 2.4. The Fourier transform satisfies the following properties.

(a) It extends to an isometric embedding L2(Zn)→ L2(T, ν).(b) Given f, g ∈ L1(Zn), one has that: where the multiplication on the left

side is multiplication in the group ring, and the multiplication on theright side is point wise multiplication of functions.

f · g = f · g(c) The transform L1(T)→ L1(Zn) given by:

f(x) =

∫T

f(ξ)ξ(x)dν(ξ)

where ν is the Haar measure on T is the adjoint of the Fourier trans-form when restricted to the class of continuous functions, and is an L2

isometry on this class.

2.2.3. Passing to abelian covers.

Proposition 2.5. Let π : Γ0 → Γ be a characteristic cover to which ϕ lifts toa continuous map ϕ0, and suppose (ϕ0)∗ acts trivially on H1(Γ0,Z). DenoteH1(Γ0,Z) ∼= Zk, K = #E(Γ0) and let t0 = TrAϕ0 ∈ Z[Zk]. We can view

Z[Zk] as a subspace of L2[Zk]. If there exists ξ ∈ (S1)k such that |t0(ξ)| > K,then there is a characteristic cover π1 : Γ1 → Γ and a lift ϕ1 : Γ1 → Γ1 suchthat (ϕ1)∗ acts with infinite order on H1(Γ1,Z).

Proof. Note that t0 is an element of L2[Zk] with finite support. If we writet0 =

∑aixi, and take ζ : Zk → C we have that t0(ξ) =

∑aiζ(xi). For any

x ∈ Zk, the function ζ → ζ(x) is continuous, and thus t0 is continuous.

Since tϕ0 is a continuous function, we can assume that ξ is a torsion elementof (S1)k, and thus corresponds to a a representation of Zk with finite image.Let r = #ξ(Zk), and let Γ1 be the cover of Γ0 associated to the kernel of the

10 ASAF HADARI

map H1(Γ0,Z) → H1(Γ0,Z/rZ). This is a characteristic cover, and thus wecan lift ϕ0 to a map ϕ1 : Γ1,→ Γ1.

The matrix (Aϕ0)ξ is a K ×K matrix with trace greater than K, and thusit has eigenvalues off of the unit circle. We deduce that the action (ϕ1)∗ onC1(Γ1,Z) has an eigenvalue off the unit circle. The Proposition now followsfrom the following claim.

Claim 2.6. Let Γ be a graph with fundamental group Fn and let ϕ : Γ→ Γ bea continous function inducing the automorphism f ∈ Aut(Fn). Suppose thatthe action of ϕ on C1(Γ,C) has an eigenvalue with absolute value > 1. Thenthe same is true for the action of ϕ on H1(Γ,C).

To see this, let U = C1(Γ,C), and let W ⊂ U be the subspace spanned byall closed paths in Γ. We have a natural identification W ∼= H1(Γ,C). Thespace U is spanned by elements of the form 1e, where e ranges over all edgesof Γ. Pick any norm ‖ · ‖ on U , and let λ > 1 be the spectral radius of theaction of ϕ on U . Then there exists an edge e such that

lim supk→∞

1

klog ‖ϕk(1e)‖ = log λ

Denote Lλ to be the direct sum of all generalized eigenspaces corresponding

to eigenvalues with absolute value λ. Setting νk = ϕk(1e)‖ϕk(1e)‖ , we have that the

distance from νk to Lλ goes to 0 as k →∞. Note that ϕk(1e) corresponds to apath in Γ, and any such path can be closed to a loop using a bounded numberof edges. Thus, the distance of νk from W goes to 0 as k → ∞. ThereforeLλ ∩W 6= {0}. Since this space is ϕ-invariant, it contains an eigenvector witheigenvalue of absolute value λ.

�

2.3. Homological shadows and extremal subgraphs. It is not obvioushow to find a ξ that satisfies the conditions of 2.5. Our plan is to find such aξ by studying the geometry of the support of t0 = TrAϕ, and using propertiesof the Fourier transform. In this section we study the shape and structure ofthe support of t0 using a tool called homological shadows.

2.3.1. Homological shadows of paths. From now on fix Γ =∨n

1 S1, a wedge

of n circles. Given a path p in Γ originating at β, let p be the lift of p to Γ

originating at β. Define the S(p) ∈ G to be S(p) =∑ciei, where ci is the

number of times p passes through the edge ei in either direction (so we alwayshave that ci ≥ 0). Given an element w ∈ π1(Γ, β) let S(w) = S(w) where w isthe path corresponding to a reduced representative of w.

Let V be the set of vertices in Γ. Fix a preferred lift β of β. This gives anidentification of V with H ∼= Zn. We have a natural inclusion H → H1(Γ,R),

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 11

which gives us an identification of V with a subset of Rn. Define the shadow ofp to be S(p) = SupportS(p) ∩ V ⊂ H ⊂ H1(Fn,R). Similarly, for an elementw ∈ π1(Γ, β) define the shadow of w, or S(w) ⊂ H1(Fn,R) to be the supportof S(w) ∩ V .

2.3.2. Homological shadows of f and ϕ. In [10], the following Theorem isproved.

Theorem 2.7. Suppose f is has a train track representative with a Perron-Frobenius transition matrix and f∗ has finite order. There exists a convexpolytope Sf ⊂ H1(Fn,R) with rational vertices such that for any w ∈ Fn withinfinite f -orbit,

lim1

kSfk(w) = Sf

where the above limit is taken in the Hausdorff topology. We call Sf theshadow of f .

Note that while the proof of the above theorem utilizes the existence of atrain track graph to calculate Sf , the result holds for any graph with freefundamental group. For the purposes of our proof we require Γ−

∨n1 S

1. Also,while we are interested mainly in Sf , for technical reasons we need to considera different shadow that is attached to ϕ. In the course of proving the abovetheorem we show the following.

Proposition 2.8. Under the above conditions, there exists a convex polytopewith rational vertices Sϕ ⊂ H1(Fn,R) called the shadow of ϕ such that forany path p in Γ with infinite ϕ orbit:

limk→∞

1

kSϕkp = Sϕ

Furthermore, Sϕ contains all points of the form tn(γ) where γ is a cycle in Tand is the convex hull of all elements of the form tn(γ), where γ is a simplecycle in T .

A priori, the shadows Sf and Sϕ do not have to be equal. They are howevercalculated in a similar manner. The shadow Sf is calculated by taking theconvex hull of tn(γ) where γ is a cycle in the train track transition graph of atrain track representative of f . If we choose a fixed vertex β in such a graph,and replace f with an element in its outer automorphism class induced by theautomorphism on π1(Γ, β) then Sϕ = Sf (see [10] for details). In the sequelwe will always assume that Sϕ = Sf .

The convex hull described above was considered by Fried in [7] for the caseof an automorphism coming from a pseudo-Anosov diffeomorphism, and in aslightly different language by Dowdall, Kapovich and Leininger ([4], [5]) andseparately by Algom-Kfir, Hironaka and Rafi ([1]) for the general Out(Fn) case.

12 ASAF HADARI

In both cases, this convex hull gives a cross-section of the dual cone to a fiberedcone (see the above references for definitions) which is a n+1 dimensional conewhen f is in the Torelli subgroup. Thus, we have the following.

Proposition 2.9. The polytope Sf is n dimensional.

2.3.3. Homological shadows and extremal subgraphs.

Definition 2.10. Let v be a vertex of Sϕ. Let Tv ⊂ T be the union of allcycles γ satisfying tn(γ) = v. We call Tv an extremal subgraph.

Extremal subgraphs have nice properties, which are consequences of thefollowing definitions, that are useful in the study of Tr(Aϕ).

Definition 2.11. Given x, y ∈ Z[H] we say that x is subordinate to y andwrite x � y if y|Support(x) = x. We say that x and y are separated and writex‖y if Support(x) ∩ Support(y) = ∅.Definition 2.12. Let T ′ ⊆ T be a subgraph. For any k, let tk[T ′] =∑

s(γ)t(γ) where the sum is taken over all cycles of length k in T ′. Notethat tk[T ′] is the trace of (Aϕ[T ′])k.Definition 2.13. Let T ′, T ′′ be subgraphs of T . We say that T ′ is subordinateto T ′′, and write T ′ � T ′′ if for all sufficiently large k, tk[T ′] � tk[T ′′].Definition 2.14. We say that two subgraphs T ′, T ′′ of T are separated, anddenote T ′‖T ′′ if for all sufficiently large k, tk[T ′] ‖ tk[T ′′].The first thing to note is the following .

Lemma 2.15. If v, w are two different vertices of of Sϕ then Tv, Tw � T andTv‖Tw.

Proof. The result follows directly from the following claim: let γ ⊂ Γ be aloop of length k, and let u be a vertex. Then t(γ) = ku ⇐⇒ γ ⊂ Tu.

Indeed, the only if direction of the claim shows that for any k, tk[Tu] =au(ku) for some number au. Applying this to the vertices v, w we get tk[Tv] =av(kv), tk[Tw] = aw(kw). Since v 6= w we have by definition that Tv‖Tw. thecoefficient of kv in tk[T ] is the sum of the signs of all loops of length k in Twith translation equal to kv.

The if direction of the claim shows that this number is equal to the sumof the signs of all loops of length k in Tv with translation equal to kv. Bydefinition, this means that Tv � T . Similarly, Tw � T .

We now turn to prove the claim. The if follows from the definition of Tu.To see the only if direction, suppose γ′ ⊂ Tu is a cycle of length k. Letω ∈ H1(Fn,R) be a cohomology class whose maximum on Sϕ is achieved atu, (such a class exists, because u is a vertex of the convex polytope Sϕ). Bymaximality, ω(t(γ′)) ≤ kω(u).

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 13

Suppose that ω(t(γ′)) < kω(u). Write γ′ = e1 . . . er. By definition of Tu,there are paths η1, . . . , ηr in Tu of lengths k1− 1, . . . , kr− 1 such that for everyi: eiηi is a cycle and ω(t(eiηi)) = kiω(u). Let η be the cycle ηr . . . η1. Letτ = ω ◦ t. Then:

τ(η) =∑i

τ(ηi) =∑i

[kiω(u)− τ(ei)] = ω(u)∑i

ki − τ(γ′)

Notice that ω(u)∑

i ki = length(η)ω(u)+kω(u), and that kω(u)−τ(γ) > 0.Thus, τ(η) > length(η)ω(u), which is a contradiction to the definition of ω.Thus, τ(γ′) = length(γ′)ω(u).

�

Observation 2.16. The definitions in this subsection, as well as Lemma 2.15do not use any topological or homological properties. Indeed, they are quitegeneral. They hold whenever we have a directed graph ∆ and a additivefunction from the cycles in ∆ to some finite dimensional rational vector spacewhich doesn’t depend on cyclic reordering of the cycles.

Extremal subgraphs behave quite well with respect to covers. One fact thatwe need is the following. If Tv, Tw are extremal subgraphs corresponding todifferent verices, π0 : Γ0 → Γ is a finite cover, and T ′v , T ′w are the lifts of Tv, Twrespectively, then T ′v‖T ′w.

2.4. Stable covers. Let π : (Γ0, β0) → (Γ, β) be a finite characteristic coverto which ϕ can be lifted to a map ϕ0, and let v be a vertex of Sϕ.

Definition 2.17. We say that the vertex v is stable in the cover Γ0 if thematrix Aϕ[Tv, π0] is not nilpotent. We say that the cover Γ0 is stable if all thevertices of Sϕ are stable in it.

Stable covers are well suited for applying Proposition 2.5. In this section weprove the following.

Proposition 2.18. Let π0 : Γ0 → Γ be a characteristic finite stable cover.Let d = dimH1(Γ0,Z). If the action of ϕ0 on H1(Γ0,Z) has finite order, thenthere exists an integer j, and an element ξ ∈ (S1)d such that |ξ(sj)| > #E(Γ0)where sj = Tr(Aϕ0)

j

First, we note that by replacing f with a power of itself we may assumethat ϕ0 acts trivially on H1(Γ0,Z). Let D be the deck group of the cover π0.We can view D and f as elements of Out(π1(Γ0)). Since ϕ lifts to Γ0, we havethat f normalizes D. By replacing it with a power of itself, we can assumethat f centralizes D. Fixing a vertex β0 ∈ V (Γ0), we can identify V (Γ0) withD. Since f centralizes D, we get that ϕ0 acts trivially on V (Γ0).

14 ASAF HADARI

This means gives that for any δ ∈ D, the maps δ ◦ ϕ0, ϕ0 ◦ δ have thesame induced action on G. Thus, if we set t0 = TrAϕ[π0], we get that t0 isD∗-invariant, where D∗ denotes the induced action of D on H1(Γ0,Z).

Observation 2.19. A stronger property of t0 holds as well. The trace t0is the sum of all the elements along the diagonal of Aϕ[π0]. The rows ofAϕ[π0] correspond to the edges of Γ0. By picking one edge in each D orbit,we get that there exists s0 ∈ Z[Zd] such that t0 =

∑δ∈D δ∗(s0). Thus, for

any h ∈ Support(t0) we get that∑

δ∈D aδh is divisible by |D|, where aδh is thecoefficient of δh in t0.

Definition 2.20. For any matrix X ∈Md[Zd], and any integer k define the ktrace of X to be:

Trk[X] =∑

h∈kH1(Γ0,Z)

ah

where ah is the coefficient of h in TrX.

Let v be a vertex of Sϕ. Pick v′ ∈ H1(Γ0,Q) such that (π0)∗v′ = v (we can

pick such a v′ since the vertices of Sϕ have rational coordinates). Let

v =∑

g ∈ Dg∗v′ ∈ H1(Γ0,Q)

be a D invariant lift of v to H1(Γ0,Z). In [10] it is shown that for any i:

Sϕi = iSϕ

Therefore, we can pick an integer i such that v ∈ H1(Γ0,Z). Thus, byreplacing f with f i, we may assume that v ∈ H1(Γ0,Z).

Denote Bv = v−1Aϕ[Tv, π0]. The matrix Bv is a translation of the matrixAϕ[Tv, π0] corresponding the the vertex v, where the translation is introducedto simplify calculations in the sequel. Since π0 is a stable cover, the vertex v isstable in π0. This means that the matrix Aϕ[Tv, π0] and hence also the matrixBv are not nilpotent. Thus Tr(Bi

v) 6= 0 for infinitely many i. We prove thefollowing stronger claim.

Lemma 2.21. For all sufficiently large k, there exist infinitely many i suchthat Trk[B

iv] 6= 0.

Proof. Let L be a lattice in Zd, and let ΨL be its characteristic function. Let

T = (S1)d. The function ΨL is not in L1. We can however still define ΨL, itsFourier transform as distribution on L1[T]. This distribution has the propertythat given a continuous g ∈ C[T]:

ΨL(g) = Ψ · g

where Ψ · g =∑

h∈L ah, and ah is the coefficient of h in g.

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 15

The distribution ΨL is simple to calculate. Indeed, we have that:

ΨL =1

|AL|∑ξ∈AL

δξ

where AL = {ξ : ξ|L = 1}, and δξ is the delta distribution at ξ.This is particularly simple to see when g is a polynomial. Suppose g =∑ahh. Then:

1

|AL|∑ξ∈AL

δξ(g) =1

|AL|∑ξ∈AL

δξ(∑h

ahh) =1

|AL|∑h

ah∑ξ∈AL

h(ξ)

The set AL is the set of all homomorphisms from H1(Γ0,Z) → C whosekernel contains L. All such homomorphisms have finite image. For any h, the

sum h(ξ) is the repeated sum of all j-th roots of unity for some j. For h ∈ L, we

have j = 1, and by definition of AL that∑

ξ∈ALh(ξ) = |AL|. For h /∈ L we have

j 6= 1 and thus∑

ξ∈ALh(ξ) = 0. Thus, 1

|AL|∑

ξ∈ALδξ(g) =

∑h∈L ah = ΨL · g.

Let Bv ∈Md[L2(T )] be the Fourier transform of Bv, here the Fourier trans-

form is taken entry by entry. The entries of Bv are then functions from (S1)d

to C. For any ξ ∈ (S1)d, we can form the matrix Bv[ξ] ∈Md(C) by plugging ξ

into each coordinate of Bv. We call this matrix the specialization of Bv at ξ.Let X ⊂ (S1)d × Cd be the set of all (ξ, z1, . . . , zd) such that (z1, . . . , zd)

are the eigenvalues of Bv[ξ], with multiplicity. Let P : X → (S1)d be theprojection P (ξ, z) = ξ. The map P is a local homeomorphism. A root isa function ρ : T → Cd, ρ = (ρ1, . . . , ρd) such that (Id, ρ) : T → T × Cd

is a section of the projection P . More explicitly, ρ is a root if for every

ξ ∈ T, (ρ1(ξ), . . . , ρd(ξ)) is the collection of eigenvalues of Bv[ξ] counted withmultiplicity.

Since P is a local diffeomorphism roots always exist, but they may notalways be continuous. However, if Y ⊂ T is a contractible subset, then we canalways find a root that restricts to a continuous function from Y to Cd. Let Zbe a finite collection of hyperplanes of hyperplanes in T such that Y = T \ Zis contractible (here a hyperplane is the image in T of an affine hyperplane inRd under the covering space map Rd → T). Without loss of generality, we canassume that Z contains no rational points in T.

Pick a root ρ = (ρ1, . . . , ρd) that is continuous on Y . Notice that for anynumber j, and any ξ ∈ T we have that

Tr Bv

j(ξ) =

d∑i=1

ρjj(ξ).

16 ASAF HADARI

Fix an integer k, and let L = kZd. We would like to use the inverse Fouriertransform to study Trk(Bv) using the functions ρ1, . . . ρd and the distribution

ΨL. Unfortunately, the isometry of the inverse Fourier transform does nothold for non-continuous functions. We address this issue in the following way.

Let U be a small neighborhood of Z (we take U to be small enough sothat it does not intersect AL.) We replace ρ1, . . . , ρd with continuous functions

ρ′1, . . . , ρ′d such that for any i, ρi = ρ′i on Y , and ρ′1 + . . .+ρ′d = Tr[Bv]. Because

ρ′1 + . . . + ρ′d = Tr[Bv], and the ρ′i are continuous, we have that∑

ΨL(ρ′i) =

Trk(Bv). Since U does not intersect AL, we have that Trk[Bv] =∑

ΨL(ρ′i).Similarly, for any integer j we have that:

Trk(Bjv) =

1

|AL|

d∑i=1

∑ξ∈AL

ρi(ξ)j

By the Vandermonde theorem, this expression is non-zero for infinitely manyj if and only if ρi|AL

6= 0 for some i. Since ρi is continuous on an open, densesubset of T , and the sets AL = AL(k) become equi-distributed as k → ∞,we get that for all sufficiently large k, ρi|AL

6= 0 for every i. The result nowfollows. �

Lemma 2.22. There exist infinitely many k such that Trk(Bv) > 0 for everyvertex v of Sϕ.

Proof. Pick a sufficiently large k which satisfies the conclusion of Lemma 2.21for all all vertices of Sϕ. As in the previous lemma, let L = kZd. Let v be avertex, and let ρ = (ρ1, . . . , ρd) be the root constructed in the previous lemmawith respect to the vertex v.

Let µv = max{|ρi(ξ)| : 1 ≤ i ≤ d, ξ ∈ AL}. Let Xv = {(i, ξ) : |ρi(ξ)| = µv},and lv = |Xv|. Choose some arbitrary order on the elements of Xv and let αvbe the vector αv = (ρi(ξ)

µv)(i,ξ)∈Xv .

Let s be the number of vertices of Sϕ. Choose some arbitrary order on thevertices of Sϕ. Let l =

∑lv, and let α be the vector α = (αv)v ∈ (S1)s. Define

a function F : (S1)s → Cs, by setting the ith coordinate of F (ξ1, . . . , ξs) to bethe sum of all the ξj corresponding to the ith vertex of Sϕ.

If α is a rational point, then there is some power j of α such that αj =(1, . . . , 1). Notice that F of this point is positive in all of its coordinates. By

replacing j with a sufficiently large multiple of itself, we get that ΨL(Trk[Bjv]) >

0, for all v. If α is not a rational point then, after replacing α with some powerof itself we get that the set {αi}∞i=1 is equdistributed in some sub-torus T ′ ⊂(S1)s that does contain a rational point and is closed under multiplication.

The function F is continuous on T ′. Since it contains a rational point, thereis an open subset V of T ′ where all the coordinates of F have real parts that

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 17

are greater than 12. Since the orbit of α is equidistributed in T ′, there are

infinitely many j such that all coordinates of F (αj) have real parts that aregreater than 1

2. For all sufficiently large j of this form, and for any vertex v

we get that ΨL(Trk[Bjv]) > 0.

�

Suppose h ∈ Zd. Notice that it is enough to prove Proposition 2.18 for the ma-trix hAϕ[π0] as opposed to Aϕ, because multiplication by h acts by isometrieson L2[Zd]. Thus, we may assume that 0 is a vertex of Sϕ.

Lemma 2.23. There exists a lattice L ⊂ Zn such that Sϕ ∩ L ⊂ V (Sϕ), and|L ∩ Sϕ| > n, where V (Sϕ) is the set of vertices of Sϕ.

Proof. Denote P = Sϕ. Let F be a face of P . We say that a face F ′ of Popposite from F if there exists α ∈ H1(Γ;R) that takes its minimum value inP precisely on F and its maximum value on F ′. Let F be a co-dimension 1face of P that has 0 as a vertex. Let F ′ be an opposite face such that thereexists α ∈ H1(Γ0,R) that takes on its minimal value of 0 precisely at F andtakes on its maximum value precisely at F ′.

We show the slightly stronger claim that there is a lattice L whose intersec-tion with P consists solely of vertices, has at least n + 1 vertices, and whoseintersection with P is contained in F ∪ F ′. We prove this by induction on n.For n = 1, the result is obvious. Suppose we have proved it up to n, and wishto prove it for n+ 1.

Denote A = V (F ), B = V (F ′). For v ∈ B, set Bv = {v′ − v|v′ ∈ B} ⊂span(F ), and let Qv be the convex hull of A and Bv. It is possible to pickv such that 0 is a vertex of Qv. Indeed, we can pick α ∈ H1(Γ0,R) whoseminimum on F is achieved at 0 and whose minimum on F ′ is achieved atprecisely one vertex. Take this vertex to be v.

Let L′ be the lattice associated to Qv and any co-dimension 1 face of Qv

by the induction hypothesis, and let L = 〈L′, v〉. By definition, the latticeL intersects P only at vertices which are contained in F ∪ F ′. Furthermore,this intersection includes 0, v and at least d − 1 other vertices. If this latticeis not n-dimensional, we can extend it by adding further linearly independentvectors without increasing its intersection with P . Thus, this lattice satisfiesthe conditions of the lemma.

�

We are now able to prove Proposition 2.18.

Proof. Pick numbers k, j satisfying the conclusion of Lemma 2.22. Let L′ be alattice in H1(Γ,Z) satisfying the condition of Lemma 2.23. Let {v1, . . . vr} =L′ ∩ Sϕ. Let v1, . . . , vr be D-invariant lifts of these vector to H1(Γ0,R). Foreach 1 ≤ i ≤ r, let Li = Tr[Bj

vi] ∩ kZd. Consider the lattice 〈L1, . . . , Lr〉. We

18 ASAF HADARI

can add additional vectors to this lattice without increasing its intersectionwith Support TrAjϕ[π0] so that it is a d dimensional lattice. Call the resultinglattice L.

Let ΨL be its characteristic function. We have that

ΨL · TrAjϕ[π0] =r∑i=1

Trk[Bjvi

]

Note that Trk[Bjvi

] > 0 for every i. Furthermore, since the set vi + kZd is Dinvariant, we have by observation 2.19 that Trk[B

jvi

] is divisible by |D|. Thus

ΨL · TrAjϕ[π0] ≥ (n+ 1)|D| > n|D| = #E(Γ0)

Since ΨL ·TrAjϕ[π0] = 1|AL|

∑ξ∈AL

TrAjϕ[π0](ξ), we get that there exists ξ ∈ AL

such that | TrAjϕ[π0](ξ)| > #E(Γ0), as required. �

2.5. Stablizing vertex subgraphs. We are now left with the problem offinding stable covers of Γ. We address this problem by finding, for each vertexv of Sϕ a cover where it is stable.

Observation 2.24. If Tv is the vertex subgraph corresponding to the stablevertex v, and T ′v is the lift of Tv to some cover π where ϕ lifts and acts triviallyon first homology, then T ′v is also stable (that is - TrAϕ[Tv, π] is not nilpotent).If we can find for each vertex v a cover πv where it is stable then by takinga common refinement we can find a cover π where f∗ has infinite order, or alllifts of all vertices are stable. If all of the covers in question are characteristic,then so is π.

Our goal in this section is to prove the following proposition.

Proposition 2.25. For any vertex v of Sϕ, there exists a finite characteristiccover π0 : Γ0 → Γ such that the lift of Tv in Γ0 is stable, or a finite character-istic cover where the homological action of a lift of ϕ has infinite order.

We begin by giving some definitions, and recalling standard facts about nilpo-tent groups.

2.5.1. Assigning subgroups to subgraphs. Let Π = η1 . . . ηk be a path of lengthk in T . This path corresponds to a subpath of ϕk(η1) in Γ whose last elementis η±1

k . Write this path as g = rη±1k . This subpath is a loop in Γ, and thus

gives a word in Fn.

Definition 2.26. Define the group element of Π or g(π) to be the elementof Fn corresponding to r if g = rηk and g otherwise.

The word g(Π) has the property that its image in H1(Fn,R) is t(Π).

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 19

Definition 2.27. Let C be a finite collection of cycles in T . Define g(C) =〈g(γ)|γ ∈ C〉 ≤ π1(Γ, β).

Let γ1, γ2 be loops based at ν. Suppose γ1 corresponds to the sub-pathwν, and γ2 corresponds to the sub-path wνν−1. In this case we have g(γ1) =g(γ2) = w. The difference between them is that s(γ1) is positive, and s(γ2)is negative. Intuitively, the two loops, γ1, γ2 correspond to a an unavoidablecancellation in the free group of part of the word ϕk(ν).

For every ν ∈ V (T ), and any g ∈ g(C), define C(ν, g,+) to be the set of allγ ∈ C originating at ν such that s(γ) = 1, g(γ) = g. Define C(ν, g,−) to bethe set of all γ ∈ C originating at ν such that s(γ) = −1, g(γ) = g.

Definition 2.28. We say that g(C) is non degenerate if there exist ν, g suchthat #C(ν, g,+) 6= #C(ν, g,−).

If v is a vertex of Sf = Sϕ, and Cv is the collection of simple cycles in Tvthen g(Cv) is non-degenerate, since otherwise v would not be a vertex of Sf .

2.5.2. Nilpotent quotients of free groups. Let F be a group. The lower centralseries of F is the the sequence of subgroups given by the recursive definitionF0 = F , Fi+1 = [F, Fi]. We denote by Ni = F/Fi. Let Li = Fi−1/Fi. Let L bea finitely generated free group. We require the following standard facts:

(a) The groups Fi are characteristic subgroups of F .(b) The groups Ni are nilpotent.(c) The groups Li are finitely generated torsion free abelian groups, and

furthermore Li = Z(Ni). If S is a generating set for F , then theset of elements of the form [a1, . . . , ai+1] (where a1, . . . , ai+1 ∈ S, and[a1, . . . , ai+1] = [a1, [a2, [a3, . . .]]]) generate Li.

(d) Let f ∈ Aut(F ) be an automorphism that acts trivially on L1. Then facts trivially on Li for every i.

2.5.3. Nilpotent-stable covers. Let v be a vertex of Sϕ, and let Tv be its vertexsubgraph. For any i ≥ 0, let pi : π1(Γ, β)→ Ni be the quotient map onto theith nilpotent quotient of Fn.

Definition 2.29. For any k ≥ 0, let tk[Tv, Ni] =∑

s(γ)πi ◦ g(γ) ∈ Z[Ni],where the sum is taken over all cycles of length k in Tv. We say that Tv is ith

level nilpotent stable if tk[Tv, Ni] 6= 0 for infinitely many k.

Lemma 2.30. There exists an i such that Tv is ith level nilpotent stable.

Proof. Since Tv is a vertex graph, for any two cycles γ1, γ2 of length k in Tvwe have that p1 ◦ g(γ1) = p1 ◦ g(γ2) = kv. Pick x ∈ Fn whose image in H is v.For any cycle γ of length k in Tv, we have that p2(x−1g(γ)) ∈ L2.

20 ASAF HADARI

Since f acts trivially on H1(Fn,Z), for any cycle γ in Tv of length k, we canwrite

p2 ◦ g(γ) = t(2)(γ)k−1∏j=0

p2 ◦ f j(x)

for some t(2)(γ) ∈ L2.Given two cycles, γ1, γ2 pf lengths l1, l2 based at the same point, we have

that g(γ1γ2) = f l1(g(γ1))g(γ2). Since L2 is central in N2, and f acts triviallyon L2, we get t(2)(γ1γ2) = t(2)(γ1)t(2)(γ2). Similarly, for any cycle γ in Tv oflength k that is a concatenation of cycles γ1, . . . , γr all based at that samepoint:

t(2)(γ) =k∏i=1

t(2)(γi)

Set

t(2)k [Tv] =

∑s(γ)t(2)(γ) ∈ Z[L2]

where the sum is taken over all based cycles of length k in Tv. We then havethat:

tk[Tv, N1] = t(2)k [Tv]

k−1∏j=0

p2 ◦ f j(x)

In contrast to the expressions of the form t(γ) that we used in previous sec-tions, the function t(2)(γ) is not necessarily invariant under a cyclic reorderingof γ. However, by the additivity property discussed above, it is invariant undercyclic reorderings that preserve the basepoint.

We define a new function bft(2) (or basepoint free translation) that assigns

to each cycle γ an element of LV (Tv)2 in the following way: for a vertex ν of

Tv and a cycle γ, define bft(2)[γ](ν) to be 0 if γ does not pass through ν andt(2)(γν) otherwise, where γν is a cyclic reordering of γ so that it is based at ν.

The function bft(2) is an additive function from cycles in Tv to a finitelygenerated torsion free abelian group whose value does not depend on cyclicreordering. As in explained in Observation 2.16, we can define extremal sub-graphs in the same way as in section 2.3.3. Thus, we have a (not necessarily

proper) vertex subgraph T (2)v ⊆ Tv such that for any cycle γ in T (2)

v of length

k, bft(2)(γ) = kv(2) for some v(2) ∈ LV (Tv)2 ⊗Q. Furthermore, if T (2)

v is ith levelnilpotent stable then so is Tv.

Let l be the minimal integer such that lv(2) ∈ LV (Tv)2 . By definition of T (2)

v

we have that kv(2) ∈ LV (Tv)2 whenever k is the length of a cycle in T (2)

v . We

get that k is divisible by l whenever k is the length of a cycle in T (2)v .

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 21

Using the same reasoning as in the previous paragraph, we can write for any

simple loop γ in T (2)v :

p3 ◦ g(γ) = t(3)(γ)

k/l−1∏j=0

p3(f jl(x))

for some t(3)(γ) ∈ L3. Proceeding analogously to the above discussion, we find

a (not necessarily proper) extremal subgraph T (3)v ⊂ T (2)

v where for any loop

γ, bft(3)(γ) = kv(3) where k is the length of γ and v(3) ∈ LV (T (2)

v )3 ⊗ Q. By

construction, if T (3)v is ith level nilpotent stable then so is T (2)

v .

Proceeding in this manner, we get a sequence of graphs . . . ⊆ T (4)v ⊆ T (3)

v ⊆T (2)v ⊆ Tv. Since Fn is residually nilpotent, there exists an i such that the

subgroup of Fn generated by the group elements assigned to all the cycles in

T (i)v based at any given vertex of this graph is cyclic. Indeed, if there were

always two such cycles whose group elements did not form a cyclic group thenby residual nilpotence one could pass to a further proper subgraph.

Since Tv is a finite graph, this process has to terminate. Since v is a persistent

vertex, we have that T (i)v is ith level nilpotent stable.

�

Lemma 2.31. Suppose that Tv is the extremal graph of the vertex v, and thatTv is ith level nilpotent stable. Then there is a characteristic cover π0 : Γ0 → Γwhere the lift of Tv is (i− 1)th level nilpotent stable, or a characteristic coverwhere the action of a lift of ϕ has infinite order action on first homology.

Proof. Let a1, . . . , ai ∈ Fn, and let p ∈ N. Explicit calculation shows that

[ap1, . . . , api ] ≡ pi[a1, . . . , ai] (mod F (i+1)

n )

(see for example [3]). Thus, if we set Kp < Fn to be the kernel of the mapFn → H1(Fn,Z/pZ), then the image of Kp

i−1 in Li−1 is pi−1Li−1. Note thatthis cover is characteristic.

Using the notation of Lemma 2.30, let

bft(i)k [T (i)

v ] =∑

s(γ)bft(i)(γ)

where the sum is taken over all loops of length k in T (i)v . Let bft

(i)k [T (i)

v , p] be the

sum we get by taking only the summands of bft(i)k [T (i)

v ] that lie in pi−1LV (T (i)

v )i−1 .

If we show that there exists a p such that for infinitely many values of k,

bft(i)k [T (i)

v , p] 6= 0, then we will have that Tv is (i− 1)th level nilpotent stable inKp.

22 ASAF HADARI

DenoteM = LV (T (i)

v )i−1 . By construction, the map bft(i) gives a homomorphism

from H1(T (i)v ,Q) to M . It can thus be extended to a homomorphism from

C1(T (i)v ) → M . Thus, if T (i)

v has m vertices, we can construct a matrix

A ∈ GLm(Z[M ]) such that t(i)k [T (i)

v ] = TrAk, for any k. We are tasked withfinding p such that for infinitely many k, Trp[A

k] 6= 0 (where Trp is the notationused in the proof of Proposition 2.18).

In Lemma 2.21 we showed that given a non-nilpotent m×m matrix A withcoefficients in Z[Zd] there exists a p such that Trp(A

k) 6= 0 for infinitely manyvalues of k (indeed, this is true for all but finitely many values of p). Applyingthis Lemma gives the result.

�

2.5.4. Concluding the proof. We now proceed to prove Proposition 2.25.

Proof. By Lemma 2.30, the graph Tv is ith level nilpotent stable for some i. Byrepeated application of Lemma 2.31, we can either find a characteristic coverwhere the homological action has infinite order, or one where the lift of Tv is1st level nilpotent stable. Since this is the same as our original definition ofbeing stable, we are done.

�

2.6. Proof of Theorem 2.1.

Proof. If f∗ has infinite order, we are done. Otherwise, by replacing f with apower of itself we can assume that f is in the Torelli group. By Proposition2.25, for each such vertex we can find a cover where it is stable. By performingthis process for every vertex, we find a stable cover of Γ. Note that we haveonly taken characteristic abelian covers at each step, so this cover is solvable.Now apply Propositions 2.18 and 2.5 to get the result.

�

2.7. Proof of Theorem 1.3. Suppose first that φ ∈ Mod(Σ) is a mutitwistabout the muticurve C. Add curves to C until it forms a pants decompositionP of Σ. Assume first that there are two pants P1, P2 ∈ P such that S = P1∪P2

is a four holed sphere in Σ, whose interior contains the curve γ ∈ C. Pick apoint p ∈ S, and let Γ be the image of π1(S, p) in π1(Σ, p) under the inclusionhomomorphism.

By a theorem of Marshall Hall ([11]), we can find a finite index subgroupK ⊂ π1(Σ, p) such that Γ < K, and Γ is a free factor in K. There is a powerφk of φ that lifts to γ, and whose restriction to a lifts of S is a Dehn twistabout a lift of the curve γ. Such a Dehn twist has an infinite order action onthe homology of S, and hence on the homology of H1(K,Z) (since Γ is a freefactor in K).

EVERY INFINITE ORDER MAPPING CLASS HAS AN INFINITE ORDER ACTION 23

Pick a finite, characteristic cover K ′ < K. We have an equivariant transfermap T : H1(K,Z) → H1(K ′,Z). The map φ lifts to the cover correspondingto K. By equivariance, some power of φ acts with infinite order on H1(K ′,Z),and hence so does φ.

Now, we return to the case where we cannot find S as required. In this case,we can find P1, P2 ∈ P such that P1 ∪ P2 is a either a surface of genus 2, or atwo holed torus. Say S is a two holed torus (the case of a genus two surfaceis nearly identical). The interior of this torus contains two disjoint simpleclosed curves γ1, γ2 that were used in the pants decomposition P . Assumethat γ1 ∈ C.

Once again, let Γ be the image of π1(S, p) in π1(Σ, p) under the inclusionhomomorphism, and let K ⊂ π1(Σ, p) be such that Γ < K, and Γ is a freefactor in K. We have a homeomorphic lift of S in the cover corresponding toK, together with lifts of γ1, γ2. Pick a curve δ in this lift intersects γ2 once, anddoes not intersect γ1. We have a surjection K → Z/2Z sending the curve δ to1. Let K1 be the kernel of this surjection. Lift the curves forming the pantsdecomposition P to a multicurve in the cover corresponding to K1, and extendthe resulting multicurve to a pants decomposition. This pants decompositioncan be taken to have two adjacent pants that form a four holed sphere, asabove.

This proves the Theorem for a multitwist. By the Nielsen-Thurston classi-fication of mapping class groups, it remains to prove the result for a mappingclass φ such that φk restricts to a pseudo-Anosov diffeomorphism of a subsur-face S ⊂ Σ and some k. Let F be the image of π1(S) in π1(Σ). By pickinga base point p ∈ S, φ induces an element of Aut(π1(Σ)) that fixes F . By[11], there exists a finite index K ≤ π1(Σ) such that F ≤ K and F is a freesub-factor of K. By passing to a power of φ we may assume that it lifts to thiscover. Thus, we may assume that φ fixes a subsurface of Σ whose fundamentalgroup is a free sub-factor. In this case the image of H1(S,Z) ⊂ H1(Σ,Z) is adirect summand.

Let ϕ ∈ Out(F ) be the element induced by φ. To prove Theorem 1.3, it isenough to prove that there is a representative f of ϕ in Aut(F ) and a finiteindex f -invariant subgroup L ≤ F such that f∗ ∈ GL(H1(L,R)) has infiniteorder. To prove this, we proceed in a similar fashion to the proof of Theorem1.4.

While the mapping class φ is a pseudo-Anosov mapping class on S, it maynot be one on S ⊂ Σ. For instance, S may have two different boundarycomponents that are identified in S, producing a simple curve fixed by φ.Let C be the multi-curve ∂S ⊂ S. By passing to a power of φ and possiblyrestricting to a subsurface, we can assume that C is a a canonical reductionsystem for φ in S (that is, the curves in C are fixed by φ, and no other curve in S

24 ASAF HADARI

has finite φ orbit). If the map φ induces an infinite order element of H1(S,R),we are done. Otherwise, by replacing it with a power we may assume that itacts trivially on H1(S,R)

In our proof of Theorem 2.1, we used the fact that the mapping class inquestion was pseudo-Anosov in one place only - to show that the convex hullof the vertices of Sϕ is a full dimensional convex polytope. We use a similaridea in this proof.

Let Mφ be the mapping torus of φ over S. That is, Mφ = S × I/ ∼ where(x, 0) ∼ (φ(x), 1). The manifold Mφ is a 3-manifold with toroidal boundarycomponents. This manifold comes equipped with a flow F given by projecting

the flow Fs(x, t) = (x, t+ s) on S×R. Let Nφ be the mapping torus of φ overS, and let F ′ be the flow on this manifold.

Given x ∈ S, and t ∈ N we can build a curve in Mφ in the following way.Flow t upwards from (x, 0). Close the resulting ray to a curve by adding theshortest possible path in S × 0. Call the resulting curve γt. Let [γt] be itsimage in H1(Mφ,R). The cone on the set of all accumulation points of 1

t[γt]

as t → ∞ is called the cone of homological directions. Denote this cone byH. Similarly, let H′ be the cone of homological directions in Nφ. In [7], Frieddefined the cones of homological directions, and describes how to calculatethem from the first return map of the flow to a cross section.

In our case, by picking a graph Γ ⊂ S that carries π1(S) and flowing upwardsfrom every point in the graph, we see that the cone H′ is a cone with crosssection Sf . Let i : S → S be the inclusion map. The map i induces asurjective map i∗ : H1(Nφ,R) → H1(Mφ,R). Since any upwards flow path inMφ corresponds to an upward flow path in Nφ, we get that H = i∗H′. Thus,H is an open cone in H1(Mφ,R).

The automorphism f ∈ Aut(π1(S)) induces an automorphism g ∈ Aut(S).Let ψ be a map from a wedge of circles to itself inducing g. Let T be atransition graph for ψ. The cone H is a cone whose cross section is the closureof the set of all tn(γ), where γ is a loop in T .

As discussed above, this cross section is a dimH1(S,R) dimensional convexpolytope. As before, for each of its vertices we can find an extremal subgraphof T , and the proof now proceeds exactly as the proof of Theorem 2.1.

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