Individual Elements of Out(Fn)
1 What are the possible growth rates for the action ofφon conjugacy classes[a]?
2 Suppose thatΦis an automorphism. Is the fixed subgroup Fix(Φ) ={a∈Fn: Φ(a) =a}finitely generated? What can its rank be?
3 How can one tell ifφis geometric? (Realized by a pseudo-Anosov homeomorphisms of a surface with one boundary component?)
4 What is the correct notion of irreducible?
5 What properites should anf :G→Grepresenting an irreducibleφhave?
6 What about the reducible case? Does it follow from the irreducible case?
Reducibility
A subgroupAofFnis afree factorif there exists a subgroupB such thatFn =A∗B.
Equivalently,Ais realized by a subgraph of a marked graph.
φ∈Out(Fn)is reducibleif it preserves (the conjugacy class[A]
of) a free factorA
Equivalently,φis represented byf :G→Gin whichf preserves a proper subgraph.
In that case eachφ|[A]is a well defined element of Out(A)
Bad news: There need not be an invariant complementary free factor.
Theorem 1 (BH)
Each irreducibleφ∈Out(Fn)is represented by an (irreducible) train track map.
Proof (Original) : Minimize the entropy.
Iff :G→Gis not a train track map then there is a procedure to find a newf :G→Gwith smaller PF eigenvalues. This stops after a finite number of iterations.
Proof (Updated [B]) : Minimize the Lipschitz constant for f :G→G.
Iteration of Conjugacy Classes
Motivate Train Track Property
Suppose thatf :G→Gis a train track map representingφand thatσ a circuit corresponding to[a].
Ifσ is legal then[a]grows exponentially with rateλ
Otherwise
σ=σ1σ2. . . σp
whereσi is legal and the indicated turns are illegal.
Can assume that the number of illegal turns inf#k(σ)is independent ofk.
The lengths of the subpaths inf(σi)that are tightened away is uniformly (independent ofσ) bounded.
Iterate to formf#k(σ)
The lengths of the subpaths inσ¯i andσi that are identified is uniformly (independent ofσandk) bounded.
P ={ρ:eachf#k(ρ)has exactly one illegal turn and uniformly bounded length}
P is a finitef#-invariant set Lemma 2
For everyσ there exists K such that f#k(σ)has a splitting into legal subpaths and periodic elements ofP for all k ≥K .
Corollary 3
Each[a]is eitherφ-periodic or grows exponentially (with growth rateλ).
Proposition 1
Each irreducibleφis represented by an irreducible train track map f :G→G such thatP has at most one periodic elementρ.
If there is such aρand if it closed then it crosses every edge of G exactly twice.
Corollary 4
IfΦrepresentsφthenFix(Φ)has rank at most one.
Corollary 5
An irreducibleφis geometric if and only if it preserves a (necessarily unique) conjugacy class
Some Theorems
Theorem 6 (BH)
(Scott Conjecture) The rank ofFix(Φ))is≤n for all Φ∈Aut(Fn).
Example 7
Φ : A7→A B7→BA C 7→CA2 Fix(Φ) =hA,BAB,¯ CACi¯
Theorem 8 (BH)
For eachφ∈Out(Fn)and conjugacy class[a]the length of φk([a])either grows polynomially of degree≤n−1or exponentially.
Subgroups of Out(Fn)
1 Does Out(Fn)satisfy the Tits Alternative? (Every finitely generated subgroup is either virtually abelian or contains a free group of rank≥2.)?
2 For whichφ, ψ∈Out(Fn)does there existN such that hφN, ψNiis free? Can one chooseNindependently ofφ, ψ?
3 What do abelian subgroups look like?
4 Is it true that every finitely generated subgroup of Out(Fn) is either virtually abelian or has infinitely generatedHb2?
5 Does every finitely generated irreducible subgroup contain an irreducible element?
Definition 9
A subgroupHof Out(Fn)isirreducibleif there is no free factor whose conjugacy class isH-invariant.
Theorem 10 (HM)
[Absolute version] IfH<Out(Fn))is finitely generated and irreducible thenHcontains an irreducible element.
