An Introduction to Out(Fn

(1)

An Introduction to Out(F

_n

) Part I (Relative) Train Track Maps

Michael Handel

Lehman College

Sao Paolo, April 2014

(2)

Disclaimer

Emphasize aspects that are most accessible to a dynamics audience.

Michael Handel An Introduction to Out(Fn)PartI(Relative) Train Track Maps

(3)

Fn

Finite alphabet{A,B, . . .}

A¯ =A⁻¹, B¯ =B⁻¹, . . .

F_n= reduced finite words in{A,A,¯ B,B, . . .}¯

(4)

IdentifyF_nwithπ1(R_n,∗).

Each element is represented by a closedpathbased at∗.

(5)

AutomorphismsΦ :F_n→F_nare represented by homotopy equivalences ofR_nthat fix the basepoint.

Example:Φ :F₂→F₂

A7→ABA B7→ABABA represented byf :R₂→R₂with the same description.

Aut(F_n)↔

homotopy equivalence ofRnpreserving∗/ homotopy rel∗

(6)

Out(Fn) =Aut(Fn)/Inn(Fn)

Out(F_n)- forget∗

Out(F_n)↔homotopy equivalence ofR_n/ homotopy

This is too restrictive.

(7)

Example:Φ :F₂→F₂

A7→B¯ B7→AB¯ has period 3

A7→B¯ 7→BA¯ 7→ABB¯ =A

B7→AB¯ 7→BB¯ A¯ = ¯A7→B butf :R₂→R₂is not periodic.

(8)

Amarked graphis a core graph equipped with a homotopy equivalenceρ:Rn→G

ρ A7→XY¯ B7→ZY¯

(9)

The markingρidentifiesπ₁(G)withπ₁(R) =F_n(up to inner automorphism) and so

Out(Fn)↔homotopy equivalence ofG/ homotopy

Continuing Example: Φ : A7→B¯ B7→AB¯ f :G→Gdefined by

X 7→Y 7→Z 7→X representsφ.

(10)

Automorphisms act on elements and subgroups ofFn.

Outer automorphisms act on conjugacy classes of elements and subgroups ofF_n.

f :G→Grepresentsφ∈Out(F_n)if it induces the same action on conjugacy classes of elements.

(11)

Other spaces withπ1(X) =F_n.

Example: A surfaceSwith connected non-empty∂S MCG(S) =Homeo(S)/ isotopy = Homeo(S)/ homotopy

Homeo(S)→Out(π₁(S)) =Out(F_n) induces

MCG(S),→Out(Fn)

(12)

Thurston classification theorem Assumeχ(S)<0 and not a pair of pants.

Definition 1

µ∈MCG(S)isreducibleif it preserves a non-emptymulticurve C.

In that case eachµ|S_i is a well defined element ofMCG(S_i)

(13)

Theorem 1 (Thurston)

Ifµis irreducible thenµis represented by apseudo-Anosov homemorphism.

The reducible case is handled by recursion.

There is a canonicalµ-invariant multicurveSwith the minimal number of complementary components such that eachµ|S_i is either trivial or irreducible.

(14)

What is the analogue for Out(F_n)?

What properties off :G→Gdo we want?

We are often interested in the action ofφon conjugacy classes [a]ofFn.

conjugacy classes[a] ↔circuitsσ ⊂G

f(σ)can be tightenedto a circuitf_#(σ)and similarly for a path with endpoints at vertices.

We want to minimize tightening.

(15)

f :G→Gis atrain track mapiff^k restricts to an immersion on each edgeE ofGfor allk ≥1.

In other words, no tightening is necessary when iterating edges.

Example 2

Positive Automorphism

A7→ABA B7→ABABA

(16)

Directionsat∗

A B A¯ B¯

A turnis an unordered pair of directions with a common basepoint.

The edge pathAABA¯ . . .takes the turns(¯A,A),(¯A,B),¯ (B,A)¯ . . .

(17)

Nondegenerate turns

(A,B) (A,A)¯ (A,B)¯ (B,A)¯ (B,B)¯ (¯A,B)¯ Degenerate turns

(A,A) (B,B) (¯A,A)¯ ( ¯B,B)¯ An edge path is immersed if and only if it that makes only non-degenerate turns.

(18)

Assuming thatf is an immersion on each edge there is an induced mapDf on directions and turns

A7→A B7→A A¯ 7→A¯ B¯ 7→A¯

Aturn is illegalif it is mapped by some iterate to a degenerate path.

Illegal turns

(A,B)7→(A,A) (¯A,B)¯ 7→(¯A,A)¯ Note: Legal turns are mapped to legal turns.

(19)

Apath is legalif it takes only legal turns.

(20)

The following are equivalent.

1 f(E)is a legal path for each edgeE.

2 f maps legal paths to legal paths.

3 f^k|E is an immersion for eachk ≥1 and each edgeE.

(21)

To eachf :G→Gthere is atransition matrixM(f).

Example 3

A7→ABA B7→ABABA

M(f) =

2 3

1 2

Iff :G→Gis a train track map then(M(f))^k =M(f^k)

(22)

There is also atransition graphΓ(f) Example 4

A7→BA B 7→A C 7→DBC D7→C E 7→EA G₁= (A,B) G₂= (A,B,C,D) G₃=G

(23)

I am going to assume that each vertex inΓ(f)is contained in at least one cycle.

Γ(f)determines afiltration

∅=G₀⊂G₁⊂. . .⊂G_N =G

by invariant subgraphs built up by starting with an invariant subgraph and then addingstrata

H_i =G_i −Gi−1

(24)

Strata correspond to equivalence classes of vertices inΓ(f).

(25)

There are two types of strata forrotationlessφ AnNEG stratumH_i has one edgeE_i and

f(E_i) =u_iE_iv_i for pathsu_i,v_i ⊂G_i−1

AnEG stratumH_r has multiple edges and its transition

submatrix has a positive iterate (and so a PF eigenvalue>1).