Dynamics on free-by-cyclic groups. Chris Leininger (UIUC) joint - - PowerPoint PPT Presentation

Dynamics on free-by-cyclic groups.

Chris Leininger (UIUC) joint with S. Dowdall and I. Kapovich August 15, 2013

Outline 1/17

FN a free group of rank N, φ ∈ Out(FN)

Outline 1/17

FN a free group of rank N, φ ∈ Out(FN) ⇒ G = Gφ = FN ⋊φ Z

Outline 1/17

FN a free group of rank N, φ ∈ Out(FN) ⇒ G = Gφ = FN ⋊φ Z

u0

− → Z

Outline 1/17

FN a free group of rank N, φ ∈ Out(FN) ⇒ G = Gφ = FN ⋊φ Z

u0

− → Z ⇒ for u “close” to u0 in PH1(G; R) = PHom(G, R),

Outline 1/17

FN a free group of rank N, φ ∈ Out(FN) ⇒ G = Gφ = FN ⋊φ Z

u0

− → Z ⇒ for u “close” to u0 in PH1(G; R) = PHom(G, R), ker(u) ∼ = FN(u), N(u) ∈ Z+ and G = ker(u) ⋊φu Z

Outline 1/17

FN a free group of rank N, φ ∈ Out(FN) ⇒ G = Gφ = FN ⋊φ Z

u0

− → Z ⇒ for u “close” to u0 in PH1(G; R) = PHom(G, R), ker(u) ∼ = FN(u), N(u) ∈ Z+ and G = ker(u) ⋊φu Z

[Neumann,Geoghegan-Mihalik-Sapir-Wise]

Outline 1/17

FN a free group of rank N, φ ∈ Out(FN) ⇒ G = Gφ = FN ⋊φ Z

u0

− → Z ⇒ for u “close” to u0 in PH1(G; R) = PHom(G, R), ker(u) ∼ = FN(u), N(u) ∈ Z+ and G = ker(u) ⋊φu Z

[Neumann,Geoghegan-Mihalik-Sapir-Wise]

Goal:

Outline 1/17

FN a free group of rank N, φ ∈ Out(FN) ⇒ G = Gφ = FN ⋊φ Z

u0

− → Z ⇒ for u “close” to u0 in PH1(G; R) = PHom(G, R), ker(u) ∼ = FN(u), N(u) ∈ Z+ and G = ker(u) ⋊φu Z

[Neumann,Geoghegan-Mihalik-Sapir-Wise]

Goal: Describe geometric, topological, and dynamical relationships between φu and φ

Outline 1/17

FN a free group of rank N, φ ∈ Out(FN) ⇒ G = Gφ = FN ⋊φ Z

u0

− → Z ⇒ for u “close” to u0 in PH1(G; R) = PHom(G, R), ker(u) ∼ = FN(u), N(u) ∈ Z+ and G = ker(u) ⋊φu Z

[Neumann,Geoghegan-Mihalik-Sapir-Wise]

Goal: Describe geometric, topological, and dynamical relationships between φu and φ Motivation from fibered hyperbolic 3–manifolds.

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2: F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

• F stretches/contracts the measures

F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

• F stretches/contracts the measures

F F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

• F stretches/contracts the measures
• λ

λ−1

• F

F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

• F stretches/contracts the measures
• λ = λ(F) = lim

n→∞

n

• length(F n(α))
• λ

λ−1

• F

F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

• F stretches/contracts the measures
• λ = λ(F) = lim

n→∞

n

• length(F n(α))

independent of α and metric

• λ

λ−1

• F

F

Motivation: Pseudo-Anosov homeomorphisms 2/17

F : S → S pseudo-Anosov on S, a closed surface of genus g ≥ 2:

• ∃ invariant, transverse measured foliations F±

S on S

• F stretches/contracts the measures
• λ = λ(F) = lim

n→∞

n

• length(F n(α))
• λ

λ−1

• F

= dilatation of F F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

M F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] M F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration M F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral M F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral... u0 =PD[S] M F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral... u0 =PD[S]

• Suspension flow ψs : M → M, 1st return = F : S → S

M F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral... u0 =PD[S]

• Suspension flow ψs : M → M, 1st return = F : S → S

M F ψ

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral... u0 =PD[S]

• Suspension flow ψs : M → M, 1st return = F : S → S
• F foliation by fibers ⇒ e = e(TF) ∈ H2(M) Euler class

M F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral... u0 =PD[S]

• Suspension flow ψs : M → M, 1st return = F : S → S
• F foliation by fibers ⇒ e = e(TF) ∈ H2(M) Euler class

M F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral... u0 =PD[S]

• Suspension flow ψs : M → M, 1st return = F : S → S
• F foliation by fibers ⇒ e = e(TF) ∈ H2(M) Euler class
• O = {γ ⊂ M | γ closed orbit of singularity of F±

S }

M F

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral... u0 =PD[S]

• Suspension flow ψs : M → M, 1st return = F : S → S
• F foliation by fibers ⇒ e = e(TF) ∈ H2(M) Euler class
• O = {γ ⊂ M | γ closed orbit of singularity of F±

S }

M F

b a a b ← γ →

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral... u0 =PD[S]

• Suspension flow ψs : M → M, 1st return = F : S → S
• F foliation by fibers ⇒ e = e(TF) ∈ H2(M) Euler class
• O = {γ ⊂ M | γ closed orbit of singularity of F±

S }

⇒ PD(e) = 1 2

• γ∈O

(2 − deg(γ))γ ∈ H1(M) M F

b a a b ← γ →

Motivation: The mapping torus 3/17

• M = MF = S × [0, 1]/(x, 1) ∼ (F(x), 0)

∼ = H3/Γ [Thurston] ⇒ S − → M

η0

− → S1 fibration

• u0 = (η0)∗ ∈ Hom(π1M, R) = H1(M)

integral... u0 =PD[S]

• Suspension flow ψs : M → M, 1st return = F : S → S
• F foliation by fibers ⇒ e = e(TF) ∈ H2(M) Euler class
• O = {γ ⊂ M | γ closed orbit of singularity of F±

S }

⇒ PD(e) = 1 2

• γ∈O

(2 − deg(γ))γ ∈ H1(M) M F

b a a b ← γ → deg(γ) = 3

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration

H1(M)

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M)

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral.

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

Theorem [Thurston,Fried]

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

Theorem [Thurston,Fried] For all integral u ∈ C

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

Theorem [Thurston,Fried] For all integral u ∈ C

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

Theorem [Thurston,Fried] For all integral u ∈ C ⇒ ∃ fibration Su − → M

ηu

− → S1 with u = (ηu)∗ =PD[Su] s.t.

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

Theorem [Thurston,Fried] For all integral u ∈ C ⇒ ∃ fibration Su − → M

ηu

− → S1 with u = (ηu)∗ =PD[Su] s.t.

• e, u = χ(Su) = −||u||T and

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

Theorem [Thurston,Fried] For all integral u ∈ C ⇒ ∃ fibration Su − → M

ηu

− → S1 with u = (ηu)∗ =PD[Su] s.t.

• e, u = χ(Su) = −||u||T and
• ψ ⋔ Su and first return Fu : Su → Su is pseudo-Anosov.

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

Theorem [Thurston,Fried] For all integral u ∈ C ⇒ ∃ fibration Su − → M

ηu

− → S1 with u = (ηu)∗ =PD[Su] s.t.

• e, u = χ(Su) = −||u||T and
• ψ ⋔ Su and first return Fu : Su → Su is pseudo-Anosov.

∃! H: C → R continuous, convex, homogeneous of degree −1 such that for all integral u ∈ C

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

Theorem [Thurston,Fried] For all integral u ∈ C ⇒ ∃ fibration Su − → M

ηu

− → S1 with u = (ηu)∗ =PD[Su] s.t.

• e, u = χ(Su) = −||u||T and
• ψ ⋔ Su and first return Fu : Su → Su is pseudo-Anosov.

∃! H: C → R continuous, convex, homogeneous of degree −1 such that for all integral u ∈ C

• log(λ(Fu)) = H(u)

H1(M)

u0

Motivation: Thurston and Fried 4/17

S − → M = MF

η0

− → S1 fibration u0 = (η0)∗ =PD[S] ∈ H1(M) integral. u0 ∈ C ⊂ H1(M), an (open) cone

• n a fibered face of || · ||T–ball,

Theorem [Thurston,Fried] For all integral u ∈ C ⇒ ∃ fibration Su − → M

ηu

− → S1 with u = (ηu)∗ =PD[Su] s.t.

• e, u = χ(Su) = −||u||T and
• ψ ⋔ Su and first return Fu : Su → Su is pseudo-Anosov.

∃! H: C → R continuous, convex, homogeneous of degree −1 such that for all integral u ∈ C

• log(λ(Fu)) = H(u) see also [Oertel, Long-Oertel, Matsumoto, McMullen]

H1(M)

u0

Motivation: Dilatation asymptotics 5/17

Corollary

H1(M)

Motivation: Dilatation asymptotics 5/17

Corollary Suppose K ⊂ C is compact and {un}∞

n=1 ⊂ R+K

all un primitive integral, un → ∞.

H1(M)

Motivation: Dilatation asymptotics 5/17

Corollary Suppose K ⊂ C is compact and {un}∞

n=1 ⊂ R+K

all un primitive integral, un → ∞.

H1(M)

Motivation: Dilatation asymptotics 5/17

Corollary Suppose K ⊂ C is compact and {un}∞

n=1 ⊂ R+K

all un primitive integral, un → ∞. Then gn = genus(Sun) → ∞ and c0 gn ≤ log(λ(Fun)) ≤ c1 gn for some 0 < c0 < c1 < ∞.

H1(M)

Motivation: Dilatation asymptotics 5/17

Corollary Suppose K ⊂ C is compact and {un}∞

n=1 ⊂ R+K

all un primitive integral, un → ∞. Then gn = genus(Sun) → ∞ and c0 gn ≤ log(λ(Fun)) ≤ c1 gn for some 0 < c0 < c1 < ∞.[Penner,McMullen]

H1(M)

Motivation: Dilatation asymptotics 5/17

Corollary Suppose K ⊂ C is compact and {un}∞

n=1 ⊂ R+K

all un primitive integral, un → ∞. Then gn = genus(Sun) → ∞ and c0 gn ≤ log(λ(Fun)) ≤ c1 gn for some 0 < c0 < c1 < ∞.[Penner,McMullen]

Theorem [Farb-L-Margalit] All pseudo-Anosov F : Sg → Sg with log(λ(F)) ≤ c/g are monodromies of fibrations of one of a finite list of fibered, finite volume hyperbolic 3–manifolds, Dehn filled along the boundary of the fiber. H1(M)

Motivation: Dilatation asymptotics 5/17

Corollary Suppose K ⊂ C is compact and {un}∞

n=1 ⊂ R+K

all un primitive integral, un → ∞. Then gn = genus(Sun) → ∞ and c0 gn ≤ log(λ(Fun)) ≤ c1 gn for some 0 < c0 < c1 < ∞.[Penner,McMullen]

Theorem [Farb-L-Margalit] All pseudo-Anosov F : Sg → Sg with log(λ(F)) ≤ c/g are monodromies of fibrations of one of a finite list of fibered, finite volume hyperbolic 3–manifolds, Dehn filled along the boundary of the fiber. See also [Agol]. H1(M)

Transition: Group theory 6/17

π1M = π1S ⋊F∗ Z.

Transition: Group theory 6/17

π1M = π1S ⋊F∗ Z. In fact, M is determined up to homeomorphism by F∗ ∈ Out(π1(S)) [Dehn-Nielsen-Baer].

Transition: Group theory 6/17

π1M = π1S ⋊F∗ Z. In fact, M is determined up to homeomorphism by F∗ ∈ Out(π1(S)) [Dehn-Nielsen-Baer]. φ ∈ Out(π1(S)) is represented by a pseudo-Anosov F if and only if φ has no nontrivial periodic conjugacy classes

Transition: Group theory 6/17

π1M = π1S ⋊F∗ Z. In fact, M is determined up to homeomorphism by F∗ ∈ Out(π1(S)) [Dehn-Nielsen-Baer]. φ ∈ Out(π1(S)) is represented by a pseudo-Anosov F if and only if φ has no nontrivial periodic conjugacy classes if and only if π1S ⋊φ Z is word-hyperbolic. [Thurston]

Transition: Group theory 6/17

π1M = π1S ⋊F∗ Z. In fact, M is determined up to homeomorphism by F∗ ∈ Out(π1(S)) [Dehn-Nielsen-Baer]. φ ∈ Out(π1(S)) is represented by a pseudo-Anosov F if and only if φ has no nontrivial periodic conjugacy classes if and only if π1S ⋊φ Z is word-hyperbolic. [Thurston] λ(F) = growth rate of word length in π1S under iteration of F∗.

Transition: Group theory 6/17

π1M = π1S ⋊F∗ Z. In fact, M is determined up to homeomorphism by F∗ ∈ Out(π1(S)) [Dehn-Nielsen-Baer]. φ ∈ Out(π1(S)) is represented by a pseudo-Anosov F if and only if φ has no nontrivial periodic conjugacy classes if and only if π1S ⋊φ Z is word-hyperbolic. [Thurston] λ(F) = growth rate of word length in π1S under iteration of F∗. Integral u ∈ Hom(π1M, R) = H1(M) is induced by a fibration over S1 if and only if ker(u) is finitely generated [Stallings]

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel]

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and
• φ is represented by an irreducible train track map.

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and
• φ is represented by an irreducible train track map.

Example:

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and
• φ is represented by an irreducible train track map.

Example:

Γ

a b c d id f d a b a b a d b a c

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and
• φ is represented by an irreducible train track map.

Example:

Γ

a b c d id f d a b a b a d b a c

• A graph Γ, π1Γ ∼

= FN,

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and
• φ is represented by an irreducible train track map.

Example:

Γ

a b c d id f d a b a b a d b a c

• A graph Γ, π1Γ ∼

= FN,

• f : Γ → Γ a h.e. and f∗ = φ

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and
• φ is represented by an irreducible train track map.

Example:

Γ

a b c d id f d a b a b a d b a c

• A graph Γ, π1Γ ∼

= FN,

• f : Γ → Γ a h.e. and f∗ = φ
• f (V Γ) ⊂ V Γ

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and
• φ is represented by an irreducible train track map.

Example:

Γ

a b c d id f d a b a b a d b a c

• A graph Γ, π1Γ ∼

= FN,

• f : Γ → Γ a h.e. and f∗ = φ
• f (V Γ) ⊂ V Γ
• f n|e is an immersion for all

n ≥ 1 and for all edges e

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and
• φ is represented by an irreducible train track map.

Example:

Γ

a b c d id f d a b a b a d b a c

• A graph Γ, π1Γ ∼

= FN,

• f : Γ → Γ a h.e. and f∗ = φ
• f (V Γ) ⊂ V Γ
• f n|e is an immersion for all

n ≥ 1 and for all edges e

• irreducible transition matrix...

Atoroidal and fully irreducible 7/17

Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out(FN) be

• Atoroidal: no nontrivial periodic conjugacy classes, and
• Fully irreducible: no nontrivial periodic free factors.

Then

• G = Gφ = FN ⋊φ Z is word-hyperbolic, and
• φ is represented by an irreducible train track map.

Example:

Γ

a b c d id f d a b a b a d b a c

• A graph Γ, π1Γ ∼

= FN,

• f : Γ → Γ a h.e. and f∗ = φ
• f (V Γ) ⊂ V Γ
• f n|e is an immersion for all

n ≥ 1 and for all edges e

• irreducible transition matrix...

Many other examples [Clay-Pettet]

Dynamics and stretch factors 8/17

Γ

a b c d id f d a b a b a d b a c

Dynamics and stretch factors 8/17

Γ

a b c d id f d a b a b a d b a c

Transition matrix A(f ) =     1 1 1 1 2 2 1 1     ,

Dynamics and stretch factors 8/17

Γ

a b c d id f d a b a b a d b a c

Transition matrix and Perron-Frobenius eigenvalue/eigenvector A(f ) =     1 1 1 1 2 2 1 1     , λ ≈ 2.4142, v ≈     .2265 .0939 .1327 .5469    

Dynamics and stretch factors 8/17

Γ

a b c d id f d a b a b a d b a c

Transition matrix and Perron-Frobenius eigenvalue/eigenvector A(f ) =     1 1 1 1 2 2 1 1     , λ ≈ 2.4142, v ≈     .2265 .0939 .1327 .5469     metric graph (Γ, dv), f ≃ fv : (Γ, dv) → (Γ, dv), affine-stretch by λ on all edges.

Dynamics and stretch factors 8/17

Γ

a b c d id f d a b a b a d b a c

Transition matrix and Perron-Frobenius eigenvalue/eigenvector A(f ) =     1 1 1 1 2 2 1 1     , λ ≈ 2.4142, v ≈     .2265 .0939 .1327 .5469     metric graph (Γ, dv), f ≃ fv : (Γ, dv) → (Γ, dv), affine-stretch by λ on all edges. λ = λ(f ) = λ(φ) = lim

n→∞

n

• length(f n(α)) = stretch factor.

Dynamics and stretch factors 8/17

Γ

a b c d id f d a b a b a d b a c

Transition matrix and Perron-Frobenius eigenvalue/eigenvector A(f ) =     1 1 1 1 2 2 1 1     , λ ≈ 2.4142, v ≈     .2265 .0939 .1327 .5469     metric graph (Γ, dv), f ≃ fv : (Γ, dv) → (Γ, dv), affine-stretch by λ on all edges. λ = λ(f ) = λ(φ) = lim

n→∞

n

• length(f n(α)) = stretch factor.

depends only on φ = f∗, not on f , α, or metric.

A model for free-by-cyclic group 9/17

Idea: Dynamics on branched surfaces in 3–manifolds

A model for free-by-cyclic group 9/17

Idea: Dynamics on branched surfaces in 3–manifolds [Williams, Christy,...,Benedetti-Petronio,...,Brinkmann-Schleimer,..]

A model for free-by-cyclic group 9/17

Idea: Dynamics on branched surfaces in 3–manifolds [Williams, Christy,...,Benedetti-Petronio,...,Brinkmann-Schleimer,..] Generalizations: Outside 3–manifolds [Gautero,Wang,...]

A model for free-by-cyclic group 9/17

Idea: Dynamics on branched surfaces in 3–manifolds [Williams, Christy,...,Benedetti-Petronio,...,Brinkmann-Schleimer,..] Generalizations: Outside 3–manifolds [Gautero,Wang,...] φ ∈ Out(FN)

A model for free-by-cyclic group 9/17

Idea: Dynamics on branched surfaces in 3–manifolds [Williams, Christy,...,Benedetti-Petronio,...,Brinkmann-Schleimer,..] Generalizations: Outside 3–manifolds [Gautero,Wang,...] φ ∈ Out(FN)

• (Xφ, ψ, A)

A model for free-by-cyclic group 9/17

Idea: Dynamics on branched surfaces in 3–manifolds [Williams, Christy,...,Benedetti-Petronio,...,Brinkmann-Schleimer,..] Generalizations: Outside 3–manifolds [Gautero,Wang,...] φ ∈ Out(FN)

• (Xφ, ψ, A)
• Xφ is a polyhedral 2–complex, K(G, 1) for G = FN ⋊φ Z.

A model for free-by-cyclic group 9/17

Idea: Dynamics on branched surfaces in 3–manifolds [Williams, Christy,...,Benedetti-Petronio,...,Brinkmann-Schleimer,..] Generalizations: Outside 3–manifolds [Gautero,Wang,...] φ ∈ Out(FN)

• (Xφ, ψ, A)
• Xφ is a polyhedral 2–complex, K(G, 1) for G = FN ⋊φ Z.
• ψ is a semi-flow on Xφ.

A model for free-by-cyclic group 9/17

Idea: Dynamics on branched surfaces in 3–manifolds [Williams, Christy,...,Benedetti-Petronio,...,Brinkmann-Schleimer,..] Generalizations: Outside 3–manifolds [Gautero,Wang,...] φ ∈ Out(FN)

• (Xφ, ψ, A)
• Xφ is a polyhedral 2–complex, K(G, 1) for G = FN ⋊φ Z.
• ψ is a semi-flow on Xφ.
• A = {[z] ∈ H1(Xφ) | z ∈ Z 1(Xφ) positive, cellular }, open cone.

A model for free-by-cyclic group 9/17

Idea: Dynamics on branched surfaces in 3–manifolds [Williams, Christy,...,Benedetti-Petronio,...,Brinkmann-Schleimer,..] Generalizations: Outside 3–manifolds [Gautero,Wang,...] φ ∈ Out(FN)

• (Xφ, ψ, A)
• Xφ is a polyhedral 2–complex, K(G, 1) for G = FN ⋊φ Z.
• ψ is a semi-flow on Xφ.
• A = {[z] ∈ H1(Xφ) | z ∈ Z 1(Xφ) positive, cellular }, open cone.
• u0 ∈ Hom(G, R) = H1(Xφ), u0(x, n) = n ⇒ u0 ∈ A.

“Fibrations”, sections, and “Euler class” 10/17

• Theorem. Fix φ ∈ Out(FN) let (Xφ, ψ, A) be as above. Then for

all u ∈ A primitive integral there exists ηu : Xφ → S1 with (ηu)∗ = u satisfying:

“Fibrations”, sections, and “Euler class” 10/17

• Theorem. Fix φ ∈ Out(FN) let (Xφ, ψ, A) be as above. Then for

all u ∈ A primitive integral there exists ηu : Xφ → S1 with (ηu)∗ = u satisfying: (1.) Γu = η−1

u (∗) ⊂ Xφ is a graph for any ∗ ∈ S1;

“Fibrations”, sections, and “Euler class” 10/17

• Theorem. Fix φ ∈ Out(FN) let (Xφ, ψ, A) be as above. Then for

all u ∈ A primitive integral there exists ηu : Xφ → S1 with (ηu)∗ = u satisfying: (1.) Γu = η−1

u (∗) ⊂ Xφ is a graph for any ∗ ∈ S1;

(2.) Γu ֒ → Xφ induces an isomorphism π1(Γu) ∼ = ker(u);

“Fibrations”, sections, and “Euler class” 10/17

• Theorem. Fix φ ∈ Out(FN) let (Xφ, ψ, A) be as above. Then for

all u ∈ A primitive integral there exists ηu : Xφ → S1 with (ηu)∗ = u satisfying: (1.) Γu = η−1

u (∗) ⊂ Xφ is a graph for any ∗ ∈ S1;

(2.) Γu ֒ → Xφ induces an isomorphism π1(Γu) ∼ = ker(u); (3.) Γu ⋔ ψ, 1st return fu : Γu → Γu has (fu)∗ = φu ∈ Out(ker(u));

“Fibrations”, sections, and “Euler class” 10/17

• Theorem. Fix φ ∈ Out(FN) let (Xφ, ψ, A) be as above. Then for

all u ∈ A primitive integral there exists ηu : Xφ → S1 with (ηu)∗ = u satisfying: (1.) Γu = η−1

u (∗) ⊂ Xφ is a graph for any ∗ ∈ S1;

(2.) Γu ֒ → Xφ induces an isomorphism π1(Γu) ∼ = ker(u); (3.) Γu ⋔ ψ, 1st return fu : Γu → Γu has (fu)∗ = φu ∈ Out(ker(u)); (4.) χ(Γu) = ǫ, u, where

“Fibrations”, sections, and “Euler class” 10/17

• Theorem. Fix φ ∈ Out(FN) let (Xφ, ψ, A) be as above. Then for

all u ∈ A primitive integral there exists ηu : Xφ → S1 with (ηu)∗ = u satisfying: (1.) Γu = η−1

u (∗) ⊂ Xφ is a graph for any ∗ ∈ S1;

(2.) Γu ֒ → Xφ induces an isomorphism π1(Γu) ∼ = ker(u); (3.) Γu ⋔ ψ, 1st return fu : Γu → Γu has (fu)∗ = φu ∈ Out(ker(u)); (4.) χ(Γu) = ǫ, u, where ǫ = 1 2

• e∈E(Xφ)

(2 − deg(e)) e ∈ H1(Xφ)

“Fibrations”, sections, and “Euler class” 10/17

• Theorem. Fix φ ∈ Out(FN) let (Xφ, ψ, A) be as above. Then for

all u ∈ A primitive integral there exists ηu : Xφ → S1 with (ηu)∗ = u satisfying: (1.) Γu = η−1

u (∗) ⊂ Xφ is a graph for any ∗ ∈ S1;

(2.) Γu ֒ → Xφ induces an isomorphism π1(Γu) ∼ = ker(u); (3.) Γu ⋔ ψ, 1st return fu : Γu → Γu has (fu)∗ = φu ∈ Out(ker(u)); (4.) χ(Γu) = ǫ, u, where ǫ = 1 2

• e∈E(Xφ)

(2 − deg(e)) e ∈ H1(Xφ) (1–3): Slightly different construction, but similar ideas as in [Gautero,Wang].

“Fibrations”, sections, and “Euler class” 10/17

• Theorem. Fix φ ∈ Out(FN) let (Xφ, ψ, A) be as above. Then for

all u ∈ A primitive integral there exists ηu : Xφ → S1 with (ηu)∗ = u satisfying: (1.) Γu = η−1

u (∗) ⊂ Xφ is a graph for any ∗ ∈ S1;

(2.) Γu ֒ → Xφ induces an isomorphism π1(Γu) ∼ = ker(u); (3.) Γu ⋔ ψ, 1st return fu : Γu → Γu has (fu)∗ = φu ∈ Out(ker(u)); (4.) χ(Γu) = ǫ, u, where ǫ = 1 2

• e∈E(Xφ)

(2 − deg(e)) e ∈ H1(Xφ) (1–3): Slightly different construction, but similar ideas as in [Gautero,Wang]. (4): linearity of u → χ(ker(u)) follows from Alexander norm [McMullen, Button, Dunfield]

Theorem [Dowdall-Kapovich-L] 11/17

Given φ ∈ Out(FN) represented by an irreducible train track map and (Xφ, ψ, A) as above,

Theorem [Dowdall-Kapovich-L] 11/17

Given φ ∈ Out(FN) represented by an irreducible train track map and (Xφ, ψ, A) as above, ∃! H: A → R continuous, convex, homogeneous of degree −1 such that for all u ∈ A ⇒:

Theorem [Dowdall-Kapovich-L] 11/17

Given φ ∈ Out(FN) represented by an irreducible train track map and (Xφ, ψ, A) as above, ∃! H: A → R continuous, convex, homogeneous of degree −1 such that for all u ∈ A ⇒: (1.) fu : Γu → Γu is an irreducible train track map representing φu = (fu)∗ ∈ Out(ker(u));

Theorem [Dowdall-Kapovich-L] 11/17

Given φ ∈ Out(FN) represented by an irreducible train track map and (Xφ, ψ, A) as above, ∃! H: A → R continuous, convex, homogeneous of degree −1 such that for all u ∈ A ⇒: (1.) fu : Γu → Γu is an irreducible train track map representing φu = (fu)∗ ∈ Out(ker(u)); (2.) log(λ(fu)) = log(λ(φu)) = H(u);

Theorem [Dowdall-Kapovich-L] 11/17

Given φ ∈ Out(FN) represented by an irreducible train track map and (Xφ, ψ, A) as above, ∃! H: A → R continuous, convex, homogeneous of degree −1 such that for all u ∈ A ⇒: (1.) fu : Γu → Γu is an irreducible train track map representing φu = (fu)∗ ∈ Out(ker(u)); (2.) log(λ(fu)) = log(λ(φu)) = H(u); (3.) If φ is fully irreducible and atoroidal, then φu is fully irreducible and atoroidal,

Theorem [Dowdall-Kapovich-L] – Remarks 12/17

φ ∈ Out(FN) fully irreducible, atoroidal, then for u ∈ A primitive integral, fu : Γu → Γu satisfies:

• fu is an irreducible train track map,
• φu = (fu)∗ is fully irreducible and atoroidal,
• log(λ(fu)) = log(λ(φu)) = H(u),

Theorem [Dowdall-Kapovich-L] – Remarks 12/17

φ ∈ Out(FN) fully irreducible, atoroidal, then for u ∈ A primitive integral, fu : Γu → Γu satisfies:

• fu is an irreducible train track map,
• φu = (fu)∗ is fully irreducible and atoroidal,
• log(λ(fu)) = log(λ(φu)) = H(u),

Remarks:

Theorem [Dowdall-Kapovich-L] – Remarks 12/17

φ ∈ Out(FN) fully irreducible, atoroidal, then for u ∈ A primitive integral, fu : Γu → Γu satisfies:

• fu is an irreducible train track map,
• φu = (fu)∗ is fully irreducible and atoroidal,
• log(λ(fu)) = log(λ(φu)) = H(u),

Remarks:

• 1. φ atoroidal implies all φu atoroidal by

[Brinkmann,Bestvina-Feighn].

Theorem [Dowdall-Kapovich-L] – Remarks 12/17

φ ∈ Out(FN) fully irreducible, atoroidal, then for u ∈ A primitive integral, fu : Γu → Γu satisfies:

• fu is an irreducible train track map,
• φu = (fu)∗ is fully irreducible and atoroidal,
• log(λ(fu)) = log(λ(φu)) = H(u),

Remarks:

• 1. φ atoroidal implies all φu atoroidal by

[Brinkmann,Bestvina-Feighn].

• 2. If we only assume φ is fully irreducible, then in general φu

will not be fully irreducible... 3–manifolds.

Small stretch factors 13/17

Corollary With the setup as above suppose K ⊂ A is compact and {un}∞

n=1 ⊂ R+K

all un primitive integral, un → ∞.

Small stretch factors 13/17

Corollary With the setup as above suppose K ⊂ A is compact and {un}∞

n=1 ⊂ R+K

all un primitive integral, un → ∞. Then N(n) = rk(ker(un)) → ∞ and c0 N(n) ≤ log(λ(φun)) ≤ c1 N(n)

Small stretch factors 13/17

Corollary With the setup as above suppose K ⊂ A is compact and {un}∞

n=1 ⊂ R+K

all un primitive integral, un → ∞. Then N(n) = rk(ker(un)) → ∞ and c0 N(n) ≤ log(λ(φun)) ≤ c1 N(n) Theorem [Algom-Kfir–Rafi] All irreducible φ ∈ Out(FN) with log(λ(φ)) ≤ c/N (over all N ≥ 2) are monodromies of “surgeries”

• n the mapping torus of one of a finite set of graph maps.

Idea of construction and proof. 14/17

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN).

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN).

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0)

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) a b c d d a b a b a d b a c

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 a b c d d a b a b a d b a c

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 a b c d d a b a b a d b a c

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... a b c d d a b a b a d b a c

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... a b c d d a b a b a d b a c

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... Difficult to perturb Mf → S1 “nicely” since fibers are not transverse to 1–cells. a b c d d a b a b a d b a c

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... Difficult to perturb Mf → S1 “nicely” since fibers are not transverse to 1–cells. Take a quotient Mf → Xφ so ψ|Γ descends to a “Stallings folding line”, c.f. [Bestvina-Feighn,Francaviglia-Martino] a b c d d a b a b a d b a c

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... Difficult to perturb Mf → S1 “nicely” since fibers are not transverse to 1–cells. Take a quotient Mf → Xφ so ψ|Γ descends to a “Stallings folding line”, c.f. [Bestvina-Feighn,Francaviglia-Martino] a b c d d a b a b a d b a c

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... Difficult to perturb Mf → S1 “nicely” since fibers are not transverse to 1–cells. Take a quotient Mf → Xφ so ψ|Γ descends to a “Stallings folding line”, c.f. [Bestvina-Feighn,Francaviglia-Martino] a b c d d a b a b a d b a c Cell structure w/ “vertical” and “skew” 1–cells, “trapazoid” 2–cells.

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... Difficult to perturb Mf → S1 “nicely” since fibers are not transverse to 1–cells. Take a quotient Mf → Xφ so ψ|Γ descends to a “Stallings folding line”, c.f. [Bestvina-Feighn,Francaviglia-Martino] a b c d d a b a b a d b a c Cell structure w/ “vertical” and “skew” 1–cells, “trapazoid” 2–cells.

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... Difficult to perturb Mf → S1 “nicely” since fibers are not transverse to 1–cells. Take a quotient Mf → Xφ so ψ|Γ descends to a “Stallings folding line”, c.f. [Bestvina-Feighn,Francaviglia-Martino] a b c d d a b a b a d b a c Cell structure w/ “vertical” and “skew” 1–cells, “trapazoid” 2–cells. η: Xφ → S1

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... Difficult to perturb Mf → S1 “nicely” since fibers are not transverse to 1–cells. Take a quotient Mf → Xφ so ψ|Γ descends to a “Stallings folding line”, c.f. [Bestvina-Feighn,Francaviglia-Martino] a b c d d a b a b a d b a c Cell structure w/ “vertical” and “skew” 1–cells, “trapazoid” 2–cells. η: Xφ → S1

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... Difficult to perturb Mf → S1 “nicely” since fibers are not transverse to 1–cells. Take a quotient Mf → Xφ so ψ|Γ descends to a “Stallings folding line”, c.f. [Bestvina-Feighn,Francaviglia-Martino] a b c d d a b a b a d b a c Cell structure w/ “vertical” and “skew” 1–cells, “trapazoid” 2–cells. η: Xφ → S1 can be perturbed to fu : Xφ → S1. for u ∈ A

Idea of construction and proof. Γ

a b c d id f d a b a b a d b a c

f : Γ → Γ an irreducible train track representative for φ ∈ Out(FN). Build Mf = Γ × [0, 1]/(x, 1) ∼ (f (x), 0) → S1 and semi-flow ψ... Difficult to perturb Mf → S1 “nicely” since fibers are not transverse to 1–cells. Take a quotient Mf → Xφ so ψ|Γ descends to a “Stallings folding line”, c.f. [Bestvina-Feighn,Francaviglia-Martino] a b c d d a b a b a d b a c Cell structure w/ “vertical” and “skew” 1–cells, “trapazoid” 2–cells. η: Xφ → S1 can be perturbed to fu : Xφ → S1. for u ∈ A

Train track map 15/17

fu an irreducible train track map?...

Train track map 15/17

fu an irreducible train track map?... Lemma For every edge e of Γ, the characteristic map σ: [0, 1] → e and the semi-flow ψ determine a map [0, 1] × [0, ∞) → Xφ by (x, t) → ψt(σ(x)). This map is locally injective.

Train track map 15/17

fu an irreducible train track map?... Lemma For every edge e of Γ, the characteristic map σ: [0, 1] → e and the semi-flow ψ determine a map [0, 1] × [0, ∞) → Xφ by (x, t) → ψt(σ(x)). This map is locally injective.

Train track map 15/17

fu an irreducible train track map?... Lemma For every edge e of Γ, the characteristic map σ: [0, 1] → e and the semi-flow ψ determine a map [0, 1] × [0, ∞) → Xφ by (x, t) → ψt(σ(x)). This map is locally injective. e

Idea of outline of ideas...16/17

ǫ, u = χ(Γu):

Idea of outline of ideas...16/17

ǫ, u = χ(Γu): ǫ, u = “Intersection number” of Γu with ǫ = 1

2

• e∈E(Xφ)(2 − deg(e)) e

Idea of outline of ideas...16/17

ǫ, u = χ(Γu): ǫ, u = “Intersection number” of Γu with ǫ = 1

2

• e∈E(Xφ)(2 − deg(e)) e

−1/2 −3/2

Idea of outline of ideas...16/17

ǫ, u = χ(Γu): ǫ, u = “Intersection number” of Γu with ǫ = 1

2

• e∈E(Xφ)(2 − deg(e)) e

−1/2 −3/2

Existence of H:

Idea of outline of ideas...16/17

ǫ, u = χ(Γu): ǫ, u = “Intersection number” of Γu with ǫ = 1

2

• e∈E(Xφ)(2 − deg(e)) e

−1/2 −3/2

Existence of H:

• Argue as Fried does (Abramov’s Theorem + variational

principal), jumping through hoops...

Idea of outline of ideas...16/17

ǫ, u = χ(Γu): ǫ, u = “Intersection number” of Γu with ǫ = 1

2

• e∈E(Xφ)(2 − deg(e)) e

−1/2 −3/2

Existence of H:

• Argue as Fried does (Abramov’s Theorem + variational

principal), jumping through hoops... φu = (fu)∗ fully irreducible:

Idea of outline of ideas...16/17

ǫ, u = χ(Γu): ǫ, u = “Intersection number” of Γu with ǫ = 1

2

• e∈E(Xφ)(2 − deg(e)) e

−1/2 −3/2

Existence of H:

• Argue as Fried does (Abramov’s Theorem + variational

principal), jumping through hoops... φu = (fu)∗ fully irreducible:

• Use characterization of full irreducibility for irreducible train

track maps of Kapovich, prove that this is inherited by fu from f . Similar ideas from lemma.

What’s next?... 17/17

Work in progress (w/ Dowdall and Kapovich)

What’s next?... 17/17

Work in progress (w/ Dowdall and Kapovich)

◮ McMullen polynomial (c.f. Teichm¨

uller polynomial of McMullen) —independently by Algom-Kfir, Hironaka, Rafi

What’s next?... 17/17

Work in progress (w/ Dowdall and Kapovich)

◮ McMullen polynomial (c.f. Teichm¨

uller polynomial of McMullen) —independently by Algom-Kfir, Hironaka, Rafi

◮ Geometric structures related to dynamics (think: McMullen

Teichm¨ uller foliation)

What’s next?... 17/17

Work in progress (w/ Dowdall and Kapovich)

◮ McMullen polynomial (c.f. Teichm¨

uller polynomial of McMullen) —independently by Algom-Kfir, Hironaka, Rafi

◮ Geometric structures related to dynamics (think: McMullen

Teichm¨ uller foliation)

◮ Twisted measured R–trees

What’s next?... 17/17

Work in progress (w/ Dowdall and Kapovich)

◮ McMullen polynomial (c.f. Teichm¨

uller polynomial of McMullen) —independently by Algom-Kfir, Hironaka, Rafi

◮ Geometric structures related to dynamics (think: McMullen

Teichm¨ uller foliation)

◮ Twisted measured R–trees ◮ ...

The end

THANKS!