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

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral.

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball,

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball,

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball,

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball, Theorem [Thurston,Fried]

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball, Theorem [Thurston,Fried] For all integral u ∈ C

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball, Theorem [Thurston,Fried] For all integral u ∈ C

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball, Theorem [Thurston,Fried] For all integral u ∈ C ⇒ η u → S 1 with u = ( η u ) ∗ = PD [ S u ] s.t. ∃ fibration S u − → M −

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball, Theorem [Thurston,Fried] For all integral u ∈ C ⇒ η u → S 1 with u = ( η u ) ∗ = PD [ S u ] s.t. ∃ fibration S u − → M − • � e , u � = χ ( S u ) = −|| u || T and

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball, Theorem [Thurston,Fried] For all integral u ∈ C ⇒ η u → S 1 with u = ( η u ) ∗ = PD [ S u ] s.t. ∃ fibration S u − → M − • � e , u � = χ ( S u ) = −|| u || T and • ψ ⋔ S u and first return F u : S u → S u is pseudo-Anosov.

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball, Theorem [Thurston,Fried] For all integral u ∈ C ⇒ η u → S 1 with u = ( η u ) ∗ = PD [ S u ] s.t. ∃ fibration S u − → M − • � e , u � = χ ( S u ) = −|| u || T and • ψ ⋔ S u and first return F u : S u → S u is pseudo-Anosov. ∃ ! H : C → R continuous, convex, homogeneous of degree − 1 such that for all integral u ∈ C

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball, Theorem [Thurston,Fried] For all integral u ∈ C ⇒ η u → S 1 with u = ( η u ) ∗ = PD [ S u ] s.t. ∃ fibration S u − → M − • � e , u � = χ ( S u ) = −|| u || T and • ψ ⋔ S u and first return F u : S u → S u is pseudo-Anosov. ∃ ! H : C → R continuous, convex, homogeneous of degree − 1 such that for all integral u ∈ C • log( λ ( F u )) = H ( u )

Motivation: Thurston and Fried 4/17 H 1 ( M ) η 0 → S 1 fibration S − → M = M F − u 0 u 0 = ( η 0 ) ∗ = PD [ S ] ∈ H 1 ( M ) integral. u 0 ∈ C ⊂ H 1 ( M ), an (open) cone on a fibered face of || · || T –ball, Theorem [Thurston,Fried] For all integral u ∈ C ⇒ η u → S 1 with u = ( η u ) ∗ = PD [ S u ] s.t. ∃ fibration S u − → M − • � e , u � = χ ( S u ) = −|| u || T and • ψ ⋔ S u and first return F u : S u → S u is pseudo-Anosov. ∃ ! H : C → R continuous, convex, homogeneous of degree − 1 such that for all integral u ∈ C • log( λ ( F u )) = H ( u ) see also [Oertel, Long-Oertel, Matsumoto, McMullen]

Motivation: Dilatation asymptotics 5/17 H 1 ( M ) Corollary

Motivation: Dilatation asymptotics 5/17 H 1 ( M ) Corollary Suppose K ⊂ C is compact and { u n } ∞ n =1 ⊂ R + K all u n primitive integral, u n → ∞ .

Motivation: Dilatation asymptotics 5/17 H 1 ( M ) Corollary Suppose K ⊂ C is compact and { u n } ∞ n =1 ⊂ R + K all u n primitive integral, u n → ∞ .

Motivation: Dilatation asymptotics 5/17 H 1 ( M ) Corollary Suppose K ⊂ C is compact and { u n } ∞ n =1 ⊂ R + K all u n primitive integral, u n → ∞ . Then g n = genus( S u n ) → ∞ and c 0 ≤ log( λ ( F u n )) ≤ c 1 g n g n for some 0 < c 0 < c 1 < ∞ .

Motivation: Dilatation asymptotics 5/17 H 1 ( M ) Corollary Suppose K ⊂ C is compact and { u n } ∞ n =1 ⊂ R + K all u n primitive integral, u n → ∞ . Then g n = genus( S u n ) → ∞ and c 0 ≤ log( λ ( F u n )) ≤ c 1 g n g n for some 0 < c 0 < c 1 < ∞ . [Penner,McMullen]

Motivation: Dilatation asymptotics 5/17 H 1 ( M ) Corollary Suppose K ⊂ C is compact and { u n } ∞ n =1 ⊂ R + K all u n primitive integral, u n → ∞ . Then g n = genus( S u n ) → ∞ and c 0 ≤ log( λ ( F u n )) ≤ c 1 g n g n for some 0 < c 0 < c 1 < ∞ . [Penner,McMullen] Theorem [Farb-L-Margalit] All pseudo-Anosov F : S g → S g 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.

Motivation: Dilatation asymptotics 5/17 H 1 ( M ) Corollary Suppose K ⊂ C is compact and { u n } ∞ n =1 ⊂ R + K all u n primitive integral, u n → ∞ . Then g n = genus( S u n ) → ∞ and c 0 ≤ log( λ ( F u n )) ≤ c 1 g n g n for some 0 < c 0 < c 1 < ∞ . [Penner,McMullen] Theorem [Farb-L-Margalit] All pseudo-Anosov F : S g → S g 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].

Transition: Group theory 6/17 π 1 M = π 1 S ⋊ F ∗ Z .

Transition: Group theory 6/17 π 1 M = π 1 S ⋊ F ∗ Z . In fact, M is determined up to homeomorphism by F ∗ ∈ Out( π 1 ( S )) [Dehn-Nielsen-Baer].

Transition: Group theory 6/17 π 1 M = π 1 S ⋊ F ∗ Z . In fact, M is determined up to homeomorphism by F ∗ ∈ Out( π 1 ( S )) [Dehn-Nielsen-Baer]. φ ∈ Out( π 1 ( S )) is represented by a pseudo-Anosov F if and only if φ has no nontrivial periodic conjugacy classes

Transition: Group theory 6/17 π 1 M = π 1 S ⋊ F ∗ Z . In fact, M is determined up to homeomorphism by F ∗ ∈ Out( π 1 ( S )) [Dehn-Nielsen-Baer]. φ ∈ Out( π 1 ( S )) is represented by a pseudo-Anosov F if and only if φ has no nontrivial periodic conjugacy classes if and only if π 1 S ⋊ φ Z is word-hyperbolic. [Thurston]

Transition: Group theory 6/17 π 1 M = π 1 S ⋊ F ∗ Z . In fact, M is determined up to homeomorphism by F ∗ ∈ Out( π 1 ( S )) [Dehn-Nielsen-Baer]. φ ∈ Out( π 1 ( S )) is represented by a pseudo-Anosov F if and only if φ has no nontrivial periodic conjugacy classes if and only if π 1 S ⋊ φ Z is word-hyperbolic. [Thurston] λ ( F ) = growth rate of word length in π 1 S under iteration of F ∗ .

Transition: Group theory 6/17 π 1 M = π 1 S ⋊ F ∗ Z . In fact, M is determined up to homeomorphism by F ∗ ∈ Out( π 1 ( S )) [Dehn-Nielsen-Baer]. φ ∈ Out( π 1 ( S )) is represented by a pseudo-Anosov F if and only if φ has no nontrivial periodic conjugacy classes if and only if π 1 S ⋊ φ Z is word-hyperbolic. [Thurston] λ ( F ) = growth rate of word length in π 1 S under iteration of F ∗ . Integral u ∈ Hom ( π 1 M , R ) = H 1 ( M ) is induced by a fibration over S 1 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( F N ) be

Atoroidal and fully irreducible 7/17 Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and

Atoroidal and fully irreducible 7/17 Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out( F N ) 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( F N ) 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( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ φ Z is word-hyperbolic, and

Atoroidal and fully irreducible 7/17 Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ 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( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ 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( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ φ Z is word-hyperbolic, and • φ is represented by an irreducible train track map . Example: a d id Γ b b a c b c f a d a a d b

Atoroidal and fully irreducible 7/17 Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ φ Z is word-hyperbolic, and • φ is represented by an irreducible train track map . - A graph Γ, π 1 Γ ∼ = F N , Example: a d id Γ b b a c b c f a d a a d b

Atoroidal and fully irreducible 7/17 Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ φ Z is word-hyperbolic, and • φ is represented by an irreducible train track map . - A graph Γ, π 1 Γ ∼ = F N , Example: a d - f : Γ → Γ a h.e. and f ∗ = φ id Γ b b a c b c f a d a a d b

Atoroidal and fully irreducible 7/17 Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ φ Z is word-hyperbolic, and • φ is represented by an irreducible train track map . - A graph Γ, π 1 Γ ∼ = F N , Example: a d - f : Γ → Γ a h.e. and f ∗ = φ id Γ b b a - f ( V Γ) ⊂ V Γ c b c f a d a a d b

Atoroidal and fully irreducible 7/17 Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ φ Z is word-hyperbolic, and • φ is represented by an irreducible train track map . - A graph Γ, π 1 Γ ∼ = F N , Example: a d - f : Γ → Γ a h.e. and f ∗ = φ id Γ b b a - f ( V Γ) ⊂ V Γ c b c - f n | e is an immersion for all f a d a a d b n ≥ 1 and for all edges e

Atoroidal and fully irreducible 7/17 Theorem [Bestvina-Feign, Brinkmann, Bestvina-Handel] Let φ ∈ Out( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ φ Z is word-hyperbolic, and • φ is represented by an irreducible train track map . - A graph Γ, π 1 Γ ∼ = F N , Example: a d - f : Γ → Γ a h.e. and f ∗ = φ id Γ b b a - f ( V Γ) ⊂ V Γ c b c - f n | e is an immersion for all f a d a a d b 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( F N ) be • Atoroidal : no nontrivial periodic conjugacy classes, and • Fully irreducible : no nontrivial periodic free factors. Then • G = G φ = F N ⋊ φ Z is word-hyperbolic, and • φ is represented by an irreducible train track map . - A graph Γ, π 1 Γ ∼ = F N , Example: a d - f : Γ → Γ a h.e. and f ∗ = φ id Γ b b a - f ( V Γ) ⊂ V Γ c b c - f n | e is an immersion for all f a d a a d b n ≥ 1 and for all edges e Many other examples [Clay-Pettet] - irreducible transition matrix...

Dynamics and stretch factors 8/17 a d id Γ b b a c b c f a d a a d b

Dynamics and stretch factors 8/17 a d id Γ b b a c b c f a d a a d b Transition matrix   0 0 0 1 1 0 0 0   A ( f ) =  ,   1 1 0 0  2 2 1 1

Dynamics and stretch factors 8/17 a d id Γ b b a c b c f a d a a d b Transition matrix and Perron-Frobenius eigenvalue/eigenvector     0 0 0 1 . 2265 1 0 0 0 . 0939     A ( f ) =  , λ ≈ 2 . 4142 , v ≈     1 1 0 0 . 1327    2 2 1 1 . 5469

Dynamics and stretch factors 8/17 a d id Γ b b a c b c f a d a a d b Transition matrix and Perron-Frobenius eigenvalue/eigenvector     0 0 0 1 . 2265 1 0 0 0 . 0939     A ( f ) =  , λ ≈ 2 . 4142 , v ≈     1 1 0 0 . 1327    2 2 1 1 . 5469 � metric graph (Γ , d v ), f ≃ f v : (Γ , d v ) → (Γ , d v ), affine-stretch by λ on all edges.

Dynamics and stretch factors 8/17 a d id Γ b b a c b c f a d a a d b Transition matrix and Perron-Frobenius eigenvalue/eigenvector     0 0 0 1 . 2265 1 0 0 0 . 0939     A ( f ) =  , λ ≈ 2 . 4142 , v ≈     1 1 0 0 . 1327    2 2 1 1 . 5469 � metric graph (Γ , d v ), f ≃ f v : (Γ , d v ) → (Γ , d v ), affine-stretch by λ on all edges. � n length ( f n ( α )) = stretch factor . λ = λ ( f ) = λ ( φ ) = lim n →∞

Dynamics and stretch factors 8/17 a d id Γ b b a c b c f a d a a d b Transition matrix and Perron-Frobenius eigenvalue/eigenvector     0 0 0 1 . 2265 1 0 0 0 . 0939     A ( f ) =  , λ ≈ 2 . 4142 , v ≈     1 1 0 0 . 1327    2 2 1 1 . 5469 � metric graph (Γ , d v ), f ≃ f v : (Γ , d v ) → (Γ , d v ), affine-stretch by λ on all edges. � n length ( f n ( α )) = stretch factor . λ = λ ( f ) = λ ( φ ) = lim n →∞ 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( F N )

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

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( F N ) � ( X φ , ψ, A ) • X φ is a polyhedral 2–complex, K ( G , 1) for G = F N ⋊ φ Z . • ψ is a semi-flow on X φ . • A = { [ z ] ∈ H 1 ( X φ ) | z ∈ Z 1 ( X φ ) 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( F N ) � ( X φ , ψ, A ) • X φ is a polyhedral 2–complex, K ( G , 1) for G = F N ⋊ φ Z . • ψ is a semi-flow on X φ . • A = { [ z ] ∈ H 1 ( X φ ) | z ∈ Z 1 ( X φ ) positive, cellular } , open cone. • u 0 ∈ Hom ( G , R ) = H 1 ( X φ ), u 0 ( x , n ) = n ⇒ u 0 ∈ A .

“Fibrations”, sections, and “Euler class” 10/17 Theorem. Fix φ ∈ Out( F N ) let ( X φ , ψ, A ) be as above. Then for all u ∈ A primitive integral there exists η u : X φ → S 1 with ( η u ) ∗ = u satisfying:

“Fibrations”, sections, and “Euler class” 10/17 Theorem. Fix φ ∈ Out( F N ) let ( X φ , ψ, A ) be as above. Then for all u ∈ A primitive integral there exists η u : X φ → S 1 with ( η u ) ∗ = u satisfying: (1. ) Γ u = η − 1 u ( ∗ ) ⊂ X φ is a graph for any ∗ ∈ S 1 ;

“Fibrations”, sections, and “Euler class” 10/17 Theorem. Fix φ ∈ Out( F N ) let ( X φ , ψ, A ) be as above. Then for all u ∈ A primitive integral there exists η u : X φ → S 1 with ( η u ) ∗ = u satisfying: (1. ) Γ u = η − 1 u ( ∗ ) ⊂ X φ is a graph for any ∗ ∈ S 1 ; → X φ induces an isomorphism π 1 (Γ u ) ∼ (2.) Γ u ֒ = ker( u );

“Fibrations”, sections, and “Euler class” 10/17 Theorem. Fix φ ∈ Out( F N ) let ( X φ , ψ, A ) be as above. Then for all u ∈ A primitive integral there exists η u : X φ → S 1 with ( η u ) ∗ = u satisfying: (1. ) Γ u = η − 1 u ( ∗ ) ⊂ X φ is a graph for any ∗ ∈ S 1 ; → X φ induces an isomorphism π 1 (Γ u ) ∼ (2.) Γ u ֒ = ker( u ); (3.) Γ u ⋔ ψ , 1 st return f u : Γ u → Γ u has ( f u ) ∗ = φ u ∈ Out(ker( u ));

“Fibrations”, sections, and “Euler class” 10/17 Theorem. Fix φ ∈ Out( F N ) let ( X φ , ψ, A ) be as above. Then for all u ∈ A primitive integral there exists η u : X φ → S 1 with ( η u ) ∗ = u satisfying: (1. ) Γ u = η − 1 u ( ∗ ) ⊂ X φ is a graph for any ∗ ∈ S 1 ; → X φ induces an isomorphism π 1 (Γ u ) ∼ (2.) Γ u ֒ = ker( u ); (3.) Γ u ⋔ ψ , 1 st return f u : Γ u → Γ u has ( f u ) ∗ = φ u ∈ Out(ker( u )); (4.) χ (Γ u ) = � ǫ, u � , where

“Fibrations”, sections, and “Euler class” 10/17 Theorem. Fix φ ∈ Out( F N ) let ( X φ , ψ, A ) be as above. Then for all u ∈ A primitive integral there exists η u : X φ → S 1 with ( η u ) ∗ = u satisfying: (1. ) Γ u = η − 1 u ( ∗ ) ⊂ X φ is a graph for any ∗ ∈ S 1 ; → X φ induces an isomorphism π 1 (Γ u ) ∼ (2.) Γ u ֒ = ker( u ); (3.) Γ u ⋔ ψ , 1 st return f u : Γ u → Γ u has ( f u ) ∗ = φ u ∈ Out(ker( u )); (4.) χ (Γ u ) = � ǫ, u � , where ǫ = 1 � (2 − deg ( e )) e ∈ H 1 ( X φ ) 2 e ∈E ( X φ )