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GAGTA7 Conference Dynamics for the splittings of free-by-cyclic groups Ilya Kapovich University of Illinois at Urbana-Champaign Based on joint work with Spencer Dowdall and Chris Leininger arXiv:1301.7739 GAGTA7 Conference, New York; May,

SLIDE 1

GAGTA7 Conference

Dynamics for the splittings

f free-by-cyclic groups

Ilya Kapovich University of Illinois at Urbana-Champaign Based on joint work with Spencer Dowdall and Chris Leininger arXiv:1301.7739 GAGTA7 Conference, New York; May, 2013

Ilya Kapovich (UIUC) May, 2013 1 / 26

SLIDE 2

Dr. Gillian Taylor: "Don’t tell me, you’re from outer space."

Further results (time permitting)

Ilya Kapovich (UIUC) May, 2013 3 / 26

SLIDE 12

Let S be a closed surface of negative Euler char, and let f : S → S be a home. Then the mapping torus Mf = S × [0, 1]/(f(x), 0) ∼ (x, 1) is a closed 3-manifold fibering η : Mf → S1 over S1 with fiber S. Mf also comes equipped with a natural flow Ψt. Also get an epimorphism η∗ : G → Z = π1(S), where G = π1(Mf) and Ker(η∗) = π1(S). If f is isotopic to a pseudo-Anosov, then Mf is hyperbolic. Q: How can Mf fiber in other ways over S1.

Ilya Kapovich (UIUC) May, 2013 4 / 26

SLIDE 13

Let S be a closed surface of negative Euler char, and let f : S → S be a home. Then the mapping torus Mf = S × [0, 1]/(f(x), 0) ∼ (x, 1) is a closed 3-manifold fibering η : Mf → S1 over S1 with fiber S. Mf also comes equipped with a natural flow Ψt. Also get an epimorphism η∗ : G → Z = π1(S), where G = π1(Mf) and Ker(η∗) = π1(S). If f is isotopic to a pseudo-Anosov, then Mf is hyperbolic. Q: How can Mf fiber in other ways over S1.

Ilya Kapovich (UIUC) May, 2013 4 / 26

SLIDE 14

Let S be a closed surface of negative Euler char, and let f : S → S be a home. Then the mapping torus Mf = S × [0, 1]/(f(x), 0) ∼ (x, 1) is a closed 3-manifold fibering η : Mf → S1 over S1 with fiber S. Mf also comes equipped with a natural flow Ψt. Also get an epimorphism η∗ : G → Z = π1(S), where G = π1(Mf) and Ker(η∗) = π1(S). If f is isotopic to a pseudo-Anosov, then Mf is hyperbolic. Q: How can Mf fiber in other ways over S1.

Ilya Kapovich (UIUC) May, 2013 4 / 26

SLIDE 15

Let S be a closed surface of negative Euler char, and let f : S → S be a home. Then the mapping torus Mf = S × [0, 1]/(f(x), 0) ∼ (x, 1) is a closed 3-manifold fibering η : Mf → S1 over S1 with fiber S. Mf also comes equipped with a natural flow Ψt. Also get an epimorphism η∗ : G → Z = π1(S), where G = π1(Mf) and Ker(η∗) = π1(S). If f is isotopic to a pseudo-Anosov, then Mf is hyperbolic. Q: How can Mf fiber in other ways over S1.

Ilya Kapovich (UIUC) May, 2013 4 / 26

SLIDE 16

Let S be a closed surface of negative Euler char, and let f : S → S be a home. Then the mapping torus Mf = S × [0, 1]/(f(x), 0) ∼ (x, 1) is a closed 3-manifold fibering η : Mf → S1 over S1 with fiber S. Mf also comes equipped with a natural flow Ψt. Also get an epimorphism η∗ : G → Z = π1(S), where G = π1(Mf) and Ker(η∗) = π1(S). If f is isotopic to a pseudo-Anosov, then Mf is hyperbolic. Q: How can Mf fiber in other ways over S1.

Ilya Kapovich (UIUC) May, 2013 4 / 26

SLIDE 17

Let S be a closed surface of negative Euler char, and let f : S → S be a home. Then the mapping torus Mf = S × [0, 1]/(f(x), 0) ∼ (x, 1) is a closed 3-manifold fibering η : Mf → S1 over S1 with fiber S. Mf also comes equipped with a natural flow Ψt. Also get an epimorphism η∗ : G → Z = π1(S), where G = π1(Mf) and Ker(η∗) = π1(S). If f is isotopic to a pseudo-Anosov, then Mf is hyperbolic. Q: How can Mf fiber in other ways over S1.

Ilya Kapovich (UIUC) May, 2013 4 / 26

SLIDE 18

Let S be a closed surface of negative Euler char, and let f : S → S be a home. Then the mapping torus Mf = S × [0, 1]/(f(x), 0) ∼ (x, 1) is a closed 3-manifold fibering η : Mf → S1 over S1 with fiber S. Mf also comes equipped with a natural flow Ψt. Also get an epimorphism η∗ : G → Z = π1(S), where G = π1(Mf) and Ker(η∗) = π1(S). If f is isotopic to a pseudo-Anosov, then Mf is hyperbolic. Q: How can Mf fiber in other ways over S1.

SLIDE 35

Summary of the results of Thurston and Fried

(1) Thurston constructed a semi-norm, called the Thurston norm n : H1(G, R) → R such that the unit ball B w.r. to n is polyhedral, so that n is linear on the cone over any open face of B. (2) The cone C mentioned above is the union of open cones over several top-dimensional open faces of B, called fibered faces. (3) For every primitive integral u ∈ C, n(u) = −χ(Su).

Ilya Kapovich (UIUC) May, 2013 7 / 26

SLIDE 36

Summary of the results of Thurston and Fried

(4) Fried showed that, assuming f is pseudo-anosov, for every primitive integral u ∈ C the fibration ηu : Mf → S1 and the fiber Su can be chosen so that Su is transversal to the flow Ψt,so that fu is the first return map,and so that fu is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B, and the cone CP

ver P, there exists a continuous convex function

H : CP → (0, ∞) such that for every primitive integral u ∈ CP we have H(u) = log λ(fu) where λ(fu) > 1 is the stretch factor of fu.

Ilya Kapovich (UIUC) May, 2013 8 / 26

SLIDE 37

Summary of the results of Thurston and Fried

(4) Fried showed that, assuming f is pseudo-anosov, for every primitive integral u ∈ C the fibration ηu : Mf → S1 and the fiber Su can be chosen so that Su is transversal to the flow Ψt,so that fu is the first return map,and so that fu is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B, and the cone CP

ver P, there exists a continuous convex function

H : CP → (0, ∞) such that for every primitive integral u ∈ CP we have H(u) = log λ(fu) where λ(fu) > 1 is the stretch factor of fu.

Ilya Kapovich (UIUC) May, 2013 8 / 26

SLIDE 38

Summary of the results of Thurston and Fried

(4) Fried showed that, assuming f is pseudo-anosov, for every primitive integral u ∈ C the fibration ηu : Mf → S1 and the fiber Su can be chosen so that Su is transversal to the flow Ψt,so that fu is the first return map,and so that fu is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B, and the cone CP

ver P, there exists a continuous convex function

H : CP → (0, ∞) such that for every primitive integral u ∈ CP we have H(u) = log λ(fu) where λ(fu) > 1 is the stretch factor of fu.

Ilya Kapovich (UIUC) May, 2013 8 / 26

SLIDE 39

Summary of the results of Thurston and Fried

(4) Fried showed that, assuming f is pseudo-anosov, for every primitive integral u ∈ C the fibration ηu : Mf → S1 and the fiber Su can be chosen so that Su is transversal to the flow Ψt,so that fu is the first return map,and so that fu is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B, and the cone CP

ver P, there exists a continuous convex function

H : CP → (0, ∞) such that for every primitive integral u ∈ CP we have H(u) = log λ(fu) where λ(fu) > 1 is the stretch factor of fu.

Ilya Kapovich (UIUC) May, 2013 8 / 26

SLIDE 40

Summary of the results of Thurston and Fried

(4) Fried showed that, assuming f is pseudo-anosov, for every primitive integral u ∈ C the fibration ηu : Mf → S1 and the fiber Su can be chosen so that Su is transversal to the flow Ψt,so that fu is the first return map,and so that fu is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B, and the cone CP

ver P, there exists a continuous convex function

Figure: A local picture of Xf

Ilya Kapovich (UIUC) May, 2013 19 / 26

SLIDE 101

Folded mapping torus

(3) Form the "full" mapping torus Mf = Γ × [0, 1]/(f(x), 0) ∼ (x, 1) and use the above folding sequence to "fold" Mf to Xf:

Figure: A local picture of Xf

Ilya Kapovich (UIUC) May, 2013 19 / 26

SLIDE 102

Folded mapping torus

a b c d d a b a b a d b a c

Figure: A global picture of Xf in our example. The 2–cells are identified as indicated by the shading and the patterns of 2–cells (white cells are not identified to any others) to produce Xf from Mf.

Ilya Kapovich (UIUC) May, 2013 20 / 26

SLIDE 103

Folded mapping torus

a b c d d a b a b a d b a c

Figure: A global picture of Xf in our example. The 2–cells are identified as indicated by the shading and the patterns of 2–cells (white cells are not identified to any others) to produce Xf from Mf.

Ilya Kapovich (UIUC) May, 2013 20 / 26

SLIDE 104

Folded mapping torus

The 1-cycle ǫ ∈ H1(Xf, R) from Thm A is defined as ǫ = 1 2

e∈E0

[2 − d(e)]e, where E0 is the subgraph of X (1)

f

consisting of all vertical and skew 1-cells e of Xf with local degree d(e) > 2. The 1-cells in E0 are oriented by "hight". The cone A consists of all u ∈ H1(Xf, R) that can be represented by "positive" 1-cocycles, that is, by cocycles z ∈ Z 1(Xf, R) such that z(e) > 0 for all e ∈ E0. Given u ∈ A and a positive cocycle z representing u, we construct a "perturbed fibration" fz : Xf → S1 such that Θz = f −1

z

(0) has the required properties.

Ilya Kapovich (UIUC) May, 2013 21 / 26

SLIDE 105

Folded mapping torus

The 1-cycle ǫ ∈ H1(Xf, R) from Thm A is defined as ǫ = 1 2

e∈E0

[2 − d(e)]e, where E0 is the subgraph of X (1)

f

consisting of all vertical and skew 1-cells e of Xf with local degree d(e) > 2. The 1-cells in E0 are oriented by "hight". The cone A consists of all u ∈ H1(Xf, R) that can be represented by "positive" 1-cocycles, that is, by cocycles z ∈ Z 1(Xf, R) such that z(e) > 0 for all e ∈ E0. Given u ∈ A and a positive cocycle z representing u, we construct a "perturbed fibration" fz : Xf → S1 such that Θz = f −1

z

(0) has the required properties.

Ilya Kapovich (UIUC) May, 2013 21 / 26

SLIDE 106

Folded mapping torus

The 1-cycle ǫ ∈ H1(Xf, R) from Thm A is defined as ǫ = 1 2

e∈E0

[2 − d(e)]e, where E0 is the subgraph of X (1)

f

consisting of all vertical and skew 1-cells e of Xf with local degree d(e) > 2. The 1-cells in E0 are oriented by "hight". The cone A consists of all u ∈ H1(Xf, R) that can be represented by "positive" 1-cocycles, that is, by cocycles z ∈ Z 1(Xf, R) such that z(e) > 0 for all e ∈ E0. Given u ∈ A and a positive cocycle z representing u, we construct a "perturbed fibration" fz : Xf → S1 such that Θz = f −1

z

(0) has the required properties.

Ilya Kapovich (UIUC) May, 2013 21 / 26

SLIDE 107

Folded mapping torus

The 1-cycle ǫ ∈ H1(Xf, R) from Thm A is defined as ǫ = 1 2

e∈E0

[2 − d(e)]e, where E0 is the subgraph of X (1)

f

consisting of all vertical and skew 1-cells e of Xf with local degree d(e) > 2. The 1-cells in E0 are oriented by "hight". The cone A consists of all u ∈ H1(Xf, R) that can be represented by "positive" 1-cocycles, that is, by cocycles z ∈ Z 1(Xf, R) such that z(e) > 0 for all e ∈ E0. Given u ∈ A and a positive cocycle z representing u, we construct a "perturbed fibration" fz : Xf → S1 such that Θz = f −1

z

(0) has the required properties.

Ilya Kapovich (UIUC) May, 2013 21 / 26

SLIDE 108

Folded mapping torus

The 1-cycle ǫ ∈ H1(Xf, R) from Thm A is defined as ǫ = 1 2

e∈E0

[2 − d(e)]e, where E0 is the subgraph of X (1)

f

consisting of all vertical and skew 1-cells e of Xf with local degree d(e) > 2. The 1-cells in E0 are oriented by "hight". The cone A consists of all u ∈ H1(Xf, R) that can be represented by "positive" 1-cocycles, that is, by cocycles z ∈ Z 1(Xf, R) such that z(e) > 0 for all e ∈ E0. Given u ∈ A and a positive cocycle z representing u, we construct a "perturbed fibration" fz : Xf → S1 such that Θz = f −1

z

(0) has the required properties.

Ilya Kapovich (UIUC) May, 2013 21 / 26

SLIDE 109

Folded mapping torus

The 1-cycle ǫ ∈ H1(Xf, R) from Thm A is defined as ǫ = 1 2

e∈E0

[2 − d(e)]e, where E0 is the subgraph of X (1)

f

consisting of all vertical and skew 1-cells e of Xf with local degree d(e) > 2. The 1-cells in E0 are oriented by "hight". The cone A consists of all u ∈ H1(Xf, R) that can be represented by "positive" 1-cocycles, that is, by cocycles z ∈ Z 1(Xf, R) such that z(e) > 0 for all e ∈ E0. Given u ∈ A and a positive cocycle z representing u, we construct a "perturbed fibration" fz : Xf → S1 such that Θz = f −1

z

(0) has the required properties.

Ilya Kapovich (UIUC) May, 2013 21 / 26

SLIDE 110

Folded mapping torus

The 1-cycle ǫ ∈ H1(Xf, R) from Thm A is defined as ǫ = 1 2

e∈E0

[2 − d(e)]e, where E0 is the subgraph of X (1)

f

consisting of all vertical and skew 1-cells e of Xf with local degree d(e) > 2. The 1-cells in E0 are oriented by "hight". The cone A consists of all u ∈ H1(Xf, R) that can be represented by "positive" 1-cocycles, that is, by cocycles z ∈ Z 1(Xf, R) such that z(e) > 0 for all e ∈ E0. Given u ∈ A and a positive cocycle z representing u, we construct a "perturbed fibration" fz : Xf → S1 such that Θz = f −1

z

(0) has the required properties.

Ilya Kapovich (UIUC) May, 2013 21 / 26

SLIDE 111

Folded mapping torus

The 1-cycle ǫ ∈ H1(Xf, R) from Thm A is defined as ǫ = 1 2

e∈E0

[2 − d(e)]e, where E0 is the subgraph of X (1)

f

consisting of all vertical and skew 1-cells e of Xf with local degree d(e) > 2. The 1-cells in E0 are oriented by "hight". The cone A consists of all u ∈ H1(Xf, R) that can be represented by "positive" 1-cocycles, that is, by cocycles z ∈ Z 1(Xf, R) such that z(e) > 0 for all e ∈ E0. Given u ∈ A and a positive cocycle z representing u, we construct a "perturbed fibration" fz : Xf → S1 such that Θz = f −1

z

(0) has the required properties.

Ilya Kapovich (UIUC) May, 2013 21 / 26

SLIDE 112

Folded mapping torus

The 1-cycle ǫ ∈ H1(Xf, R) from Thm A is defined as ǫ = 1 2

e∈E0

[2 − d(e)]e, where E0 is the subgraph of X (1)

f

consisting of all vertical and skew 1-cells e of Xf with local degree d(e) > 2. The 1-cells in E0 are oriented by "hight". The cone A consists of all u ∈ H1(Xf, R) that can be represented by "positive" 1-cocycles, that is, by cocycles z ∈ Z 1(Xf, R) such that z(e) > 0 for all e ∈ E0. Given u ∈ A and a positive cocycle z representing u, we construct a "perturbed fibration" fz : Xf → S1 such that Θz = f −1

z

(0) has the required properties.

Ilya Kapovich (UIUC) May, 2013 21 / 26

SLIDE 113

Additional results

It follows from Thms A and B that if φ ∈ Out(FN) is hyperbolic and fully irreducible, then the function D : A → (0, ∞), D(u) := H(u)ǫ(u) is continuous and constant on open rays starting at the origin.Hence D(u) is bounded by positive constants above and below on the cone

ver any compact subset of A. From here we obtain:

Cor 1 Let φ ∈ Out(FN) be hyperbolic and fully irreducible and such that b1(Gφ) ≥ 2. Then there exist c2 > c1 > 0 and Ni → ∞ as i → ∞ such that Gφ splits as Gφ = FNi ⋊φi Z, where φi ∈ Out(FNi) are hyperbolic fully irreducible and such that c1 Ni ≤ log λ(φi) ≤ c2 Ni . C.f. a similar result of McMullen for 3-manifolds and complimentary results of Algom-Kfir and Rafi for free-by-cyclic groups.

Ilya Kapovich (UIUC) May, 2013 22 / 26

SLIDE 114

Additional results

It follows from Thms A and B that if φ ∈ Out(FN) is hyperbolic and fully irreducible, then the function D : A → (0, ∞), D(u) := H(u)ǫ(u) is continuous and constant on open rays starting at the origin.Hence D(u) is bounded by positive constants above and below on the cone

ver any compact subset of A. From here we obtain:

Cor 1 Let φ ∈ Out(FN) be hyperbolic and fully irreducible and such that b1(Gφ) ≥ 2. Then there exist c2 > c1 > 0 and Ni → ∞ as i → ∞ such that Gφ splits as Gφ = FNi ⋊φi Z, where φi ∈ Out(FNi) are hyperbolic fully irreducible and such that c1 Ni ≤ log λ(φi) ≤ c2 Ni . C.f. a similar result of McMullen for 3-manifolds and complimentary results of Algom-Kfir and Rafi for free-by-cyclic groups.

Ilya Kapovich (UIUC) May, 2013 22 / 26

SLIDE 115

Additional results

It follows from Thms A and B that if φ ∈ Out(FN) is hyperbolic and fully irreducible, then the function D : A → (0, ∞), D(u) := H(u)ǫ(u) is continuous and constant on open rays starting at the origin.Hence D(u) is bounded by positive constants above and below on the cone

ver any compact subset of A. From here we obtain:

Cor 1 Let φ ∈ Out(FN) be hyperbolic and fully irreducible and such that b1(Gφ) ≥ 2. Then there exist c2 > c1 > 0 and Ni → ∞ as i → ∞ such that Gφ splits as Gφ = FNi ⋊φi Z, where φi ∈ Out(FNi) are hyperbolic fully irreducible and such that c1 Ni ≤ log λ(φi) ≤ c2 Ni . C.f. a similar result of McMullen for 3-manifolds and complimentary results of Algom-Kfir and Rafi for free-by-cyclic groups.

Ilya Kapovich (UIUC) May, 2013 22 / 26

SLIDE 116

Additional results

It follows from Thms A and B that if φ ∈ Out(FN) is hyperbolic and fully irreducible, then the function D : A → (0, ∞), D(u) := H(u)ǫ(u) is continuous and constant on open rays starting at the origin.Hence D(u) is bounded by positive constants above and below on the cone

ver any compact subset of A. From here we obtain:

Cor 1 Let φ ∈ Out(FN) be hyperbolic and fully irreducible and such that b1(Gφ) ≥ 2. Then there exist c2 > c1 > 0 and Ni → ∞ as i → ∞ such that Gφ splits as Gφ = FNi ⋊φi Z, where φi ∈ Out(FNi) are hyperbolic fully irreducible and such that c1 Ni ≤ log λ(φi) ≤ c2 Ni . C.f. a similar result of McMullen for 3-manifolds and complimentary results of Algom-Kfir and Rafi for free-by-cyclic groups.

Ilya Kapovich (UIUC) May, 2013 22 / 26

SLIDE 117

Additional results

It follows from Thms A and B that if φ ∈ Out(FN) is hyperbolic and fully irreducible, then the function D : A → (0, ∞), D(u) := H(u)ǫ(u) is continuous and constant on open rays starting at the origin.Hence D(u) is bounded by positive constants above and below on the cone

ver any compact subset of A. From here we obtain:

Cor 1 Let φ ∈ Out(FN) be hyperbolic and fully irreducible and such that b1(Gφ) ≥ 2. Then there exist c2 > c1 > 0 and Ni → ∞ as i → ∞ such that Gφ splits as Gφ = FNi ⋊φi Z, where φi ∈ Out(FNi) are hyperbolic fully irreducible and such that c1 Ni ≤ log λ(φi) ≤ c2 Ni . C.f. a similar result of McMullen for 3-manifolds and complimentary results of Algom-Kfir and Rafi for free-by-cyclic groups.

Ilya Kapovich (UIUC) May, 2013 22 / 26

SLIDE 118

Additional results

It follows from Thms A and B that if φ ∈ Out(FN) is hyperbolic and fully irreducible, then the function D : A → (0, ∞), D(u) := H(u)ǫ(u) is continuous and constant on open rays starting at the origin.Hence D(u) is bounded by positive constants above and below on the cone

ver any compact subset of A. From here we obtain:

Cor 1 Let φ ∈ Out(FN) be hyperbolic and fully irreducible and such that b1(Gφ) ≥ 2. Then there exist c2 > c1 > 0 and Ni → ∞ as i → ∞ such that Gφ splits as Gφ = FNi ⋊φi Z, where φi ∈ Out(FNi) are hyperbolic fully irreducible and such that c1 Ni ≤ log λ(φi) ≤ c2 Ni . C.f. a similar result of McMullen for 3-manifolds and complimentary results of Algom-Kfir and Rafi for free-by-cyclic groups.

Ilya Kapovich (UIUC) May, 2013 22 / 26

SLIDE 119

Additional results

It follows from Thms A and B that if φ ∈ Out(FN) is hyperbolic and fully irreducible, then the function D : A → (0, ∞), D(u) := H(u)ǫ(u) is continuous and constant on open rays starting at the origin.Hence D(u) is bounded by positive constants above and below on the cone

ver any compact subset of A. From here we obtain:

Cor 1 Let φ ∈ Out(FN) be hyperbolic and fully irreducible and such that b1(Gφ) ≥ 2. Then there exist c2 > c1 > 0 and Ni → ∞ as i → ∞ such that Gφ splits as Gφ = FNi ⋊φi Z, where φi ∈ Out(FNi) are hyperbolic fully irreducible and such that c1 Ni ≤ log λ(φi) ≤ c2 Ni . C.f. a similar result of McMullen for 3-manifolds and complimentary results of Algom-Kfir and Rafi for free-by-cyclic groups.

Ilya Kapovich (UIUC) May, 2013 22 / 26

SLIDE 120

Additional results

An element φ ∈ Out(FN) is called reducible if there exists a free product decomposition FN = A1 ∗ · · · ∗ Ak ∗ C where k ≥ 1, and each Ai is a proper free factor of FN, such that φ permutes the conjugacy classes of A1, . . . , Ak. An element φ ∈ Out(FN) is called irreducible if it is not reducible. Fact: The definitions imply that φ ∈ Out(FN) is fully irreducible if and

nly if φn is irreducible for all n = 0.

The proof of Thm B produces, as a side result, the following: Cor 2. Let φ ∈ Out(FN) be hyperbolic.Then φ is irreducible if and only if φ is fully irreducible. The statement of Cor 2 is well-known to be false for non-hyperbolic automorphisms.

Ilya Kapovich (UIUC) May, 2013 23 / 26

SLIDE 121

Additional results

An element φ ∈ Out(FN) is called reducible if there exists a free product decomposition FN = A1 ∗ · · · ∗ Ak ∗ C where k ≥ 1, and each Ai is a proper free factor of FN, such that φ permutes the conjugacy classes of A1, . . . , Ak. An element φ ∈ Out(FN) is called irreducible if it is not reducible. Fact: The definitions imply that φ ∈ Out(FN) is fully irreducible if and

nly if φn is irreducible for all n = 0.

The proof of Thm B produces, as a side result, the following: Cor 2. Let φ ∈ Out(FN) be hyperbolic.Then φ is irreducible if and only if φ is fully irreducible. The statement of Cor 2 is well-known to be false for non-hyperbolic automorphisms.

Ilya Kapovich (UIUC) May, 2013 23 / 26

SLIDE 122

Additional results

An element φ ∈ Out(FN) is called reducible if there exists a free product decomposition FN = A1 ∗ · · · ∗ Ak ∗ C where k ≥ 1, and each Ai is a proper free factor of FN, such that φ permutes the conjugacy classes of A1, . . . , Ak. An element φ ∈ Out(FN) is called irreducible if it is not reducible. Fact: The definitions imply that φ ∈ Out(FN) is fully irreducible if and

nly if φn is irreducible for all n = 0.

The proof of Thm B produces, as a side result, the following: Cor 2. Let φ ∈ Out(FN) be hyperbolic.Then φ is irreducible if and only if φ is fully irreducible. The statement of Cor 2 is well-known to be false for non-hyperbolic automorphisms.

Ilya Kapovich (UIUC) May, 2013 23 / 26

SLIDE 123

Additional results

An element φ ∈ Out(FN) is called reducible if there exists a free product decomposition FN = A1 ∗ · · · ∗ Ak ∗ C where k ≥ 1, and each Ai is a proper free factor of FN, such that φ permutes the conjugacy classes of A1, . . . , Ak. An element φ ∈ Out(FN) is called irreducible if it is not reducible. Fact: The definitions imply that φ ∈ Out(FN) is fully irreducible if and

nly if φn is irreducible for all n = 0.

The proof of Thm B produces, as a side result, the following: Cor 2. Let φ ∈ Out(FN) be hyperbolic.Then φ is irreducible if and only if φ is fully irreducible. The statement of Cor 2 is well-known to be false for non-hyperbolic automorphisms.

Ilya Kapovich (UIUC) May, 2013 23 / 26

SLIDE 124

Additional results

An element φ ∈ Out(FN) is called reducible if there exists a free product decomposition FN = A1 ∗ · · · ∗ Ak ∗ C where k ≥ 1, and each Ai is a proper free factor of FN, such that φ permutes the conjugacy classes of A1, . . . , Ak. An element φ ∈ Out(FN) is called irreducible if it is not reducible. Fact: The definitions imply that φ ∈ Out(FN) is fully irreducible if and

nly if φn is irreducible for all n = 0.

The proof of Thm B produces, as a side result, the following: Cor 2. Let φ ∈ Out(FN) be hyperbolic.Then φ is irreducible if and only if φ is fully irreducible. The statement of Cor 2 is well-known to be false for non-hyperbolic automorphisms.

Ilya Kapovich (UIUC) May, 2013 23 / 26

SLIDE 125

Additional results

An element φ ∈ Out(FN) is called reducible if there exists a free product decomposition FN = A1 ∗ · · · ∗ Ak ∗ C where k ≥ 1, and each Ai is a proper free factor of FN, such that φ permutes the conjugacy classes of A1, . . . , Ak. An element φ ∈ Out(FN) is called irreducible if it is not reducible. Fact: The definitions imply that φ ∈ Out(FN) is fully irreducible if and

nly if φn is irreducible for all n = 0.

The proof of Thm B produces, as a side result, the following: Cor 2. Let φ ∈ Out(FN) be hyperbolic.Then φ is irreducible if and only if φ is fully irreducible. The statement of Cor 2 is well-known to be false for non-hyperbolic automorphisms.

Ilya Kapovich (UIUC) May, 2013 23 / 26

SLIDE 126

Additional results

An element φ ∈ Out(FN) is called reducible if there exists a free product decomposition FN = A1 ∗ · · · ∗ Ak ∗ C where k ≥ 1, and each Ai is a proper free factor of FN, such that φ permutes the conjugacy classes of A1, . . . , Ak. An element φ ∈ Out(FN) is called irreducible if it is not reducible. Fact: The definitions imply that φ ∈ Out(FN) is fully irreducible if and

nly if φn is irreducible for all n = 0.

The proof of Thm B produces, as a side result, the following: Cor 2. Let φ ∈ Out(FN) be hyperbolic.Then φ is irreducible if and only if φ is fully irreducible. The statement of Cor 2 is well-known to be false for non-hyperbolic automorphisms.

Ilya Kapovich (UIUC) May, 2013 23 / 26

SLIDE 127

Additional results

An element φ ∈ Out(FN) is called reducible if there exists a free product decomposition FN = A1 ∗ · · · ∗ Ak ∗ C where k ≥ 1, and each Ai is a proper free factor of FN, such that φ permutes the conjugacy classes of A1, . . . , Ak. An element φ ∈ Out(FN) is called irreducible if it is not reducible. Fact: The definitions imply that φ ∈ Out(FN) is fully irreducible if and

nly if φn is irreducible for all n = 0.

In our running example, φ ∈ Aut(F(x1, x2, x3)) is indeed hyperbolic and fully irreducible. For G = Gφ = F(x1, x2, x3) ⋊φ t we have u0 ∈ Hom(G, R), u0(wtn) = n, so that u0(xi) = 0 and u0(t) = 1. The PI element u ∈ Hom(G, R) given by u(xi) = 1, u(t) = 1, is "positive" in our sense and u ∈ A. Thus ker(u) is free and finitely generated. Here is a picture of the graph Θu ⊆ Xf in this case:

Ilya Kapovich (UIUC) May, 2013 24 / 26

SLIDE 134

b b 3 3 9 5 4 2 9 14 1 d a c a d 11 3 13 12 7 4 15 6 9 8 9 11 15 4 5 6 10 1 a a b d c a b 8 w y x x y w w y x x y w v v t t t v s s q r r p v t p r w q p w y x Ilya Kapovich (UIUC) May, 2013 25 / 26

SLIDE 135

w t y x q r p s v 1 5 7 2 13 3 4 15 12 14 9 10 8 11 6

Ilya Kapovich (UIUC) May, 2013 26 / 26