GAGTA7 Conference Dynamics for the splittings of free-by-cyclic - PowerPoint PPT Presentation

Fibrations of 3-manifolds Facts: (1) We have η ∗ ∈ Hom ( G , R ) = H 1 ( G , R ) = H 1 ( M f , R ) with η ∗ ( G ) = Z and ker ( η ∗ ) = π 1 ( S ) finitely generated. (2) Fibrations M f → S 1 naturally correspond to elements u ∈ Hom ( G , R ) = H 1 ( G , R ) which are primitive integral (i.e. u ( G ) = Z ) and have f.g. kernel ker ( u ) . In this case ker ( u ) = π 1 ( S u ) where S u is the fiber for the fibration η u : M f → S 1 corresponding to u . (3)Every such u has a corresponding monodromy homeo f u : S u → S u so that M f also splits as the mapping torus of f u . Ilya Kapovich (UIUC) May, 2013 5 / 26

Fibrations of 3-manifolds Facts: (1) We have η ∗ ∈ Hom ( G , R ) = H 1 ( G , R ) = H 1 ( M f , R ) with η ∗ ( G ) = Z and ker ( η ∗ ) = π 1 ( S ) finitely generated. (2) Fibrations M f → S 1 naturally correspond to elements u ∈ Hom ( G , R ) = H 1 ( G , R ) which are primitive integral (i.e. u ( G ) = Z ) and have f.g. kernel ker ( u ) . In this case ker ( u ) = π 1 ( S u ) where S u is the fiber for the fibration η u : M f → S 1 corresponding to u . (3)Every such u has a corresponding monodromy homeo f u : S u → S u so that M f also splits as the mapping torus of f u . Ilya Kapovich (UIUC) May, 2013 5 / 26

Fibrations of 3-manifolds Facts: (1) We have η ∗ ∈ Hom ( G , R ) = H 1 ( G , R ) = H 1 ( M f , R ) with η ∗ ( G ) = Z and ker ( η ∗ ) = π 1 ( S ) finitely generated. (2) Fibrations M f → S 1 naturally correspond to elements u ∈ Hom ( G , R ) = H 1 ( G , R ) which are primitive integral (i.e. u ( G ) = Z ) and have f.g. kernel ker ( u ) . In this case ker ( u ) = π 1 ( S u ) where S u is the fiber for the fibration η u : M f → S 1 corresponding to u . (3)Every such u has a corresponding monodromy homeo f u : S u → S u so that M f also splits as the mapping torus of f u . Ilya Kapovich (UIUC) May, 2013 5 / 26

Fibrations of 3-manifolds Facts: (1) We have η ∗ ∈ Hom ( G , R ) = H 1 ( G , R ) = H 1 ( M f , R ) with η ∗ ( G ) = Z and ker ( η ∗ ) = π 1 ( S ) finitely generated. (2) Fibrations M f → S 1 naturally correspond to elements u ∈ Hom ( G , R ) = H 1 ( G , R ) which are primitive integral (i.e. u ( G ) = Z ) and have f.g. kernel ker ( u ) . In this case ker ( u ) = π 1 ( S u ) where S u is the fiber for the fibration η u : M f → S 1 corresponding to u . (3)Every such u has a corresponding monodromy homeo f u : S u → S u so that M f also splits as the mapping torus of f u . Ilya Kapovich (UIUC) May, 2013 5 / 26

Fibrations of 3-manifolds Facts: (1) We have η ∗ ∈ Hom ( G , R ) = H 1 ( G , R ) = H 1 ( M f , R ) with η ∗ ( G ) = Z and ker ( η ∗ ) = π 1 ( S ) finitely generated. (2) Fibrations M f → S 1 naturally correspond to elements u ∈ Hom ( G , R ) = H 1 ( G , R ) which are primitive integral (i.e. u ( G ) = Z ) and have f.g. kernel ker ( u ) . In this case ker ( u ) = π 1 ( S u ) where S u is the fiber for the fibration η u : M f → S 1 corresponding to u . (3)Every such u has a corresponding monodromy homeo f u : S u → S u so that M f also splits as the mapping torus of f u . Ilya Kapovich (UIUC) May, 2013 5 / 26

Fibrations of 3-manifolds (4) There exists an open cone C ⊆ Hom ( G , R ) = H 1 ( G , R ) containing η ∗ such that a primitive integral u ∈ H 1 ( G , Z ) has a f.g. kernel (and thus defines a fibration of M f if and only if u ∈ C . (5) Thus M f fibers in infinitely many ways iff b 1 ( M f ) ≥ 2. (6) If f is homotopic to pseudo-anosov then M f is hyperbolic and hence for every other primitive integral u ∈ C the monodromy f u is (up to homotopy) pseudo-Anosov. Ilya Kapovich (UIUC) May, 2013 6 / 26

Fibrations of 3-manifolds (4) There exists an open cone C ⊆ Hom ( G , R ) = H 1 ( G , R ) containing η ∗ such that a primitive integral u ∈ H 1 ( G , Z ) has a f.g. kernel (and thus defines a fibration of M f if and only if u ∈ C . (5) Thus M f fibers in infinitely many ways iff b 1 ( M f ) ≥ 2. (6) If f is homotopic to pseudo-anosov then M f is hyperbolic and hence for every other primitive integral u ∈ C the monodromy f u is (up to homotopy) pseudo-Anosov. Ilya Kapovich (UIUC) May, 2013 6 / 26

Fibrations of 3-manifolds (4) There exists an open cone C ⊆ Hom ( G , R ) = H 1 ( G , R ) containing η ∗ such that a primitive integral u ∈ H 1 ( G , Z ) has a f.g. kernel (and thus defines a fibration of M f if and only if u ∈ C . (5) Thus M f fibers in infinitely many ways iff b 1 ( M f ) ≥ 2. (6) If f is homotopic to pseudo-anosov then M f is hyperbolic and hence for every other primitive integral u ∈ C the monodromy f u is (up to homotopy) pseudo-Anosov. Ilya Kapovich (UIUC) May, 2013 6 / 26

Fibrations of 3-manifolds (4) There exists an open cone C ⊆ Hom ( G , R ) = H 1 ( G , R ) containing η ∗ such that a primitive integral u ∈ H 1 ( G , Z ) has a f.g. kernel (and thus defines a fibration of M f if and only if u ∈ C . (5) Thus M f fibers in infinitely many ways iff b 1 ( M f ) ≥ 2. (6) If f is homotopic to pseudo-anosov then M f is hyperbolic and hence for every other primitive integral u ∈ C the monodromy f u is (up to homotopy) pseudo-Anosov. Ilya Kapovich (UIUC) May, 2013 6 / 26

Fibrations of 3-manifolds (4) There exists an open cone C ⊆ Hom ( G , R ) = H 1 ( G , R ) containing η ∗ such that a primitive integral u ∈ H 1 ( G , Z ) has a f.g. kernel (and thus defines a fibration of M f if and only if u ∈ C . (5) Thus M f fibers in infinitely many ways iff b 1 ( M f ) ≥ 2. (6) If f is homotopic to pseudo-anosov then M f is hyperbolic and hence for every other primitive integral u ∈ C the monodromy f u is (up to homotopy) pseudo-Anosov. Ilya Kapovich (UIUC) May, 2013 6 / 26

Summary of the results of Thurston and Fried (1) Thurston constructed a semi-norm, called the Thurston norm n : H 1 ( 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 ( S u ) . Ilya Kapovich (UIUC) May, 2013 7 / 26

Summary of the results of Thurston and Fried (1) Thurston constructed a semi-norm, called the Thurston norm n : H 1 ( 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 ( S u ) . Ilya Kapovich (UIUC) May, 2013 7 / 26

Summary of the results of Thurston and Fried (1) Thurston constructed a semi-norm, called the Thurston norm n : H 1 ( 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 ( S u ) . Ilya Kapovich (UIUC) May, 2013 7 / 26

Summary of the results of Thurston and Fried (1) Thurston constructed a semi-norm, called the Thurston norm n : H 1 ( 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 ( S u ) . Ilya Kapovich (UIUC) May, 2013 7 / 26

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 η u : M f → S 1 and the fiber S u can be chosen so that S u is transversal to the flow Ψ t ,so that f u is the first return map,and so that f u is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B , and the cone C P over P , there exists a continuous convex function H : C P → ( 0 , ∞ ) such that for every primitive integral u ∈ C P we have H ( u ) = log λ ( f u ) where λ ( f u ) > 1 is the stretch factor of f u . Ilya Kapovich (UIUC) May, 2013 8 / 26

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 η u : M f → S 1 and the fiber S u can be chosen so that S u is transversal to the flow Ψ t ,so that f u is the first return map,and so that f u is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B , and the cone C P over P , there exists a continuous convex function H : C P → ( 0 , ∞ ) such that for every primitive integral u ∈ C P we have H ( u ) = log λ ( f u ) where λ ( f u ) > 1 is the stretch factor of f u . Ilya Kapovich (UIUC) May, 2013 8 / 26

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 η u : M f → S 1 and the fiber S u can be chosen so that S u is transversal to the flow Ψ t ,so that f u is the first return map,and so that f u is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B , and the cone C P over P , there exists a continuous convex function H : C P → ( 0 , ∞ ) such that for every primitive integral u ∈ C P we have H ( u ) = log λ ( f u ) where λ ( f u ) > 1 is the stretch factor of f u . Ilya Kapovich (UIUC) May, 2013 8 / 26

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 η u : M f → S 1 and the fiber S u can be chosen so that S u is transversal to the flow Ψ t ,so that f u is the first return map,and so that f u is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B , and the cone C P over P , there exists a continuous convex function H : C P → ( 0 , ∞ ) such that for every primitive integral u ∈ C P we have H ( u ) = log λ ( f u ) where λ ( f u ) > 1 is the stretch factor of f u . Ilya Kapovich (UIUC) May, 2013 8 / 26

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 η u : M f → S 1 and the fiber S u can be chosen so that S u is transversal to the flow Ψ t ,so that f u is the first return map,and so that f u is already pseudo-anosov (without homotopy). (5) Fried also proved that for any fibered face P of B , and the cone C P over P , there exists a continuous convex function H : C P → ( 0 , ∞ ) such that for every primitive integral u ∈ C P we have H ( u ) = log λ ( f u ) where λ ( f u ) > 1 is the stretch factor of f u . Ilya Kapovich (UIUC) May, 2013 8 / 26

Free-by-cyclic groups Let F N be free of rank N ≥ 2 and let φ ∈ Aut ( F N ) (or Out ( F N ) ).We can form the mapping torus group G = G φ = F N ⋊ φ Z = � F N , t | twt − 1 = φ ( w ) , w ∈ F N � We get a natural epimorphism u 0 : G → Z , u ( t n w ) = n , with ker ( u ) = F N .Thus u 0 ∈ Hom ( G , R ) = H 1 ( G , R ) is a primitive integral (PI) element (i.e. u 0 ( G ) = Z ) with a f.g. kernel. It is known, for general cohomological reasons, that for any PI element u ∈ Hom ( G , R ) = H 1 ( G , R ) if ker ( u ) is finitely generated then ker ( u ) is actually free and we get a splitting G = ker ( u ) ⋊ φ u Z of G as a (f.g. free)-by-(infinite cyclic) group where φ u ∈ Aut ( ker ( u )) is the associated monodromy . Any other splitting of G as a (f.g. free)-by-(infinite cyclic) group arises in this way. Ilya Kapovich (UIUC) May, 2013 9 / 26

Free-by-cyclic groups Let F N be free of rank N ≥ 2 and let φ ∈ Aut ( F N ) (or Out ( F N ) ).We can form the mapping torus group G = G φ = F N ⋊ φ Z = � F N , t | twt − 1 = φ ( w ) , w ∈ F N � We get a natural epimorphism u 0 : G → Z , u ( t n w ) = n , with ker ( u ) = F N .Thus u 0 ∈ Hom ( G , R ) = H 1 ( G , R ) is a primitive integral (PI) element (i.e. u 0 ( G ) = Z ) with a f.g. kernel. It is known, for general cohomological reasons, that for any PI element u ∈ Hom ( G , R ) = H 1 ( G , R ) if ker ( u ) is finitely generated then ker ( u ) is actually free and we get a splitting G = ker ( u ) ⋊ φ u Z of G as a (f.g. free)-by-(infinite cyclic) group where φ u ∈ Aut ( ker ( u )) is the associated monodromy . Any other splitting of G as a (f.g. free)-by-(infinite cyclic) group arises in this way. Ilya Kapovich (UIUC) May, 2013 9 / 26

Free-by-cyclic groups Let F N be free of rank N ≥ 2 and let φ ∈ Aut ( F N ) (or Out ( F N ) ).We can form the mapping torus group G = G φ = F N ⋊ φ Z = � F N , t | twt − 1 = φ ( w ) , w ∈ F N � We get a natural epimorphism u 0 : G → Z , u ( t n w ) = n , with ker ( u ) = F N .Thus u 0 ∈ Hom ( G , R ) = H 1 ( G , R ) is a primitive integral (PI) element (i.e. u 0 ( G ) = Z ) with a f.g. kernel. It is known, for general cohomological reasons, that for any PI element u ∈ Hom ( G , R ) = H 1 ( G , R ) if ker ( u ) is finitely generated then ker ( u ) is actually free and we get a splitting G = ker ( u ) ⋊ φ u Z of G as a (f.g. free)-by-(infinite cyclic) group where φ u ∈ Aut ( ker ( u )) is the associated monodromy . Any other splitting of G as a (f.g. free)-by-(infinite cyclic) group arises in this way. Ilya Kapovich (UIUC) May, 2013 9 / 26

Free-by-cyclic groups Let F N be free of rank N ≥ 2 and let φ ∈ Aut ( F N ) (or Out ( F N ) ).We can form the mapping torus group G = G φ = F N ⋊ φ Z = � F N , t | twt − 1 = φ ( w ) , w ∈ F N � We get a natural epimorphism u 0 : G → Z , u ( t n w ) = n , with ker ( u ) = F N .Thus u 0 ∈ Hom ( G , R ) = H 1 ( G , R ) is a primitive integral (PI) element (i.e. u 0 ( G ) = Z ) with a f.g. kernel. It is known, for general cohomological reasons, that for any PI element u ∈ Hom ( G , R ) = H 1 ( G , R ) if ker ( u ) is finitely generated then ker ( u ) is actually free and we get a splitting G = ker ( u ) ⋊ φ u Z of G as a (f.g. free)-by-(infinite cyclic) group where φ u ∈ Aut ( ker ( u )) is the associated monodromy . Any other splitting of G as a (f.g. free)-by-(infinite cyclic) group arises in this way. Ilya Kapovich (UIUC) May, 2013 9 / 26

Free-by-cyclic groups Let F N be free of rank N ≥ 2 and let φ ∈ Aut ( F N ) (or Out ( F N ) ).We can form the mapping torus group G = G φ = F N ⋊ φ Z = � F N , t | twt − 1 = φ ( w ) , w ∈ F N � We get a natural epimorphism u 0 : G → Z , u ( t n w ) = n , with ker ( u ) = F N .Thus u 0 ∈ Hom ( G , R ) = H 1 ( G , R ) is a primitive integral (PI) element (i.e. u 0 ( G ) = Z ) with a f.g. kernel. It is known, for general cohomological reasons, that for any PI element u ∈ Hom ( G , R ) = H 1 ( G , R ) if ker ( u ) is finitely generated then ker ( u ) is actually free and we get a splitting G = ker ( u ) ⋊ φ u Z of G as a (f.g. free)-by-(infinite cyclic) group where φ u ∈ Aut ( ker ( u )) is the associated monodromy . Any other splitting of G as a (f.g. free)-by-(infinite cyclic) group arises in this way. Ilya Kapovich (UIUC) May, 2013 9 / 26

Free-by-cyclic groups Let F N be free of rank N ≥ 2 and let φ ∈ Aut ( F N ) (or Out ( F N ) ).We can form the mapping torus group G = G φ = F N ⋊ φ Z = � F N , t | twt − 1 = φ ( w ) , w ∈ F N � We get a natural epimorphism u 0 : G → Z , u ( t n w ) = n , with ker ( u ) = F N .Thus u 0 ∈ Hom ( G , R ) = H 1 ( G , R ) is a primitive integral (PI) element (i.e. u 0 ( G ) = Z ) with a f.g. kernel. It is known, for general cohomological reasons, that for any PI element u ∈ Hom ( G , R ) = H 1 ( G , R ) if ker ( u ) is finitely generated then ker ( u ) is actually free and we get a splitting G = ker ( u ) ⋊ φ u Z of G as a (f.g. free)-by-(infinite cyclic) group where φ u ∈ Aut ( ker ( u )) is the associated monodromy . Any other splitting of G as a (f.g. free)-by-(infinite cyclic) group arises in this way. Ilya Kapovich (UIUC) May, 2013 9 / 26

Free-by-cyclic groups Let F N be free of rank N ≥ 2 and let φ ∈ Aut ( F N ) (or Out ( F N ) ).We can form the mapping torus group G = G φ = F N ⋊ φ Z = � F N , t | twt − 1 = φ ( w ) , w ∈ F N � We get a natural epimorphism u 0 : G → Z , u ( t n w ) = n , with ker ( u ) = F N .Thus u 0 ∈ Hom ( G , R ) = H 1 ( G , R ) is a primitive integral (PI) element (i.e. u 0 ( G ) = Z ) with a f.g. kernel. It is known, for general cohomological reasons, that for any PI element u ∈ Hom ( G , R ) = H 1 ( G , R ) if ker ( u ) is finitely generated then ker ( u ) is actually free and we get a splitting G = ker ( u ) ⋊ φ u Z of G as a (f.g. free)-by-(infinite cyclic) group where φ u ∈ Aut ( ker ( u )) is the associated monodromy . Any other splitting of G as a (f.g. free)-by-(infinite cyclic) group arises in this way. Ilya Kapovich (UIUC) May, 2013 9 / 26

Free-by-cyclic groups It is also known (e.g. by the theory of the Bieri-Neumann-Strebel invariant of f.p. groups) that there exists an open cone C G ⊆ Hom ( G , R ) = H 1 ( G , R ) , containing u 0 , such that a PI element u ∈ H 1 ( G , R ) has a f.g. kernel (and thus defines a free-by-cyclic splitting of G ) iff u ∈ C G . Thus G φ splits as a (f.g. free)-by-(infinite cyclic) group in infinitely many ways iff b 1 ( G φ ) ≥ 2. Ilya Kapovich (UIUC) May, 2013 10 / 26

Free-by-cyclic groups It is also known (e.g. by the theory of the Bieri-Neumann-Strebel invariant of f.p. groups) that there exists an open cone C G ⊆ Hom ( G , R ) = H 1 ( G , R ) , containing u 0 , such that a PI element u ∈ H 1 ( G , R ) has a f.g. kernel (and thus defines a free-by-cyclic splitting of G ) iff u ∈ C G . Thus G φ splits as a (f.g. free)-by-(infinite cyclic) group in infinitely many ways iff b 1 ( G φ ) ≥ 2. Ilya Kapovich (UIUC) May, 2013 10 / 26

Free-by-cyclic groups It is also known (e.g. by the theory of the Bieri-Neumann-Strebel invariant of f.p. groups) that there exists an open cone C G ⊆ Hom ( G , R ) = H 1 ( G , R ) , containing u 0 , such that a PI element u ∈ H 1 ( G , R ) has a f.g. kernel (and thus defines a free-by-cyclic splitting of G ) iff u ∈ C G . Thus G φ splits as a (f.g. free)-by-(infinite cyclic) group in infinitely many ways iff b 1 ( G φ ) ≥ 2. Ilya Kapovich (UIUC) May, 2013 10 / 26

Free-by-cyclic groups It is also known (e.g. by the theory of the Bieri-Neumann-Strebel invariant of f.p. groups) that there exists an open cone C G ⊆ Hom ( G , R ) = H 1 ( G , R ) , containing u 0 , such that a PI element u ∈ H 1 ( G , R ) has a f.g. kernel (and thus defines a free-by-cyclic splitting of G ) iff u ∈ C G . Thus G φ splits as a (f.g. free)-by-(infinite cyclic) group in infinitely many ways iff b 1 ( G φ ) ≥ 2. Ilya Kapovich (UIUC) May, 2013 10 / 26

Free group automorphisms The notion of being pseudo-anosov has two distinct analogs in the Out ( F N ) context. The first such analog is geometric in nature: An element φ ∈ Out ( F N ) is scalled hyperbolic if G φ = F N ⋊ φ Z is word-hyperbolic. The second, much more important, analog is dynamical in nature: An element φ ∈ Out ( F N ) is called fully irreducible if no positive power of φ preserves the conjugacy class of a proper free factor of F N . The best analog of p.A. is an element of Out ( F N ) which is both hyperbolic and fully irreducible. Ilya Kapovich (UIUC) May, 2013 11 / 26

Free group automorphisms The notion of being pseudo-anosov has two distinct analogs in the Out ( F N ) context. The first such analog is geometric in nature: An element φ ∈ Out ( F N ) is scalled hyperbolic if G φ = F N ⋊ φ Z is word-hyperbolic. The second, much more important, analog is dynamical in nature: An element φ ∈ Out ( F N ) is called fully irreducible if no positive power of φ preserves the conjugacy class of a proper free factor of F N . The best analog of p.A. is an element of Out ( F N ) which is both hyperbolic and fully irreducible. Ilya Kapovich (UIUC) May, 2013 11 / 26

Free group automorphisms The notion of being pseudo-anosov has two distinct analogs in the Out ( F N ) context. The first such analog is geometric in nature: An element φ ∈ Out ( F N ) is scalled hyperbolic if G φ = F N ⋊ φ Z is word-hyperbolic. The second, much more important, analog is dynamical in nature: An element φ ∈ Out ( F N ) is called fully irreducible if no positive power of φ preserves the conjugacy class of a proper free factor of F N . The best analog of p.A. is an element of Out ( F N ) which is both hyperbolic and fully irreducible. Ilya Kapovich (UIUC) May, 2013 11 / 26

Free group automorphisms The notion of being pseudo-anosov has two distinct analogs in the Out ( F N ) context. The first such analog is geometric in nature: An element φ ∈ Out ( F N ) is scalled hyperbolic if G φ = F N ⋊ φ Z is word-hyperbolic. The second, much more important, analog is dynamical in nature: An element φ ∈ Out ( F N ) is called fully irreducible if no positive power of φ preserves the conjugacy class of a proper free factor of F N . The best analog of p.A. is an element of Out ( F N ) which is both hyperbolic and fully irreducible. Ilya Kapovich (UIUC) May, 2013 11 / 26

Free group automorphisms The notion of being pseudo-anosov has two distinct analogs in the Out ( F N ) context. The first such analog is geometric in nature: An element φ ∈ Out ( F N ) is scalled hyperbolic if G φ = F N ⋊ φ Z is word-hyperbolic. The second, much more important, analog is dynamical in nature: An element φ ∈ Out ( F N ) is called fully irreducible if no positive power of φ preserves the conjugacy class of a proper free factor of F N . The best analog of p.A. is an element of Out ( F N ) which is both hyperbolic and fully irreducible. Ilya Kapovich (UIUC) May, 2013 11 / 26

Free group automorphisms The notion of being pseudo-anosov has two distinct analogs in the Out ( F N ) context. The first such analog is geometric in nature: An element φ ∈ Out ( F N ) is scalled hyperbolic if G φ = F N ⋊ φ Z is word-hyperbolic. The second, much more important, analog is dynamical in nature: An element φ ∈ Out ( F N ) is called fully irreducible if no positive power of φ preserves the conjugacy class of a proper free factor of F N . The best analog of p.A. is an element of Out ( F N ) which is both hyperbolic and fully irreducible. Ilya Kapovich (UIUC) May, 2013 11 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms To work with an element φ of Out ( F N ) one usually uses a topological representative of φ , that is, a "nice" continuous graph-map f : Γ → Γ such that: * Γ is a finite connected graph with F N = π 1 (Γ) * f is a homotopy equivalence such that f ∗ = φ (in the appropriate sense) * f ( V Γ) ⊆ V Γ * For every edge e of Γ f ( e ) is a PL edge-path in Γ . A map f as above is called a train-track map if for every e ∈ E Γ and every n ≥ 1 the path f n ( e ) is immersed (i.e. contains no backtracks). Fact: If φ ∈ Out ( F N ) is fully irreducible then φ admits a train-track rep f : Γ → Γ with an irreducible transition matrix A ( f ) . The spectral radius λ ( f ) ≥ 1 of A ( f ) , called the stretch factor of φ , does not depend on the choice of such f . We denote λ ( φ ) = λ ( f ) . Ilya Kapovich (UIUC) May, 2013 12 / 26

Free group automorphisms Note: If φ ∈ Out ( F N ) is hyperbolic then G = G φ = F N ⋊ φ Z is word-hyperbolic.Therefore for any PI element u ∈ C G and the corresponding splitting G = ker ( u ) ⋊ φ u Z the automorphism φ u of ker ( u ) is again hyperbolic. However, if φ is fully irreducible, there is no longer any ’a priori’ reason why φ u must also be fully irreducible. Moreover, there are counterexamples to this conclusion when φ is fully irreducible but not hyperbolic. Ilya Kapovich (UIUC) May, 2013 13 / 26

Free group automorphisms Note: If φ ∈ Out ( F N ) is hyperbolic then G = G φ = F N ⋊ φ Z is word-hyperbolic.Therefore for any PI element u ∈ C G and the corresponding splitting G = ker ( u ) ⋊ φ u Z the automorphism φ u of ker ( u ) is again hyperbolic. However, if φ is fully irreducible, there is no longer any ’a priori’ reason why φ u must also be fully irreducible. Moreover, there are counterexamples to this conclusion when φ is fully irreducible but not hyperbolic. Ilya Kapovich (UIUC) May, 2013 13 / 26

Free group automorphisms Note: If φ ∈ Out ( F N ) is hyperbolic then G = G φ = F N ⋊ φ Z is word-hyperbolic.Therefore for any PI element u ∈ C G and the corresponding splitting G = ker ( u ) ⋊ φ u Z the automorphism φ u of ker ( u ) is again hyperbolic. However, if φ is fully irreducible, there is no longer any ’a priori’ reason why φ u must also be fully irreducible. Moreover, there are counterexamples to this conclusion when φ is fully irreducible but not hyperbolic. Ilya Kapovich (UIUC) May, 2013 13 / 26

Free group automorphisms Note: If φ ∈ Out ( F N ) is hyperbolic then G = G φ = F N ⋊ φ Z is word-hyperbolic.Therefore for any PI element u ∈ C G and the corresponding splitting G = ker ( u ) ⋊ φ u Z the automorphism φ u of ker ( u ) is again hyperbolic. However, if φ is fully irreducible, there is no longer any ’a priori’ reason why φ u must also be fully irreducible. Moreover, there are counterexamples to this conclusion when φ is fully irreducible but not hyperbolic. Ilya Kapovich (UIUC) May, 2013 13 / 26

Free group automorphisms Note: If φ ∈ Out ( F N ) is hyperbolic then G = G φ = F N ⋊ φ Z is word-hyperbolic.Therefore for any PI element u ∈ C G and the corresponding splitting G = ker ( u ) ⋊ φ u Z the automorphism φ u of ker ( u ) is again hyperbolic. However, if φ is fully irreducible, there is no longer any ’a priori’ reason why φ u must also be fully irreducible. Moreover, there are counterexamples to this conclusion when φ is fully irreducible but not hyperbolic. Ilya Kapovich (UIUC) May, 2013 13 / 26

Summary of the main results Thm A. Given any φ ∈ Out ( F N ) and a "nice" top rep f : Γ → Γ of φ (which always exists and is easy to find), we construct a folded mapping torus X f of f with the following properties: (1) X f is a compact 2-complex which is a K ( G , 1 ) for G = G φ . Moreover, X f inherits a semi-flow Ψ t from the mapping torus M f = Γ × [ 0 , 1 ] / ( f ( x ) , 0 ) ∼ ( x , 1 ) . (2) We construct an open cone A ⊆ H 1 ( G , R ) containing u 0 , such that every PI element u ∈ A has ker ( u ) f.g. (and hence free) (3) Given any PI element u ∈ A we construct a "fibration" f u : X f → S 1 such that Θ u := f − 1 u ( 0 ) is a finite graph which is a section of Ψ t , that π 1 (Θ u ) = ker ( u ) with the inclusion Θ u → X f being π 1 -injective, and such that the first return map f u : Θ u → Θ u is a homotopy equivalence representing the monodromy φ u of the splitting G = ker ( u ) ⋊ φ u Z (4) We construct a cellular 1-cycle ǫ ∈ H 1 ( X f , R ) such that for every PI element u ∈ A ǫ ( u ) = − χ (Γ) = rk ( ker ( u )) − 1 . Ilya Kapovich (UIUC) May, 2013 14 / 26

Summary of the main results Thm A. Given any φ ∈ Out ( F N ) and a "nice" top rep f : Γ → Γ of φ (which always exists and is easy to find), we construct a folded mapping torus X f of f with the following properties: (1) X f is a compact 2-complex which is a K ( G , 1 ) for G = G φ . Moreover, X f inherits a semi-flow Ψ t from the mapping torus M f = Γ × [ 0 , 1 ] / ( f ( x ) , 0 ) ∼ ( x , 1 ) . (2) We construct an open cone A ⊆ H 1 ( G , R ) containing u 0 , such that every PI element u ∈ A has ker ( u ) f.g. (and hence free) (3) Given any PI element u ∈ A we construct a "fibration" f u : X f → S 1 such that Θ u := f − 1 u ( 0 ) is a finite graph which is a section of Ψ t , that π 1 (Θ u ) = ker ( u ) with the inclusion Θ u → X f being π 1 -injective, and such that the first return map f u : Θ u → Θ u is a homotopy equivalence representing the monodromy φ u of the splitting G = ker ( u ) ⋊ φ u Z (4) We construct a cellular 1-cycle ǫ ∈ H 1 ( X f , R ) such that for every PI element u ∈ A ǫ ( u ) = − χ (Γ) = rk ( ker ( u )) − 1 . Ilya Kapovich (UIUC) May, 2013 14 / 26

Summary of the main results Thm A. Given any φ ∈ Out ( F N ) and a "nice" top rep f : Γ → Γ of φ (which always exists and is easy to find), we construct a folded mapping torus X f of f with the following properties: (1) X f is a compact 2-complex which is a K ( G , 1 ) for G = G φ . Moreover, X f inherits a semi-flow Ψ t from the mapping torus M f = Γ × [ 0 , 1 ] / ( f ( x ) , 0 ) ∼ ( x , 1 ) . (2) We construct an open cone A ⊆ H 1 ( G , R ) containing u 0 , such that every PI element u ∈ A has ker ( u ) f.g. (and hence free) (3) Given any PI element u ∈ A we construct a "fibration" f u : X f → S 1 such that Θ u := f − 1 u ( 0 ) is a finite graph which is a section of Ψ t , that π 1 (Θ u ) = ker ( u ) with the inclusion Θ u → X f being π 1 -injective, and such that the first return map f u : Θ u → Θ u is a homotopy equivalence representing the monodromy φ u of the splitting G = ker ( u ) ⋊ φ u Z (4) We construct a cellular 1-cycle ǫ ∈ H 1 ( X f , R ) such that for every PI element u ∈ A ǫ ( u ) = − χ (Γ) = rk ( ker ( u )) − 1 . Ilya Kapovich (UIUC) May, 2013 14 / 26

Summary of the main results Thm A. Given any φ ∈ Out ( F N ) and a "nice" top rep f : Γ → Γ of φ (which always exists and is easy to find), we construct a folded mapping torus X f of f with the following properties: (1) X f is a compact 2-complex which is a K ( G , 1 ) for G = G φ . Moreover, X f inherits a semi-flow Ψ t from the mapping torus M f = Γ × [ 0 , 1 ] / ( f ( x ) , 0 ) ∼ ( x , 1 ) . (2) We construct an open cone A ⊆ H 1 ( G , R ) containing u 0 , such that every PI element u ∈ A has ker ( u ) f.g. (and hence free) (3) Given any PI element u ∈ A we construct a "fibration" f u : X f → S 1 such that Θ u := f − 1 u ( 0 ) is a finite graph which is a section of Ψ t , that π 1 (Θ u ) = ker ( u ) with the inclusion Θ u → X f being π 1 -injective, and such that the first return map f u : Θ u → Θ u is a homotopy equivalence representing the monodromy φ u of the splitting G = ker ( u ) ⋊ φ u Z (4) We construct a cellular 1-cycle ǫ ∈ H 1 ( X f , R ) such that for every PI element u ∈ A ǫ ( u ) = − χ (Γ) = rk ( ker ( u )) − 1 . Ilya Kapovich (UIUC) May, 2013 14 / 26

Summary of the main results Thm A. Given any φ ∈ Out ( F N ) and a "nice" top rep f : Γ → Γ of φ (which always exists and is easy to find), we construct a folded mapping torus X f of f with the following properties: (1) X f is a compact 2-complex which is a K ( G , 1 ) for G = G φ . Moreover, X f inherits a semi-flow Ψ t from the mapping torus M f = Γ × [ 0 , 1 ] / ( f ( x ) , 0 ) ∼ ( x , 1 ) . (2) We construct an open cone A ⊆ H 1 ( G , R ) containing u 0 , such that every PI element u ∈ A has ker ( u ) f.g. (and hence free) (3) Given any PI element u ∈ A we construct a "fibration" f u : X f → S 1 such that Θ u := f − 1 u ( 0 ) is a finite graph which is a section of Ψ t , that π 1 (Θ u ) = ker ( u ) with the inclusion Θ u → X f being π 1 -injective, and such that the first return map f u : Θ u → Θ u is a homotopy equivalence representing the monodromy φ u of the splitting G = ker ( u ) ⋊ φ u Z (4) We construct a cellular 1-cycle ǫ ∈ H 1 ( X f , R ) such that for every PI element u ∈ A ǫ ( u ) = − χ (Γ) = rk ( ker ( u )) − 1 . Ilya Kapovich (UIUC) May, 2013 14 / 26

Summary of the main results Thm A. Given any φ ∈ Out ( F N ) and a "nice" top rep f : Γ → Γ of φ (which always exists and is easy to find), we construct a folded mapping torus X f of f with the following properties: (1) X f is a compact 2-complex which is a K ( G , 1 ) for G = G φ . Moreover, X f inherits a semi-flow Ψ t from the mapping torus M f = Γ × [ 0 , 1 ] / ( f ( x ) , 0 ) ∼ ( x , 1 ) . (2) We construct an open cone A ⊆ H 1 ( G , R ) containing u 0 , such that every PI element u ∈ A has ker ( u ) f.g. (and hence free) (3) Given any PI element u ∈ A we construct a "fibration" f u : X f → S 1 such that Θ u := f − 1 u ( 0 ) is a finite graph which is a section of Ψ t , that π 1 (Θ u ) = ker ( u ) with the inclusion Θ u → X f being π 1 -injective, and such that the first return map f u : Θ u → Θ u is a homotopy equivalence representing the monodromy φ u of the splitting G = ker ( u ) ⋊ φ u Z (4) We construct a cellular 1-cycle ǫ ∈ H 1 ( X f , R ) such that for every PI element u ∈ A ǫ ( u ) = − χ (Γ) = rk ( ker ( u )) − 1 . Ilya Kapovich (UIUC) May, 2013 14 / 26

Summary of the main results Thm A. Given any φ ∈ Out ( F N ) and a "nice" top rep f : Γ → Γ of φ (which always exists and is easy to find), we construct a folded mapping torus X f of f with the following properties: (1) X f is a compact 2-complex which is a K ( G , 1 ) for G = G φ . Moreover, X f inherits a semi-flow Ψ t from the mapping torus M f = Γ × [ 0 , 1 ] / ( f ( x ) , 0 ) ∼ ( x , 1 ) . (2) We construct an open cone A ⊆ H 1 ( G , R ) containing u 0 , such that every PI element u ∈ A has ker ( u ) f.g. (and hence free) (3) Given any PI element u ∈ A we construct a "fibration" f u : X f → S 1 such that Θ u := f − 1 u ( 0 ) is a finite graph which is a section of Ψ t , that π 1 (Θ u ) = ker ( u ) with the inclusion Θ u → X f being π 1 -injective, and such that the first return map f u : Θ u → Θ u is a homotopy equivalence representing the monodromy φ u of the splitting G = ker ( u ) ⋊ φ u Z (4) We construct a cellular 1-cycle ǫ ∈ H 1 ( X f , R ) such that for every PI element u ∈ A ǫ ( u ) = − χ (Γ) = rk ( ker ( u )) − 1 . Ilya Kapovich (UIUC) May, 2013 14 / 26

Summary of the main results Thm A. Given any φ ∈ Out ( F N ) and a "nice" top rep f : Γ → Γ of φ (which always exists and is easy to find), we construct a folded mapping torus X f of f with the following properties: (1) X f is a compact 2-complex which is a K ( G , 1 ) for G = G φ . Moreover, X f inherits a semi-flow Ψ t from the mapping torus M f = Γ × [ 0 , 1 ] / ( f ( x ) , 0 ) ∼ ( x , 1 ) . (2) We construct an open cone A ⊆ H 1 ( G , R ) containing u 0 , such that every PI element u ∈ A has ker ( u ) f.g. (and hence free) (3) Given any PI element u ∈ A we construct a "fibration" f u : X f → S 1 such that Θ u := f − 1 u ( 0 ) is a finite graph which is a section of Ψ t , that π 1 (Θ u ) = ker ( u ) with the inclusion Θ u → X f being π 1 -injective, and such that the first return map f u : Θ u → Θ u is a homotopy equivalence representing the monodromy φ u of the splitting G = ker ( u ) ⋊ φ u Z (4) We construct a cellular 1-cycle ǫ ∈ H 1 ( X f , R ) such that for every PI element u ∈ A ǫ ( u ) = − χ (Γ) = rk ( ker ( u )) − 1 . Ilya Kapovich (UIUC) May, 2013 14 / 26

Summary of the main results Notes: (1) Gautero had earlier constructed another complex Y , with a semi-flow, such that (3) above holds. (2) The existence of a linear functional τ : A → R satisfying (4) can be extracted, with a bit of work, from the results of McMullen, Button and Dunfield about the Alexander norm on f.p. groups.Thm A provides a geometric realization of the Alexander norm on A via the 1-cycle ǫ ∈ H 1 ( G φ , R ) . Thus for every PI element u ∈ A we get || u || A = ǫ ( u ) , where || . || A is the Alexander norm. Ilya Kapovich (UIUC) May, 2013 15 / 26

Summary of the main results Notes: (1) Gautero had earlier constructed another complex Y , with a semi-flow, such that (3) above holds. (2) The existence of a linear functional τ : A → R satisfying (4) can be extracted, with a bit of work, from the results of McMullen, Button and Dunfield about the Alexander norm on f.p. groups.Thm A provides a geometric realization of the Alexander norm on A via the 1-cycle ǫ ∈ H 1 ( G φ , R ) . Thus for every PI element u ∈ A we get || u || A = ǫ ( u ) , where || . || A is the Alexander norm. Ilya Kapovich (UIUC) May, 2013 15 / 26

Summary of the main results Notes: (1) Gautero had earlier constructed another complex Y , with a semi-flow, such that (3) above holds. (2) The existence of a linear functional τ : A → R satisfying (4) can be extracted, with a bit of work, from the results of McMullen, Button and Dunfield about the Alexander norm on f.p. groups.Thm A provides a geometric realization of the Alexander norm on A via the 1-cycle ǫ ∈ H 1 ( G φ , R ) . Thus for every PI element u ∈ A we get || u || A = ǫ ( u ) , where || . || A is the Alexander norm. Ilya Kapovich (UIUC) May, 2013 15 / 26

Summary of the main results Notes: (1) Gautero had earlier constructed another complex Y , with a semi-flow, such that (3) above holds. (2) The existence of a linear functional τ : A → R satisfying (4) can be extracted, with a bit of work, from the results of McMullen, Button and Dunfield about the Alexander norm on f.p. groups.Thm A provides a geometric realization of the Alexander norm on A via the 1-cycle ǫ ∈ H 1 ( G φ , R ) . Thus for every PI element u ∈ A we get || u || A = ǫ ( u ) , where || . || A is the Alexander norm. Ilya Kapovich (UIUC) May, 2013 15 / 26

Summary of the main results Notes: (1) Gautero had earlier constructed another complex Y , with a semi-flow, such that (3) above holds. (2) The existence of a linear functional τ : A → R satisfying (4) can be extracted, with a bit of work, from the results of McMullen, Button and Dunfield about the Alexander norm on f.p. groups.Thm A provides a geometric realization of the Alexander norm on A via the 1-cycle ǫ ∈ H 1 ( G φ , R ) . Thus for every PI element u ∈ A we get || u || A = ǫ ( u ) , where || . || A is the Alexander norm. Ilya Kapovich (UIUC) May, 2013 15 / 26

Summary of the main results Thm B. Let φ ∈ Out ( F N ) be hyperbolic and fully irreducible, let f : Γ → Γ be a train-track map representing φ and let u 0 : G φ → Z be the homomorphism associated to the splitting G = G φ = F N ⋊ φ Z . Let A ⊆ H 1 ( G , R ) = Hom ( G , R ) , X f and Ψ t be as constructed in Thm A. Then: (1) For any PI element u ∈ A the section Θ u in Thm A can be chosen so that the first-return map f u : Θ u → Θ u is a train-track map (w/o further homotopy) representing the monodromy φ u ∈ Out ( ker ( u )) . (2) For every PI element u ∈ A the monodromy φ u is again hyperbolic and fully irreducible. (3) There exists a continuous, homogeneous of degree − 1, convex function H : A → ( 0 , ∞ ) such that for every PI element u ∈ A we have H ( u ) = log λ ( f u ) = log λ ( φ u ) and H ( u ) is equal to the topological entropy of f u : Θ u → Θ u . Ilya Kapovich (UIUC) May, 2013 16 / 26

Summary of the main results Thm B. Let φ ∈ Out ( F N ) be hyperbolic and fully irreducible, let f : Γ → Γ be a train-track map representing φ and let u 0 : G φ → Z be the homomorphism associated to the splitting G = G φ = F N ⋊ φ Z . Let A ⊆ H 1 ( G , R ) = Hom ( G , R ) , X f and Ψ t be as constructed in Thm A. Then: (1) For any PI element u ∈ A the section Θ u in Thm A can be chosen so that the first-return map f u : Θ u → Θ u is a train-track map (w/o further homotopy) representing the monodromy φ u ∈ Out ( ker ( u )) . (2) For every PI element u ∈ A the monodromy φ u is again hyperbolic and fully irreducible. (3) There exists a continuous, homogeneous of degree − 1, convex function H : A → ( 0 , ∞ ) such that for every PI element u ∈ A we have H ( u ) = log λ ( f u ) = log λ ( φ u ) and H ( u ) is equal to the topological entropy of f u : Θ u → Θ u . Ilya Kapovich (UIUC) May, 2013 16 / 26

Summary of the main results Thm B. Let φ ∈ Out ( F N ) be hyperbolic and fully irreducible, let f : Γ → Γ be a train-track map representing φ and let u 0 : G φ → Z be the homomorphism associated to the splitting G = G φ = F N ⋊ φ Z . Let A ⊆ H 1 ( G , R ) = Hom ( G , R ) , X f and Ψ t be as constructed in Thm A. Then: (1) For any PI element u ∈ A the section Θ u in Thm A can be chosen so that the first-return map f u : Θ u → Θ u is a train-track map (w/o further homotopy) representing the monodromy φ u ∈ Out ( ker ( u )) . (2) For every PI element u ∈ A the monodromy φ u is again hyperbolic and fully irreducible. (3) There exists a continuous, homogeneous of degree − 1, convex function H : A → ( 0 , ∞ ) such that for every PI element u ∈ A we have H ( u ) = log λ ( f u ) = log λ ( φ u ) and H ( u ) is equal to the topological entropy of f u : Θ u → Θ u . Ilya Kapovich (UIUC) May, 2013 16 / 26

Summary of the main results Thm B. Let φ ∈ Out ( F N ) be hyperbolic and fully irreducible, let f : Γ → Γ be a train-track map representing φ and let u 0 : G φ → Z be the homomorphism associated to the splitting G = G φ = F N ⋊ φ Z . Let A ⊆ H 1 ( G , R ) = Hom ( G , R ) , X f and Ψ t be as constructed in Thm A. Then: (1) For any PI element u ∈ A the section Θ u in Thm A can be chosen so that the first-return map f u : Θ u → Θ u is a train-track map (w/o further homotopy) representing the monodromy φ u ∈ Out ( ker ( u )) . (2) For every PI element u ∈ A the monodromy φ u is again hyperbolic and fully irreducible. (3) There exists a continuous, homogeneous of degree − 1, convex function H : A → ( 0 , ∞ ) such that for every PI element u ∈ A we have H ( u ) = log λ ( f u ) = log λ ( φ u ) and H ( u ) is equal to the topological entropy of f u : Θ u → Θ u . Ilya Kapovich (UIUC) May, 2013 16 / 26

Summary of the main results Thm B. Let φ ∈ Out ( F N ) be hyperbolic and fully irreducible, let f : Γ → Γ be a train-track map representing φ and let u 0 : G φ → Z be the homomorphism associated to the splitting G = G φ = F N ⋊ φ Z . Let A ⊆ H 1 ( G , R ) = Hom ( G , R ) , X f and Ψ t be as constructed in Thm A. Then: (1) For any PI element u ∈ A the section Θ u in Thm A can be chosen so that the first-return map f u : Θ u → Θ u is a train-track map (w/o further homotopy) representing the monodromy φ u ∈ Out ( ker ( u )) . (2) For every PI element u ∈ A the monodromy φ u is again hyperbolic and fully irreducible. (3) There exists a continuous, homogeneous of degree − 1, convex function H : A → ( 0 , ∞ ) such that for every PI element u ∈ A we have H ( u ) = log λ ( f u ) = log λ ( φ u ) and H ( u ) is equal to the topological entropy of f u : Θ u → Θ u . Ilya Kapovich (UIUC) May, 2013 16 / 26

Summary of the main results Thm B. Let φ ∈ Out ( F N ) be hyperbolic and fully irreducible, let f : Γ → Γ be a train-track map representing φ and let u 0 : G φ → Z be the homomorphism associated to the splitting G = G φ = F N ⋊ φ Z . Let A ⊆ H 1 ( G , R ) = Hom ( G , R ) , X f and Ψ t be as constructed in Thm A. Then: (1) For any PI element u ∈ A the section Θ u in Thm A can be chosen so that the first-return map f u : Θ u → Θ u is a train-track map (w/o further homotopy) representing the monodromy φ u ∈ Out ( ker ( u )) . (2) For every PI element u ∈ A the monodromy φ u is again hyperbolic and fully irreducible. (3) There exists a continuous, homogeneous of degree − 1, convex function H : A → ( 0 , ∞ ) such that for every PI element u ∈ A we have H ( u ) = log λ ( f u ) = log λ ( φ u ) and H ( u ) is equal to the topological entropy of f u : Θ u → Θ u . Ilya Kapovich (UIUC) May, 2013 16 / 26

Folded mapping torus About the construction of X f in Thm A: (1) Start with a "nice" f : Γ → Γ representing φ . Ex: Let Γ denote the graph in Figure 17 below. Γ has four edges, oriented as shown and labeled { a , b , c , d } = E + Γ . Consider a graph-map f : Γ → Γ under which the edges of Γ map to the combinatorial edge paths f ( a ) = d , f ( b ) = a , f ( c ) = b − 1 a , and f ( d ) = ba − 1 db − 1 ac . a d ι Γ ∆ b b a c c b a a d a f ′ b d Figure: An example graph-map. Left: Original graph Γ . Right: ∆ is the subdivided graph Γ with labels. Our example f : Γ → Γ is the is the composition of the “identity” ι : Γ → ∆ with the map f ′ : ∆ → Γ that sends edges to edges preserving labels and orientations. Ilya Kapovich (UIUC) May, 2013 17 / 26

Folded mapping torus About the construction of X f in Thm A: (1) Start with a "nice" f : Γ → Γ representing φ . Ex: Let Γ denote the graph in Figure 17 below. Γ has four edges, oriented as shown and labeled { a , b , c , d } = E + Γ . Consider a graph-map f : Γ → Γ under which the edges of Γ map to the combinatorial edge paths f ( a ) = d , f ( b ) = a , f ( c ) = b − 1 a , and f ( d ) = ba − 1 db − 1 ac . a d ι Γ ∆ b b a c c b a a d a f ′ b d Figure: An example graph-map. Left: Original graph Γ . Right: ∆ is the subdivided graph Γ with labels. Our example f : Γ → Γ is the is the composition of the “identity” ι : Γ → ∆ with the map f ′ : ∆ → Γ that sends edges to edges preserving labels and orientations. Ilya Kapovich (UIUC) May, 2013 17 / 26

Folded mapping torus About the construction of X f in Thm A: (1) Start with a "nice" f : Γ → Γ representing φ . Ex: Let Γ denote the graph in Figure 17 below. Γ has four edges, oriented as shown and labeled { a , b , c , d } = E + Γ . Consider a graph-map f : Γ → Γ under which the edges of Γ map to the combinatorial edge paths f ( a ) = d , f ( b ) = a , f ( c ) = b − 1 a , and f ( d ) = ba − 1 db − 1 ac . a d ι Γ ∆ b b a c c b a a d a f ′ b d Figure: An example graph-map. Left: Original graph Γ . Right: ∆ is the subdivided graph Γ with labels. Our example f : Γ → Γ is the is the composition of the “identity” ι : Γ → ∆ with the map f ′ : ∆ → Γ that sends edges to edges preserving labels and orientations. Ilya Kapovich (UIUC) May, 2013 17 / 26

Folded mapping torus About the construction of X f in Thm A: (1) Start with a "nice" f : Γ → Γ representing φ . Ex: Let Γ denote the graph in Figure 17 below. Γ has four edges, oriented as shown and labeled { a , b , c , d } = E + Γ . Consider a graph-map f : Γ → Γ under which the edges of Γ map to the combinatorial edge paths f ( a ) = d , f ( b ) = a , f ( c ) = b − 1 a , and f ( d ) = ba − 1 db − 1 ac . a d ι Γ ∆ b b a c c b a a d a f ′ b d Figure: An example graph-map. Left: Original graph Γ . Right: ∆ is the subdivided graph Γ with labels. Our example f : Γ → Γ is the is the composition of the “identity” ι : Γ → ∆ with the map f ′ : ∆ → Γ that sends edges to edges preserving labels and orientations. Ilya Kapovich (UIUC) May, 2013 17 / 26

Folded mapping torus This f is a train-track map representing φ ∈ Out ( F ( x , y , z )) where x = b − 1 a , y = a − 1 d and z = c and φ is given by φ ( x ) = y , φ ( y ) = y − 1 x − 1 yxz and φ ( z ) = x . (2) Choose a sequence of Stallings folds corresponding to f . Γ ∆ a d b c b a c b d a d a a b d d b b a c b a d a a a c b d a b Figure: Two combinatorial Stallings folds Ilya Kapovich (UIUC) May, 2013 18 / 26

Folded mapping torus This f is a train-track map representing φ ∈ Out ( F ( x , y , z )) where x = b − 1 a , y = a − 1 d and z = c and φ is given by φ ( x ) = y , φ ( y ) = y − 1 x − 1 yxz and φ ( z ) = x . (2) Choose a sequence of Stallings folds corresponding to f . Γ ∆ a d b c b a c b d a d a a b d d b b a c b a d a a a c b d a b Figure: Two combinatorial Stallings folds Ilya Kapovich (UIUC) May, 2013 18 / 26

Folded mapping torus This f is a train-track map representing φ ∈ Out ( F ( x , y , z )) where x = b − 1 a , y = a − 1 d and z = c and φ is given by φ ( x ) = y , φ ( y ) = y − 1 x − 1 yxz and φ ( z ) = x . (2) Choose a sequence of Stallings folds corresponding to f . Γ ∆ a d b c b a c b d a d a a b d d b b a c b a d a a a c b d a b Figure: Two combinatorial Stallings folds Ilya Kapovich (UIUC) May, 2013 18 / 26

Folded mapping torus This f is a train-track map representing φ ∈ Out ( F ( x , y , z )) where x = b − 1 a , y = a − 1 d and z = c and φ is given by φ ( x ) = y , φ ( y ) = y − 1 x − 1 yxz and φ ( z ) = x . (2) Choose a sequence of Stallings folds corresponding to f . Γ ∆ a d b c b a c b d a d a a b d d b b a c b a d a a a c b d a b Figure: Two combinatorial Stallings folds Ilya Kapovich (UIUC) May, 2013 18 / 26

Folded mapping torus (3) Form the "full" mapping torus M f = Γ × [ 0 , 1 ] / ( f ( x ) , 0 ) ∼ ( x , 1 ) and use the above folding sequence to "fold" M f to X f : Figure: A local picture of X f Ilya Kapovich (UIUC) May, 2013 19 / 26