## Geometry & Topology

### Dynamics on free-by-cyclic groups

#### Abstract

Given a free-by-cyclic group $G = FN ⋊ φℤ$ determined by any outer automorphism $φ ∈ Out(FN)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$–complex $X$ called the folded mapping torus of $f$, and equip it with a semiflow. We show that $X$ enjoys many similar properties to those proven by Thurston and Fried for the mapping torus of a pseudo-Anosov homeomorphism. In particular, we construct an open, convex cone $A⊂ H1(X; ℝ) = Hom(G; ℝ)$ containing the homomorphism $u0: G → ℤ$ having $ker(u0) = FN$, a homology class $ϵ ∈ H1(X; ℝ)$, and a continuous, convex, homogeneous of degree $− 1$ function $ℌ: A→ ℝ$ with the following properties. Given any primitive integral class $u ∈A$ there is a graph $Θu ⊂ X$ such that:

1. The inclusion $Θu → X$ is $π1$–injective and $π1(Θu) = ker(u)$.
2. $u(ϵ) = χ(Θu)$.
3. $Θu ⊂ X$ is a section of the semiflow and the first return map to $Θu$ is an expanding irreducible train track map representing $φu ∈ Out(ker(u))$ such that $G = ker(u) ⋊ φuℤ$.
4. The logarithm of the stretch factor of $φu$ is precisely $ℌ(u)$.
5. If $φ$ was further assumed to be hyperbolic and fully irreducible then for every primitive integral $u ∈A$ the automorphism $φu$ of $ker(u)$ is also hyperbolic and fully irreducible.

#### Article information

Source
Geom. Topol., Volume 19, Number 5 (2015), 2801-2899.

Dates
Revised: 30 December 2014
Accepted: 26 January 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858850

Digital Object Identifier
doi:10.2140/gt.2015.19.2801

Mathematical Reviews number (MathSciNet)
MR3416115

Zentralblatt MATH identifier
1364.20026

Subjects

#### Citation

Dowdall, Spencer; Kapovich, Ilya; Leininger, Christopher J. Dynamics on free-by-cyclic groups. Geom. Topol. 19 (2015), no. 5, 2801--2899. doi:10.2140/gt.2015.19.2801. https://projecteuclid.org/euclid.gt/1510858850

#### References

• L M Abramov, The entropy of a derived automorphism, Dokl. Akad. Nauk SSSR 128 (1959) 647–650
• I Agol, Ideal triangulations of pseudo-Anosov mapping tori, from: “Topology and geometry in dimension three”, (W Li, L Bartolini, J Johnson, F Luo, R Myers, J H Rubinstein, editors), Contemp. Math. 560, Amer. Math. Soc. (2011) 1–17
• Y Algom-Kfir, E Hironaka, K Rafi, Digraphs and cycle polynomials for free-by-cyclic groups, Geom. Topol. 19 (2015) 1111–1154
• Y Algom-Kfir, K Rafi, Mapping tori of small dilatation expanding train-track maps, Topology Appl. 180 (2015) 44–63
• L Alsedà, F Mañosas, P Mumbrú, Minimizing topological entropy for continuous maps on graphs, Ergodic Theory Dynam. Systems 20 (2000) 1559–1576
• M Bestvina, N Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997) 445–470
• M Bestvina, M Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992) 85–101
• M Bestvina, M Feighn, Hyperbolicity of the complex of free factors, Adv. Math. 256 (2014) 104–155
• M Bestvina, M Handel, Train tracks and automorphisms of free groups, Ann. of Math. 135 (1992) 1–51
• R Bieri, W D Neumann, R Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987) 451–477
• R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer, New York (1982)
• R Bowen, Topological entropy and axiom ${\rm A}$, from: “Global Analysis”, Proc. Symp. Pure Math. 14, Amer. Math. Soc. (1970) 23–41
• P Brinkmann, Hyperbolic automorphisms of free groups, Geom. Funct. Anal. 10 (2000) 1071–1089
• P Brinkmann, S Schleimer, Computing triangulations of mapping tori of surface homeomorphisms, Experiment. Math. 10 (2001) 571–581
• J O Button, Mapping tori with first Betti number at least two, J. Math. Soc. Japan 59 (2007) 351–370
• D Calegari, J Maher, Statistics and compression of scl, Ergodic Theory Dynam. Systems 35 (2015) 64–110
• M Clay, A Pettet, Twisting out fully irreducible automorphisms, Geom. Funct. Anal. 20 (2010) 657–689
• M Denker, C Grillenberger, K Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics 527, Springer, Berlin (1976)
• S Dowdall, I Kapovich, C J Leininger, McMullen polynomials and Lipschitz flows for free-by-cyclic groups
• S Dowdall, I Kapovich, C J Leininger, Dynamics on free-by-cyclic groups (2013)
• S Dowdall, I Kapovich, C J Leininger, Unbounded asymmetry of stretch factors, C. R. Math. Acad. Sci. Paris 352 (2014) 885–887
• T Downarowicz, Entropy in dynamical systems, New Mathematical Monographs 18, Cambridge Univ. Press (2011)
• N M Dunfield, Alexander and Thurston norms of fibered $3$–manifolds, Pacific J. Math. 200 (2001) 43–58
• B Farb, C J Leininger, D Margalit, Small dilatation pseudo-Anosov homeomorphisms and $3$–manifolds, Adv. Math. 228 (2011) 1466–1502
• M Farber, R Geĭgan, D Shyutts, Closed $1$–forms in topology and geometric group theory, Uspekhi Mat. Nauk 65 (2010) 145–176 In Russian; translated in Russian Math. Surveys 65 (2010) 143–172
• A Fathi, F Laudenbach, V Poénaru (editors), Travaux de Thurston sur les surfaces, Astérisque 66–67, Soc. Math. France, Paris (1979)
• M Feighn, M Handel, Mapping tori of free group automorphisms are coherent, Ann. of Math. 149 (1999) 1061–1077
• S Francaviglia, A Martino, Metric properties of outer space, Publ. Mat. 55 (2011) 433–473
• D Fried, Flow equivalence, hyperbolic systems and a new zeta function for flows, Comment. Math. Helv. 57 (1982) 237–259
• D Fried, The geometry of cross sections to flows, Topology 21 (1982) 353–371
• D Gabai, Foliations and the topology of $3$–manifolds, J. Differential Geom. 18 (1983) 445–503
• F Gautero, Dynamical $2$–complexes, Geometriae Dedicata 88 (2001) 283–319
• F Gautero, Feuilletages de $2$–complexes, Ann. Fac. Sci. Toulouse Math. 10 (2001) 619–638
• F Gautero, Cross sections to semi-flows on $2$–complexes, Ergodic Theory Dynam. Systems 23 (2003) 143–174
• F Gautero, Combinatorial mapping-torus, branched surfaces and free group automorphisms, Ann. Sc. Norm. Super. Pisa Cl. Sci. 6 (2007) 405–440
• R Geoghegan, M L Mihalik, M Sapir, D T Wise, Ascending HNN extensions of finitely generated free groups are Hopfian, Bull. London Math. Soc. 33 (2001) 292–298
• M Handel, L Mosher, The expansion factors of an outer automorphism and its inverse, Trans. Amer. Math. Soc. 359 (2007) 3185–3208
• A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
• I Kapovich, Algorithmic detectability of iwip automorphisms, Bull. Lond. Math. Soc. 46 (2014) 279–290
• I Kapovich, M Lustig, Cannon–Thurston fibers for iwip automorphisms of $F_N$, J. Lond. Math. Soc. 91 (2015) 203–224
• I Kapovich, A Myasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248 (2002) 608–668
• I Kapovich, K Rafi, On hyperbolicity of free splitting and free factor complexes, Groups Geom. Dyn. 8 (2014) 391–414
• G Levitt, $1$–formes fermées singulières et groupe fondamental, Invent. Math. 88 (1987) 635–667
• D D Long, U Oertel, Hyperbolic surface bundles over the circle, from: “Progress in knot theory and related topics”, (M Boileau, M Domergue, Y Mathieu, K Millett, editors), Travaux en Cours 56, Hermann, Paris (1997) 121–142
• C T McMullen, Polynomial invariants for fibered $3$–manifolds and Teichmüller geodesics for foliations, Ann. Sci. École Norm. Sup. 33 (2000) 519–560
• C T McMullen, The Alexander polynomial of a $3$–manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. 35 (2002) 153–171
• L Mosher, Dynamical systems and the homology norm of a $3$–manifold, I: Efficient intersection of surfaces and flows, Duke Math. J. 65 (1992) 449–500
• L Mosher, Dynamical systems and the homology norm of a $3$–manifold, II, Invent. Math. 107 (1992) 243–281
• W D Neumann, Normal subgroups with infinite cyclic quotient, Math. Sci. 4 (1979) 143–148
• U Oertel, Homology branched surfaces: Thurston's norm on $H\sb 2(M\sp 3)$, from: “Low-dimensional topology and Kleinian groups”, (D B A Epstein, editor), London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press (1986) 253–272
• U Oertel, Affine laminations and their stretch factors, Pacific J. Math. 182 (1998) 303–328
• J-P Otal, The hyperbolization theorem for fibered $3$–manifolds, SMF/AMS Texts and Monographs 7, Amer. Math. Soc. (2001)
• C Pfaff, Constructing and classifying fully irreducible outer automorphisms of free groups
• C Pfaff, Ideal Whitehead graphs in $Out(F_r)$, II: The complete graph in each rank, J. Homotopy Relat. Struct. 10 (2015) 275–301
• I Rivin, Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J. 142 (2008) 353–379
• I Rivin, Zariski density and genericity, Int. Math. Res. Not. 2010 (2010) 3649–3657
• M Scharlemann, Sutured manifolds and generalized Thurston norms, J. Differential Geom. 29 (1989) 557–614
• J Stallings, On fibering certain $3$–manifolds, from: “Topology of $3$–manifolds and related topics”, Prentice-Hall, Englewood Cliffs, NJ (1962) 95–100
• J R Stallings, Topology of finite graphs, Invent. Math. 71 (1983) 551–565
• W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 339, Amer. Math. Soc. (1986)
• W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988) 417–431
• D Tischler, On fibering certain foliated manifolds over $S\sp{1}$, Topology 9 (1970) 153–154
• Z Wang, Mapping tori of outer automorphisms of free groups, PhD thesis, Rutgers, NJ (2002) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/305500756 {\unhbox0