Illinois Journal of Mathematics

Stretching factors, metrics and train tracks for free products

Stefano Francaviglia and Armando Martino

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of $G$-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices.

We describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths.

We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed (in particular, answering to some questions raised in Axis in outer space (2011) concerning the axis bundle of irreducible automorphisms).

Finally, we include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory.

A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.

Article information

Source
Illinois J. Math., Volume 59, Number 4 (2015), 859-899.

Dates
Received: 11 November 2015
Revised: 25 August 2016
First available in Project Euclid: 27 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1488186013

Digital Object Identifier
doi:10.1215/ijm/1488186013

Mathematical Reviews number (MathSciNet)
MR3628293

Zentralblatt MATH identifier
1382.20031

Subjects
Primary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations 20E36: Automorphisms of infinite groups [For automorphisms of finite groups, see 20D45] 20E08: Groups acting on trees [See also 20F65]

Citation

Francaviglia, Stefano; Martino, Armando. Stretching factors, metrics and train tracks for free products. Illinois J. Math. 59 (2015), no. 4, 859--899. doi:10.1215/ijm/1488186013. https://projecteuclid.org/euclid.ijm/1488186013


Export citation

References

  • Y. Agolm-Kfir and M. Bestvina, Asymmetry of outer space, Geom. Dedicata 156 (2012), 81–92.
  • R. Alperin and H. Bass, Length functions of group actions on $\Lambda$-trees. Combinatorial Group Theory and Topology, Sel. Pap. Conf., alta/Utah 1984, Ann. of Math. Stud. 111 (1987), 265–378.
  • M. Bestvina, A Bers-like proof of the existence of train tracks for free group automorphisms, Fund. Math. 214 (2011), no. 1, 1–12.
  • M. Bestvina, M. Feighn and M. Handel, Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal. 7 (1997), no. 7, 215–244.
  • M. Bestvina, M. Feighn and M. Handel, The tits alternative for $\operatorname{Out}(F n )$. I: Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151 (2000), no. 2, 517–623.
  • M. Bestvina, M. Feighn and M. Handel, Solvable subgroups of $\operatorname{Out}(F n )$ are virtually Abelian, Geom. Dedicata 104 (2004), 71–96.
  • M. Bestvina, M. Feighn and M. Handel, The tits alternative for $\operatorname{Out}(F n )$. II: A Kolchin type theorem, Ann. of Math. (2) 161 (2005), no. 1, 1–59.
  • M. Bestvina and M. Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), no. 1, 1–51.
  • D. J. Collins and E. C. Turner, Efficient representatives for automorphisms of free products, Michigan Math. J. 41 (1994), no. 3, 443–464.
  • T. Coulbois and A. Hilion, Ergodic currents dual to a real tree, preprint; available at \arxivurlarXiv:1302.3766.
  • T. Coulbois, A. Hilion and M. Lustig, Non-unique ergodicity, observers' topology and the dual algebraic lamination for $\mathbb R$-trees, Illinois J. Math. 51 (2007), no. 3, 897–911.
  • T. Coulbois, A. Hilion and M. Lustig, $\mathbb R$-Trees and laminations for free groups I: Algebraic laminations, J. Lond. Math. Soc. 78 (2008), no. 3, 723–736.
  • T. Coulbois, A. Hilion and M. Lustig, $\mathbb R$-Trees and laminations for free groups II: The dual lamination of an $\mathbb R$-tree, J. Lond. Math. Soc. 78 (2008), no. 3, 737–754.
  • T. Coulbois, A. Hilion and M. Lustig, $\mathbb R$-Trees and laminations for free groups III: Currents and dual $\mathbb R$-tree metrics, J. Lond. Math. Soc. 78 (2008), no. 3, 755–766.
  • T. Coulbois, A. Hilion and M. Lustig, $\mathbb R$-Trees, dual laminations, and compact systems of partial isometries, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 2, 345–368.
  • T. Coulbois, A. Hilion and P. Reynolds, Indecomposable $F_N$-trees and minimal laminations, preprint; available at \arxivurlarXiv:1110.3506.
  • M. Culler and J. W. Morgan, Group actions on $\mathbb{R}$-trees, Proc. Lond. Math. Soc. (3) 55 (1987), no. 3, 57–604.
  • M. Foster, Deformation and rigidity of simplicial group actions on trees, Geom. Topol. 6 (2002), 219–267.
  • S. Francaviglia, Geodesic currents and length compactness for automorphisms of free groups, Trans. Amer. Math. Soc. 361 (2009), 161–176.
  • S. Francaviglia and A. Martino, Metric properties of outer space, Publ. Mat. 55 (2011), 433–473.
  • S. Francaviglia and A. Martino, The isometry group of outer space, Adv. Mat. (3-4) 231 (2012), 1940–1973.
  • A. Hadari, Homological shadows of attracting laminations, preprint; available at \arxivurlarXiv:1305.1613.
  • M. Handel and L. Mosher, Axis in outer space, Mem. Am. Math. Soc., vol. 1004, 2011.
  • I. Kapovich, Currents on free groups, topological and asymptotic aspects of group theory (R. Grigorchuk, M. Mihalik, M. Sapir and Z. Sunik, eds.), AMS Contemporary Mathematics Series, vol. 394, 2006, pp. 149–176.
  • I. Kapovich, Clusters, currents and Whitehead's algorithm, Exp. Math. 16 (2007), no. 1, 67–76.
  • I. Kapovich and M. Lustig, The actions of $\operatorname{Out}(F_k)$ on the boundary of outer space and on the space of currents: Minimal sets and equivariant incompatibili, Ergodic Theory Dynam. Systems 27 (2007), no. 3, 827–847.
  • I. Kapovich and M. Lustig, Intersection form, laminations and currents on free groups, Geom. Funct. Anal. 19 (2010), no. 5, 1426–1467.
  • I. Kapovich and M. Lustig, Invariant laminations for irreducible automorphisms of free groups, Quarterly J. Math. (2010), to appear; published online Jan 30, 2014, Q. J. Math.
  • I. Kapovich and T. Nagnibeda, The Patterson–Sullivan embedding and minimal volume entropy for outer space, Geom. Funct. Anal. 17 (2007), no. 4, 1201–1236.
  • I. Kapovich and T. Nagnibeda, Geometric entropy of geodesic currents on free groups, Dynamical numbers: Interplay between dynamical systems and number theory, Contemporary Mathematics Series, American Mathematical Society, Providence, 2010, pp. 149–176.
  • I. Kapovich and T. Nagnibeda, Subset currents on free groups, Geom. Dedicata 166, (2013), 307–348. pp. 307–348.
  • S. Meinert, The Lipschitz metric on deformation spaces of $G$-trees, preprint; available at \arxivurlarXiv:1312.1829.
  • L. Mosher and C. Pfaff, Lone Axes in Outer Space, preprint; available at \arxivurlarXiv:1311.6855.
  • M. Sykiotis, Stable representatives for symmetric automorphismsofgroupsandthegeneralform of the Scott conjecture, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2405–2441.
  • G. Vincente and G. Levitt, Deformation spaces of trees, Groups Geom. Dyn. 1 (2007), 135–181.
  • G. Vincente and G. Levitt, The outer space of a free product, Proc. Lond. Math. Soc. (3) 94 (2007), 695–714.