Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-30T20:35:30.316Z Has data issue: false hasContentIssue false
  • English
  • Français

ℝ-trees, dual laminations and compact systems of partial isometries

Published online by Cambridge University Press:  01 September 2009

THIERRY COULBOIS ,
ARNAUD HILION  and
MARTIN LUSTIG
THIERRY COULBOIS
Affiliation:
LATP, Université Aix-Marseille III, Marseille, France. e-mail: [email protected], [email protected], [email protected]
ARNAUD HILION
Affiliation:
LATP, Université Aix-Marseille III, Marseille, France. e-mail: [email protected], [email protected], [email protected]
MARTIN LUSTIG
Affiliation:
LATP, Université Aix-Marseille III, Marseille, France. e-mail: [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let FN be a free group of finite rank N ≥ 2, and let T be an ℝ-tree with a very small, minimal action of FN with dense orbits. For any basis of FN there exists a heart ⊂ (= the metric completion of T) which is a compact subtree that has the property that the dynamical system of partial isometries ai : , for each ai ∈ , defines a tree which contains an isometric copy of T as minimal subtree.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 2 , September 2009 , pp. 345 - 368
Copyright
Copyright © Cambridge Philosophical Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[BF95]Bestvina, M. and Feighn, M.Stable actions of groups on real trees. Invent. Math. 121 (2) (1995), 287–321.CrossRefGoogle Scholar
[BFH00]Bestvina, M., Feighn, M. and Handel, M.The Tits alternative for Out(F n). I. Dynamics of exponentially-growing automorphisms. Ann. of Math. 151 (2000), 517–623.Google Scholar
[BH04]Buzzi, J. and Hubert, P.Piecewise monotone maps without periodic points: rigidity, measures and complexity. Ergodic Theory Dynam. Systems 24 (2004), 383–405.CrossRefGoogle Scholar
[Bow99]Bowditch, B.Treelike structures arising from continua and convergence groups, Mem. Amer. Math. Soc. 662 (1999).Google Scholar
[Cou08]Coulbois, T. Fractal trees for irreducible automorphisms of free groups. In preparation, 2008.Google Scholar
[CH08]Coulbois, T. and Hilion, A. Index of an action of a free group on an ℝ-tree. In preparation, 2008.Google Scholar
[CHL05]Coulbois, T., Hilion, A. and Lustig, M.Non-uniquely ergodic ℝ-trees are topologically determined by their algebraic lamination. Illinois J. Math. 51 (2007), 897–911.CrossRefGoogle Scholar
[CHL-I]Coulbois, T., Hilion, A. and Lustig, M.ℝ-trees and laminations for free groups I: Algebraic laminations. J. London Math Soc. 78 (2008), 723–736.Google Scholar
[CHL-II]Coulbois, T., Hilion, A. and Lustig, M.ℝ-trees and laminations for free groups II: The lamination associated to an ℝ-tree. J. London Math Soc. 78 (2008), 737–754.Google Scholar
[CL95]Cohen, M. and Lustig, M.Very small group actions on R-trees and Dehn twist automorphisms. Topology 34 (1995), 575–617.CrossRefGoogle Scholar
[CM87]Culler, M. and Morgan, J.Group actions on R-trees. Proc. London Math. Soc. 55 (1987), 571–604.CrossRefGoogle Scholar
[Gab97]Gaboriau, D.Générateurs indépendants pour les systèmes d'isométries de dimension un. Ann. Inst. Fourier (Grenoble) 47 (1) (1997), 101–122.Google Scholar
[GL95]Gaboriau, D. and Levitt, G.The rank of actions on R-trees. Ann. Sci. École Norm. Sup. 28 (1995), 549–570.CrossRefGoogle Scholar
[GLP94]Gaboriau, D., Levitt, G. and Paulin, F.Pseudogroups of isometries of R and Rips' theorem on free actions on R-trees. Israel J. Math. 87 (1–3) (1994), 403–428.CrossRefGoogle Scholar
[HM06]Handel, M. and Mosher, L. Axes in Outer Space. ArXiv:math/0605355.Google Scholar
[KL07]Kapovich, I. and Lustig, M. Intersection form, laminations and currents on free groups ArXiv:math/0711.4337.Google Scholar
[LL03]Levitt, G. and Lustig, M.Irreducible automorphisms of F n have north-south dynamics on compactified outer space. J. Inst. Math. Jussieu 2 (2003), 59–72.CrossRefGoogle Scholar
[LL08]Levitt, G. and Lustig, M. Automorphisms of free groups have asymptotically periodic dynamics J. Reine Angew. Math. J. Reine Angew. Math., to appear 2008, arXiv:math/0407437.Google Scholar
[LP97]Levitt, G. and Paulin, F.Geometric group actions on trees. Amer. J. Math. 119 (1) (1997), 83–102.Google Scholar
[MS91]Morgan, J. W. and Shalen, P. B.Free actions of surface groups on R-trees. Topology 30 (2) (1991), 143–154.CrossRefGoogle Scholar
[Pau88]Paulin, F.Topologie de Gromov équivariante, structures hyperboliques et arbres réels. Invent. Math. 94 (1) (1988), 53–80.Google Scholar
[Sela97]Sela, Z.Acylindrical accessibility for groups. Invent. Math. 129 (3) (1997), 527–565.CrossRefGoogle Scholar
[Sko96]Skora, R.Splittings of surfaces. J. Amer. Math. Soc. 9 (1996), 605–616.Google Scholar