Pseudogroups of isometries of ℝ and Rips’ theorem on free actions on ℝ-trees

Gaboriau, Damien; Levitt, Gilbert; Paulin, Frédéric

doi:10.1007/bf02773004

Cited by 95 publications

(154 citation statements)

References 18 publications

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“…[LP] First, let's fix notations and terminology (see [LP,GLP1,Pau4]). Let < S||R > be a finite presentation of Γ.…”

Section: Approximation By a Geometric Actionmentioning

confidence: 99%

“…Like in [GLP1,BF2,Pau4], we are first going to perform some Rips moves on our resolution. These moves are transformations that preserve the action (TΣ Each generator for which none of its bases is reduced to {c} has a unique corresponding generator in D ; and given a singleton having a base equal to {c}, change this base to any of {c i }.…”

Section: Pure Components Of σmentioning

confidence: 99%

“…According to [Lev1] or [GLP1] (see rk. p. 418) and thanks to homogeneousness, given any compact set K in • D, X −ε has the same orbits in restriction to K for sufficiently small ε.…”

Section: Segment-closed Property ([Glp1]) Assume That τ Is An Isometmentioning

confidence: 99%

“…Let Γ be a finitely presented group. Every minimal stable action of decomposition into minimal and simplicial components (see [GLP1,BF2]). This provides a graph of actions on R-trees such that its non trivial vertex actions correspond to its minimal components.…”

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confidence: 99%

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Approximations of stable actions on $ \Bbb {R} $-trees

Guirardel

1998

Comment. Math. Helv.

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Abstract. This article shows how to approximate a stable action of a finitely presented group on an R-tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an R-tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on R-trees and on the dynamical study of exotic components by D. Gaboriau. Mathematics Subject Classification (1991). 20E08, 20E06, 20F32, 05C05.Keywords. Stable action, R-trees, Rips theorem, approximation, splitting, Bass-Serre.M. Bestvina and M. Feighn introduced the notion of stable action of a group on an R-tree ([BF2]) by slightly weakening some conditions by E. Rips or by H. Gillet and P. Shalen. Roughly speaking, stable actions are such that the stabilizer of an arc must stabilize when this arc gets smaller and smaller (see section 1 for a formal definition). This condition is true in usual cases: any action with trivial arc stabilizer, the actions coming from iteration of automorphisms of free groups or from degeneracy of hyperbolic structures, and every small action of a hyperbolic group is stable (see section 1).The main theorem about stable actions is Rips's theorem (see [BF2]): if a finitely presented group Γ has a non trivial stable action on an R-tree, then Γ splits over a group C which is an extension of Z k by a subgroup of Γ fixing an arc in T .On the other hand, M. Cohen and M. Lustig have introduced very small actions on R-trees and have shown that the set of free actions of the free group F n on simplicial R-trees is dense in the space of very small actions of F n on simplicial Rtrees ([CL]). Then, M. Bestvina and M. Feighn showed that every very small action of F n on an R-tree can be approximated by a very small action on a simplicial R-tree ([BF3]). This showed that the closure of M. Culler and K. Vogtmann's outer space is the projectivised set of very small actions of F n on R-trees.Our theorem is both a refinement of E. Rips' splitting theorem, and a generalisation of M. Bestvina and M. Feighn's approximation theorem: Theorem 1. Let Γ be a finitely presented group. Every minimal stable action of

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“…[LP] First, let's fix notations and terminology (see [LP,GLP1,Pau4]). Let < S||R > be a finite presentation of Γ.…”

Section: Approximation By a Geometric Actionmentioning

confidence: 99%

Section: Pure Components Of σmentioning

confidence: 99%

“…According to [Lev1] or [GLP1] (see rk. p. 418) and thanks to homogeneousness, given any compact set K in • D, X −ε has the same orbits in restriction to K for sufficiently small ε.…”

Section: Segment-closed Property ([Glp1]) Assume That τ Is An Isometmentioning

confidence: 99%

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confidence: 99%

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Approximations of stable actions on $ \Bbb {R} $-trees

Guirardel

1998

Comment. Math. Helv.

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“…Sur chacun d'eux, soit toute orbite est finie, soit toute orbite est dense (la composante est alors dite minimale). Les composantes minimales se répartissent en 3 types [GLP1] : -homogènes, les orbites sont les mêmes que celles d'un groupe d'isométries (Définition 3.2), -échanges d'intervalles, -exotiques, dont l'existence a été mise en évidence par G. Levitt [Lev3].…”

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Générateurs indépendants pour les systèmes d'isométries de dimension un

Gaboriau¹

1997

Annales De L’institut Fourier

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Hyperbolic Manifolds and Discrete Groups

Kapovich¹

2010

313

355

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Many of the original research and survey monographs in pure and been groundbreaking and have come to be regarded as foundational to the subject. Through the MBC Series, a select number of these modern classics, entirely uncorrected, are being re-released in accessible to new generations of students, scholars, and researchers. paperback (and as eBooks) to ensure that these treasures remain applied mathematics published by Birkhäuser in recent decades have

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Pseudogroups of isometries of ℝ and Rips’ theorem on free actions on ℝ-trees

Cited by 95 publications

References 18 publications

Approximations of stable actions on $ \Bbb {R} $-trees

Approximations of stable actions on $ \Bbb {R} $-trees

Générateurs indépendants pour les systèmes d'isométries de dimension un

Hyperbolic Manifolds and Discrete Groups

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