Actions of finitely generated groups on ℝ -trees

Vincent Guirardel[1]

  • [1] Université Paul Sabatier Institut de Mathématiques de Toulouse, UMR 5219 31062 Toulouse cedex 9 (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 1, page 159-211
  • ISSN: 0373-0956

Abstract

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We study actions of finitely generated groups on ℝ -trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on 2 -orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements.The proof relies on an extended version of Scott’s Lemma of independent interest. This statement claims that if a group G is a direct limit of groups having suitably compatible splittings, then G splits.

How to cite

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Guirardel, Vincent. "Actions of finitely generated groups on $\mathbb{R}$-trees." Annales de l’institut Fourier 58.1 (2008): 159-211. <http://eudml.org/doc/10308>.

@article{Guirardel2008,
abstract = {We study actions of finitely generated groups on $\mathbb\{R\}$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on $2$-orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements.The proof relies on an extended version of Scott’s Lemma of independent interest. This statement claims that if a group $G$ is a direct limit of groups having suitably compatible splittings, then $G$ splits.},
affiliation = {Université Paul Sabatier Institut de Mathématiques de Toulouse, UMR 5219 31062 Toulouse cedex 9 (France)},
author = {Guirardel, Vincent},
journal = {Annales de l’institut Fourier},
keywords = {R-tree; splitting of group; Rips theory; -trees; splittings of groups},
language = {eng},
number = {1},
pages = {159-211},
publisher = {Association des Annales de l’institut Fourier},
title = {Actions of finitely generated groups on $\mathbb\{R\}$-trees},
url = {http://eudml.org/doc/10308},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Guirardel, Vincent
TI - Actions of finitely generated groups on $\mathbb{R}$-trees
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 1
SP - 159
EP - 211
AB - We study actions of finitely generated groups on $\mathbb{R}$-trees under some stability hypotheses. We prove that either the group splits over some controlled subgroup (fixing an arc in particular), or the action can be obtained by gluing together actions of simple types: actions on simplicial trees, actions on lines, and actions coming from measured foliations on $2$-orbifolds. This extends results by Sela and Rips-Sela. However, their results are misstated, and we give a counterexample to their statements.The proof relies on an extended version of Scott’s Lemma of independent interest. This statement claims that if a group $G$ is a direct limit of groups having suitably compatible splittings, then $G$ splits.
LA - eng
KW - R-tree; splitting of group; Rips theory; -trees; splittings of groups
UR - http://eudml.org/doc/10308
ER -

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